Spectral theory of ordinary differential equations

<h2 id="introduction">Introduction</h2>
<a href="/facts/Spectral_theory/8sAEEIMM">Spectral theory</a> for second order ordinary differential equations on a compact interval was developed by <a href="/facts/Jacques_Charles_Fran%25C3%25A7ois_Sturm/N9F0StnF">Jacques Charles François Sturm</a> and <a href="/facts/Joseph_Liouville/U22o7wtK">Joseph Liouville</a> in the nineteenth century and is now known as <a href="/facts/Sturm%25E2%2580%2593Liouville_theory/xwvX8d9d">Sturm–Liouville theory</a>. In modern language, it is an application of the <a href="/facts/Spectral_theorem/Vqc04B21">spectral theorem</a> for <a href="/facts/Compact_operator/KNkcK4mE">compact operators</a> due to <a href="/facts/David_Hilbert/1vhlHn4b">David Hilbert</a>. In his dissertation, published in 1910, <a href="/facts/Hermann_Weyl/4C5HhyGU">Hermann Weyl</a> extended this theory to second order ordinary differential equations with <a href="/facts/Singularity_(mathematics)/Y5sMQY4I">singularities</a> at the endpoints of the interval, now allowed to be infinite or semi-infinite. He simultaneously developed a spectral theory adapted to these special operators and introduced <a href="/facts/Boundary_condition/QEfmUqmP">boundary conditions</a> in terms of his celebrated dichotomy between limit points and limit circles.
In the 1920s, <a href="/facts/John_von_Neumann/aUk6urof">John von Neumann</a> established a general spectral theorem for <a href="/facts/Unbounded_operator/5u2njqHt">unbounded</a> <a href="/facts/Self-adjoint_operator/VrH1nEAK">self-adjoint operators</a>, which <a href="/facts/Kunihiko_Kodaira/u73B5W0D">Kunihiko Kodaira</a> used to streamline Weyl's method. Kodaira also generalised Weyl's method to singular ordinary differential equations of even order and obtained a simple formula for the <a href="/facts/Spectral_theorem/Vqc04B21">spectral measure</a>. The same formula had also been obtained independently by <a href="/facts/E._C._Titchmarsh/607FjJwB">E. C. Titchmarsh</a> in 1946 (scientific communication between <a href="/facts/Japan/U5mz0SRT">Japan</a> and the <a href="/facts/United_Kingdom/kuPAILt2">United Kingdom</a> had been interrupted by <a href="/facts/World_War_II/2efV5L1U">World War II</a>). Titchmarsh had followed the method of the German mathematician <a href="/facts/Emil_Hilb/Hnr5FR36">Emil Hilb</a>, who derived the eigenfunction expansions using <a href="/facts/Complex_function_theory/hgCoQlH0">complex function theory</a> instead of <a href="/facts/Operator_theory/Ays1ofel">operator theory</a>. Other methods avoiding the spectral theorem were later developed independently by Levitan, Levinson and Yoshida, who used the fact that the <a href="/facts/Spectral_theorem/Vqc04B21">resolvent</a> of the singular differential operator could be approximated by <a href="/facts/Compact_operator/KNkcK4mE">compact</a> resolvents corresponding to <a href="/facts/Sturm%25E2%2580%2593Liouville_problem/xwvX8d9d">Sturm–Liouville problems</a> for proper subintervals. Another method was found by <a href="/facts/Mark_Grigoryevich_Krein/cJCx4hBB">Mark Grigoryevich Krein</a>; his use of direction functionals was subsequently generalised by Izrail Glazman to arbitrary ordinary differential equations of even order.
Weyl applied his theory to <a href="/facts/Carl_Friedrich_Gauss/f4eSMBD7">Carl Friedrich Gauss</a>'s <a href="/facts/Hypergeometric_differential_equation/yTEC6NBJ">hypergeometric differential equation</a>, thus obtaining a far-reaching generalisation of the transform formula of <a href="/facts/Gustav_Ferdinand_Mehler/MRXaDdqT">Gustav Ferdinand Mehler</a> (1881) for the <a href="/facts/Legendre_function/sDo3ra2G">Legendre differential equation</a>, rediscovered by the Russian physicist <a href="/facts/Vladimir_Fock/IeLmHFxd">Vladimir Fock</a> in 1943, and usually called the <a href="/facts/Mehler%25E2%2580%2593Fock_transform/Q6Sd5yWL">Mehler–Fock transform</a>. The corresponding ordinary differential operator is the radial part of the <a href="/facts/Laplacian_operator/kov9u3PT">Laplacian operator</a> on 2-dimensional <a href="/facts/Hyperbolic_space/kI7tnwdk">hyperbolic space</a>. More generally, the <a href="/facts/Plancherel_theorem/IlpoAySd">Plancherel theorem</a> for <a href="/facts/SL(2%2cR)/xjKM9JLJ">SL(2,R)</a> of <a href="/facts/Harish_Chandra/f9De1zCJ">Harish Chandra</a> and <a href="/facts/I._M._Gelfand/TJeJouKs">Gelfand</a>–<a href="/facts/M._A._Naimark/5TEptr3F">Naimark</a> can be deduced from Weyl's theory for the hypergeometric equation, as can the theory of <a href="/facts/Zonal_spherical_function/myzsFsrU">spherical functions</a> for the <a href="/facts/Isometry_group/X5uyVIdJ">isometry groups</a> of higher dimensional hyperbolic spaces. Harish Chandra's later development of the Plancherel theorem for general real <a href="/facts/Semisimple_Lie_group/m8jA3Bcf">semisimple Lie groups</a> was strongly influenced by the methods Weyl developed for eigenfunction expansions associated with singular ordinary differential equations. Equally importantly the theory also laid the mathematical foundations for the analysis of the <a href="/facts/Schr%25C3%25B6dinger_equation/3csgSIWF">Schrödinger equation</a> and <a href="/facts/Scattering_matrix/NffJvlpp">scattering matrix</a> in <a href="/facts/Quantum_mechanics/BRc3Mzgr">quantum mechanics</a>.

<h2 id="solutions-of-ordinary-differential-equations">Solutions of ordinary differential equations</h2>
Main article: <a href="/facts/Ordinary_differential_equations/ygKGQ8kD">Ordinary differential equations</a>
<h3>Reduction to standard form</h3>
Let D be the second order differential operator on (a, b) given by

D
        f
        (
        x
        )
        =
        −
        p
        (
        x
        )
        
          f
          ″
        
        (
        x
        )
        +
        r
        (
        x
        )
        
          f
          ′
        
        (
        x
        )
        +
        q
        (
        x
        )
        f
        (
        x
        )
        ,
      
    
    {\displaystyle Df(x)=-p(x)f''(x)+r(x)f'(x)+q(x)f(x),}

where p is a strictly positive continuously differentiable function and q and r are continuous real-valued functions.
For x0 in (a, b), define the <a href="/facts/Sturm%25E2%2580%2593Liouville_theory/xwvX8d9d">Liouville transformation</a> ψ by

ψ
        (
        x
        )
        =
        
          ∫
          
            
              x
              
                0
              
            
          
          
            x
          
        
        p
        (
        t
        
          )
          
            −
            1
            
              /
            
            2
          
        
        
        d
        t
      
    
    {\displaystyle \psi (x)=\int _{x_{0}}^{x}p(t)^{-1/2}\,dt}

If

U
        :
        
          L
          
            2
          
        
        (
        a
        ,
        b
        )
        ↦
        
          L
          
            2
          
        
        (
        ψ
        (
        a
        )
        ,
        ψ
        (
        b
        )
        )
      
    
    {\displaystyle U:L^{2}(a,b)\mapsto L^{2}(\psi (a),\psi (b))}

is the <a href="/facts/Unitary_operator/rkypfQQ9">unitary operator</a> defined by

(
        U
        f
        )
        (
        ψ
        (
        x
        )
        )
        =
        f
        (
        x
        )
        ×
        
          
            (
            
              
                ψ
                ′
              
              (
              x
              )
            
            )
          
          
            −
            1
            
              /
            
            2
          
        
        ,
         
         
        ∀
        x
        ∈
        (
        a
        ,
        b
        )
      
    
    {\displaystyle (Uf)(\psi (x))=f(x)\times \left(\psi '(x)\right)^{-1/2},\ \ \forall x\in (a,b)}

then

U
        
          
            
              d
            
            
              
                d
              
              x
            
          
        
        
          U
          
            −
            1
          
        
        g
        =
        
          g
          ′
        
        
          ψ
          ′
        
        +
        
          
            1
            2
          
        
        g
        
          
            
              ψ
              ″
            
            
              ψ
              ′
            
          
        
      
    
    {\displaystyle U{\frac {\mathrm {d} }{\mathrm {d} x}}U^{-1}g=g'\psi '+{\frac {1}{2}}g{\frac {\psi ''}{\psi '}}}

and

U
                
                  
                    
                      
                        d
                      
                      
                        2
                      
                    
                    
                      
                        d
                      
                      
                        x
                        
                          2
                        
                      
                    
                  
                
                
                  U
                  
                    −
                    1
                  
                
                g
              
              
                
                =
                
                  (
                  
                    U
                    
                      
                        
                          d
                        
                        
                          
                            d
                          
                          x
                        
                      
                    
                    
                      U
                      
                        −
                        1
                      
                    
                  
                  )
                
                ×
                
                  (
                  
                    U
                    
                      
                        
                          d
                        
                        
                          
                            d
                          
                          x
                        
                      
                    
                    
                      U
                      
                        −
                        1
                      
                    
                  
                  )
                
                g
              
            
            
              
              
                
                =
                
                  
                    
                      d
                    
                    
                      
                        d
                      
                      ψ
                    
                  
                
                
                  [
                  
                    
                      g
                      ′
                    
                    
                      ψ
                      ′
                    
                    +
                    
                      
                        1
                        2
                      
                    
                    g
                    
                      
                        
                          ψ
                          ″
                        
                        
                          ψ
                          ′
                        
                      
                    
                  
                  ]
                
                ⋅
                
                  ψ
                  ′
                
                +
                
                  
                    1
                    2
                  
                
                
                  [
                  
                    
                      g
                      ′
                    
                    
                      ψ
                      ′
                    
                    +
                    
                      
                        1
                        2
                      
                    
                    g
                    
                      
                        
                          ψ
                          ″
                        
                        
                          ψ
                          ′
                        
                      
                    
                  
                  ]
                
                ⋅
                
                  
                    
                      ψ
                      ″
                    
                    
                      ψ
                      ′
                    
                  
                
              
            
            
              
              
                
                =
                
                  g
                  ″
                
                
                  ψ
                  
                    ′
                    
                      2
                    
                  
                
                +
                2
                
                  g
                  ′
                
                
                  ψ
                  ″
                
                +
                
                  
                    1
                    2
                  
                
                g
                ⋅
                
                  [
                  
                    
                      
                        
                          ψ
                          ‴
                        
                        
                          ψ
                          ′
                        
                      
                    
                    −
                    
                      
                        1
                        2
                      
                    
                    
                      
                        
                          ψ
                          
                            ″
                            
                              2
                            
                          
                        
                        
                          ψ
                          
                            ′
                            
                              2
                            
                          
                        
                      
                    
                  
                  ]
                
              
            
          
        
      
    
    {\displaystyle {\begin{aligned}U{\frac {\mathrm {d} ^{2}}{\mathrm {d} x^{2}}}U^{-1}g&=\left(U{\frac {\mathrm {d} }{\mathrm {d} x}}U^{-1}\right)\times \left(U{\frac {\mathrm {d} }{\mathrm {d} x}}U^{-1}\right)g\\[1ex]&={\frac {\mathrm {d} }{\mathrm {d} \psi }}\left[g'\psi '+{\frac {1}{2}}g{\frac {\psi ''}{\psi '}}\right]\cdot \psi '+{\frac {1}{2}}\left[g'\psi '+{\frac {1}{2}}g{\frac {\psi ''}{\psi '}}\right]\cdot {\frac {\psi ''}{\psi '}}\\[1ex]&=g''\psi '^{2}+2g'\psi ''+{\frac {1}{2}}g\cdot \left[{\frac {\psi '''}{\psi '}}-{\frac {1}{2}}{\frac {\psi ''^{2}}{\psi '^{2}}}\right]\end{aligned}}}

Hence,

U
        D
        
          U
          
            −
            1
          
        
        g
        =
        −
        
          g
          ″
        
        +
        R
        
          g
          ′
        
        +
        Q
        g
        ,
      
    
    {\displaystyle UDU^{-1}g=-g''+Rg'+Qg,}

where

R
        =
        
          
            
              
                p
                ′
              
              +
              r
            
            
              p
              
                1
                
                  /
                
                2
              
            
          
        
      
    
    {\displaystyle R={\frac {p'+r}{p^{1/2}}}}

and

Q
        =
        q
        −
        
          
            
              r
              
                p
                ′
              
            
            
              4
              p
            
          
        
        +
        
          
            
              p
              ″
            
            4
          
        
        −
        
          
            
              5
              
                p
                
                  ′
                  
                    2
                  
                
              
            
            
              16
              p
            
          
        
      
    
    {\displaystyle Q=q-{\frac {rp'}{4p}}+{\frac {p''}{4}}-{\frac {5p'^{2}}{16p}}}

The term in g′ can be removed using an <a href="/facts/Euler/z2fsbjgj">Euler</a> <a href="/facts/Integrating_factor/5tKufbqg">integrating factor</a>. If S′/S = R/2, then h = Sg satisfies

(
        S
        U
        D
        
          U
          
            −
            1
          
        
        
          S
          
            −
            1
          
        
        )
        h
        =
        −
        
          h
          ″
        
        +
        V
        h
        ,
      
    
    {\displaystyle (SUDU^{-1}S^{-1})h=-h''+Vh,}

where the <a href="/facts/Potential/8Fcbtnt9">potential</a> V is given by

V
        =
        Q
        +
        
          
            
              S
              ″
            
            S
          
        
      
    
    {\displaystyle V=Q+{\frac {S''}{S}}}

The differential operator can thus always be reduced to one of the form <a class="footnote-ref" id="fnref:1" href="#fn:1">1</a>

D
        f
        =
        −
        
          f
          ″
        
        +
        q
        f
        .
      
    
    {\displaystyle Df=-f''+qf.}

<h3>Existence theorem</h3>
The following is a version of the classical <a href="/facts/Picard%25E2%2580%2593Lindel%25C3%25B6f_theorem/yKHSWoMJ">Picard existence theorem</a> for second order differential equations with values in a <a href="/facts/Banach_space/sDEIk7EC">Banach space</a> E.<a class="footnote-ref" id="fnref:2" href="#fn:2">2</a>
Let α, β be arbitrary elements of E, A a <a href="/facts/Bounded_operator/LYmDTIqR">bounded operator</a> on E and q a continuous function on [a, b].
Then, for c = a or c = b, the differential equation

D
        f
        =
        A
        f
      
    
    {\displaystyle Df=Af}

has a unique solution f in C2([a,b], E) satisfying the initial conditions

f
        (
        c
        )
        =
        β
        
        ,
        
        
          f
          ′
        
        (
        c
        )
        =
        α
        .
      
    
    {\displaystyle f(c)=\beta \,,\;f'(c)=\alpha .}

In fact a solution of the differential equation with these initial conditions is equivalent to a solution
of the <a href="/facts/Integral_equation/LCu4Imfq">integral equation</a>

f
        =
        h
        +
        T
        f
      
    
    {\displaystyle f=h+Tf}

with T the bounded linear map on C([a,b], E) defined by

T
        f
        (
        x
        )
        =
        
          ∫
          
            c
          
          
            x
          
        
        K
        (
        x
        ,
        y
        )
        f
        (
        y
        )
        
        d
        y
        ,
      
    
    {\displaystyle Tf(x)=\int _{c}^{x}K(x,y)f(y)\,dy,}

where K is the <a href="/facts/Volterra_kernel/oDUJXYxR">Volterra kernel</a>

K
        (
        x
        ,
        t
        )
        =
        (
        x
        −
        t
        )
        (
        q
        (
        t
        )
        −
        A
        )
      
    
    {\displaystyle K(x,t)=(x-t)(q(t)-A)}

and

h
        (
        x
        )
        =
        α
        (
        x
        −
        c
        )
        +
        β
        .
      
    
    {\displaystyle h(x)=\alpha (x-c)+\beta .}

Since ‖Tk‖ tends to 0, this integral equation has a unique solution given by the <a href="/facts/Neumann_series/E6e3gyAK">Neumann series</a>

f
        =
        (
        I
        −
        T
        
          )
          
            −
            1
          
        
        h
        =
        h
        +
        T
        h
        +
        
          T
          
            2
          
        
        h
        +
        
          T
          
            3
          
        
        h
        +
        ⋯
      
    
    {\displaystyle f=(I-T)^{-1}h=h+Th+T^{2}h+T^{3}h+\cdots }

This iterative scheme is often called Picard iteration after the French mathematician <a href="/facts/Charles_%25C3%2589mile_Picard/fq6m32JE">Charles Émile Picard</a>.

<h3>Fundamental eigenfunctions</h3>
If f is twice continuously differentiable (i.e. C2) on (a, b) satisfying Df = λf, then f is called an <a href="/facts/Eigenfunction/62EJS0XB">eigenfunction</a> of D with <a href="/facts/Eigenvalue/8TjEoT8u">eigenvalue</a> λ.

<ul><li>In the case of a compact interval [a, b] and q continuous on [a, b], the existence theorem implies that for c = a or c = b and every complex number λ there a unique C2 eigenfunction fλ on [a, b] with fλ(c) and f′λ(c) prescribed. Moreover, for each x in [a, b], fλ(x) and f′λ(x) are <a href="/facts/Holomorphic_function/kVzthtEf">holomorphic functions</a> of λ.</li>
<li>For an arbitrary interval (a, b) and q continuous on (a, b), the existence theorem implies that for c in (a, b) and every complex number λ there a unique C2 eigenfunction fλ on (a, b) with fλ(c) and f′λ(c) prescribed. Moreover, for each x in (a, b), fλ(x) and f′λ(x) are <a href="/facts/Holomorphic_function/kVzthtEf">holomorphic functions</a> of λ.</li></ul>
<h3>Green's formula</h3>
If f and g are C2 functions on (a, b), the <a href="/facts/Wronskian/sJNmIF5C">Wronskian</a> W(f, g) is defined by

W
        (
        f
        ,
        g
        )
        (
        x
        )
        =
        f
        (
        x
        )
        
          g
          ′
        
        (
        x
        )
        −
        
          f
          ′
        
        (
        x
        )
        g
        (
        x
        )
        .
      
    
    {\displaystyle W(f,g)(x)=f(x)g'(x)-f'(x)g(x).}

<a href="/facts/Green%2527s_identities/YBQPLqdS">Green's formula</a> - which in this one-dimensional case is a simple integration by parts - states that for x, y in (a, b)

∫
          
            x
          
          
            y
          
        
        (
        D
        f
        )
        g
        −
        f
        (
        D
        g
        )
        
        d
        t
        =
        W
        (
        f
        ,
        g
        )
        (
        y
        )
        −
        W
        (
        f
        ,
        g
        )
        (
        x
        )
        .
      
    
    {\displaystyle \int _{x}^{y}(Df)g-f(Dg)\,dt=W(f,g)(y)-W(f,g)(x).}

When q is continuous and f, g are C2 on the compact interval [a, b], this formula also holds for x = a or y = b.
When f and g are eigenfunctions for the same eigenvalue, then

d
            
              d
              x
            
          
        
        W
        (
        f
        ,
        g
        )
        =
        0
        ,
      
    
    {\displaystyle {\frac {d}{dx}}W(f,g)=0,}

so that W(f, g) is independent of x.

<h2 id="classical-sturmliouville-theory">Classical Sturm–Liouville theory</h2>
Main article: <a href="/facts/Sturm%25E2%2580%2593Liouville_theory/xwvX8d9d">Sturm–Liouville theory</a>
Let [a, b] be a finite closed interval, q a real-valued continuous function on [a, b] and let H0 be the space of C2 functions f on [a, b] satisfying the <a href="/facts/Robin_boundary_condition/tEoxCAfR">Robin boundary conditions</a>

{
            
              
                
                  cos
                  ⁡
                  α
                  
                  f
                  (
                  a
                  )
                  −
                  sin
                  ⁡
                  α
                  
                  
                    f
                    ′
                  
                  (
                  a
                  )
                  =
                  0
                  ,
                
              
              
                
                  cos
                  ⁡
                  β
                  
                  f
                  (
                  b
                  )
                  −
                  sin
                  ⁡
                  β
                  
                  
                    f
                    ′
                  
                  (
                  b
                  )
                  =
                  0
                  ,
                
              
            
            
          
        
      
    
    {\displaystyle {\begin{cases}\cos \alpha \,f(a)-\sin \alpha \,f'(a)=0,\\[0.5ex]\cos \beta \,f(b)-\sin \beta \,f'(b)=0,\end{cases}}}

with <a href="/facts/Inner_product/HoyjElEy">inner product</a>

(
        f
        ,
        g
        )
        =
        
          ∫
          
            a
          
          
            b
          
        
        f
        (
        x
        )
        
          
            
              g
              (
              x
              )
            
            ¯
          
        
        
        d
        x
        .
      
    
    {\displaystyle (f,g)=\int _{a}^{b}f(x){\overline {g(x)}}\,dx.}

In practice usually one of the two standard boundary conditions:

<ul><li><a href="/facts/Dirichlet_boundary_condition/m9wsgG89">Dirichlet boundary condition</a> f(c) = 0</li>
<li><a href="/facts/Neumann_boundary_condition/0e0YBW4G">Neumann boundary condition</a> f′(c) = 0</li></ul>
is imposed at each endpoint c = a, b.
The differential operator D given by

D
        f
        =
        −
        
          f
          ″
        
        +
        q
        f
      
    
    {\displaystyle Df=-f''+qf}

acts on H0. A function f in H0 is called an <a href="/facts/Eigenfunction/62EJS0XB">eigenfunction</a> of D (for the above choice of boundary values) if Df = λ f for some complex number λ, the corresponding <a href="/facts/Eigenvalue/8TjEoT8u">eigenvalue</a>. By Green's formula, D is formally <a href="/facts/Self-adjoint/rKqGtlpy">self-adjoint</a> on H0, since the Wronskian W(f, g) vanishes if both f, g satisfy the boundary conditions:

(
        D
        f
        ,
        g
        )
        =
        (
        f
        ,
        D
        g
        )
        ,
        
        
           for 
        
        f
        ,
        g
        ∈
        
          H
          
            0
          
        
        .
      
    
    {\displaystyle (Df,g)=(f,Dg),\quad {\text{ for }}f,g\in H_{0}.}

As a consequence, exactly as for a <a href="/facts/Hermitian_matrix/qArPblYo">self-adjoint matrix</a> in finite dimensions,

<ul><li>the eigenvalues of D are real;</li>
<li>the <a href="/facts/Eigenspace/8TjEoT8u">eigenspaces</a> for distinct eigenvalues are <a href="/facts/Orthogonal/JVIjYPog">orthogonal</a>.</li></ul>
It turns out that the eigenvalues can be described by the <a href="/facts/Courant_minimax_principle/P0MRtGPv">maximum-minimum</a> principle of <a href="/facts/John_Strutt%2c_3rd_Baron_Rayleigh/HcJyYdk5">Rayleigh</a>–<a href="/facts/Walter_Ritz/jl9VvugR">Ritz</a><a class="footnote-ref" id="fnref:3" href="#fn:3">3</a> (see below). In fact it is easy to see a priori that the eigenvalues are bounded below because the operator D is itself bounded below on H0:

(
 D
 f
 ,
 f
 )
 ≥
 M
 (
 f
 ,
 f
 )
 
 
 {\displaystyle (Df,f)\geq M(f,f)}
 
 for some finite (possibly negative) constant 
 
 
 
 M
 
 
 {\displaystyle M}
 
.
In fact, integrating by parts,

(
        D
        f
        ,
        f
        )
        =
        
          
            [
            
              −
              
                f
                ′
              
              
                
                  f
                  ¯
                
              
            
            ]
          
          
            a
          
          
            b
          
        
        +
        ∫
        
          |
        
        
          f
          ′
        
        
          
            |
          
          
            2
          
        
        +
        ∫
        q
        
          |
        
        f
        
          
            |
          
          
            2
          
        
        .
      
    
    {\displaystyle (Df,f)=\left[-f'{\overline {f}}\right]_{a}^{b}+\int |f'|^{2}+\int q|f|^{2}.}

For Dirichlet or Neumann boundary conditions, the first term vanishes and the inequality holds with M = inf q.
For general Robin boundary conditions the first term can be estimated using an elementary Peter-Paul version of <a href="/facts/Sobolev_inequality/jxJvCHML">Sobolev's inequality</a>:

<blockquote>"Given ε > 0, there is constant R > 0 such that |f(x)|2 ≤ ε (f′, f′) + R (f, f) for all f in C1[a, b]."</blockquote>
In fact, since

|
        
        f
        (
        b
        )
        −
        f
        (
        x
        )
        
          |
        
        ≤
        (
        b
        −
        a
        
          )
          
            1
            
              /
            
            2
          
        
        ⋅
        ‖
        
          f
          ′
        
        
          ‖
          
            2
          
        
        ,
      
    
    {\displaystyle |f(b)-f(x)|\leq (b-a)^{1/2}\cdot \|f'\|_{2},}

only an estimate for f(b) is needed and this follows by replacing f(x) in the above inequality by (x − a)n·(b − a)−n·f(x) for n sufficiently large.

<h3>Green's function (regular case)</h3>
From the theory of ordinary differential equations, there are unique fundamental eigenfunctions φλ(x), χλ(x) such that

<ul><li>D φλ = λ φλ, φλ(a) = sin α, φλ'(a) = cos α</li>
<li>D χλ = λ χλ, χλ(b) = sin β, χλ'(b) = cos β</li></ul>
which at each point, together with their first derivatives, depend holomorphically on λ. Let

ω
        (
        λ
        )
        =
        W
        (
        
          ϕ
          
            λ
          
        
        ,
        
          χ
          
            λ
          
        
        )
        ,
      
    
    {\displaystyle \omega (\lambda )=W(\phi _{\lambda },\chi _{\lambda }),}

be an <a href="/facts/Entire_function/XNluKzu7">entire holomorphic function</a>.
This function ω(λ) plays the role of the <a href="/facts/Characteristic_polynomial/7m8roDqo">characteristic polynomial</a> of D. Indeed, the uniqueness of the fundamental eigenfunctions implies that its zeros are precisely the eigenvalues of D and that each non-zero eigenspace is one-dimensional. In particular there are at most countably many eigenvalues of D and, if there are infinitely many, they must tend to infinity. It turns out that the zeros of ω(λ) also have mutilplicity one (see below).
If λ is not an eigenvalue of D on H0, define the <a href="/facts/Green%2527s_function/rKxKKPcp">Green's function</a> by

G
          
            λ
          
        
        (
        x
        ,
        y
        )
        =
        
          
            {
            
              
                
                  
                    ϕ
                    
                      λ
                    
                  
                  (
                  x
                  )
                  
                    χ
                    
                      λ
                    
                  
                  (
                  y
                  )
                  
                    /
                  
                  ω
                  (
                  λ
                  )
                
                
                  
                     for 
                  
                  x
                  ≥
                  y
                
              
              
                
                  
                    χ
                    
                      λ
                    
                  
                  (
                  x
                  )
                  
                    ϕ
                    
                      λ
                    
                  
                  (
                  y
                  )
                  
                    /
                  
                  ω
                  (
                  λ
                  )
                
                
                  
                     for 
                  
                  y
                  ≥
                  x
                  .
                
              
            
            
          
        
      
    
    {\displaystyle G_{\lambda }(x,y)={\begin{cases}\phi _{\lambda }(x)\chi _{\lambda }(y)/\omega (\lambda )&{\text{ for }}x\geq y\\[1ex]\chi _{\lambda }(x)\phi _{\lambda }(y)/\omega (\lambda )&{\text{ for }}y\geq x.\end{cases}}}

This kernel defines an operator on the inner product space C[a,b] via

(
        
          G
          
            λ
          
        
        f
        )
        (
        x
        )
        =
        
          ∫
          
            a
          
          
            b
          
        
        
          G
          
            λ
          
        
        (
        x
        ,
        y
        )
        f
        (
        y
        )
        
        d
        y
        .
      
    
    {\displaystyle (G_{\lambda }f)(x)=\int _{a}^{b}G_{\lambda }(x,y)f(y)\,dy.}

Since Gλ(x,y) is continuous on [a, b] × [a, b], it defines a <a href="/facts/Hilbert%25E2%2580%2593Schmidt_operator/Xhy17yn6">Hilbert–Schmidt operator</a> on the Hilbert space completion H of C[a, b] = H1 (or equivalently of the dense subspace H0), taking values in H1. This operator carries H1 into H0. When λ is real, Gλ(x,y) = Gλ(y,x) is also real, so defines a self-adjoint operator on H. Moreover,

<ul><li>Gλ (D − λ) = I on H0</li>
<li>Gλ carries H1 into H0, and (D − λ) Gλ = I on H1.</li></ul>
Thus the operator Gλ can be identified with the <a href="/facts/Resolvent_formalism/IKaKtcef">resolvent</a> (D − λ)−1.

<h3>Spectral theorem</h3>

Theorem—The eigenvalues of D are real of multiplicity one and form an increasing sequence λ1 < λ2 < ⋯ tending to infinity.
The corresponding normalised eigenfunctions form an orthonormal basis of H0.
The k-th eigenvalue of D is given by the <a href="/facts/Courant_minimax_principle/P0MRtGPv">minimax principle</a>

λ
          
            k
          
        
        =
        
          max
          
            dim
            ⁡
            G
            =
            k
            −
            1
          
        
        
        
          min
          
            f
            ⊥
            G
          
        
        
          
            
              (
              D
              f
              ,
              f
              )
            
            
              (
              f
              ,
              f
              )
            
          
        
        .
      
    
    {\displaystyle \lambda _{k}=\max _{\dim G=k-1}\,\min _{f\perp G}{(Df,f) \over (f,f)}.}

In particular if q1 ≤ q2, then

λ
          
            k
          
        
        (
        
          D
          
            1
          
        
        )
        ≤
        
          λ
          
            k
          
        
        (
        
          D
          
            2
          
        
        )
        .
      
    
    {\displaystyle \lambda _{k}(D_{1})\leq \lambda _{k}(D_{2}).}

In fact let T = Gλ for λ large and negative. Then T defines a <a href="/facts/Compact_operator/KNkcK4mE">compact self-adjoint operator</a> on the Hilbert space H.
By the <a href="/facts/Spectral_theorem/Vqc04B21">spectral theorem</a> for compact self-adjoint operators, H has an orthonormal basis consisting of eigenvectors ψn of T with Tψn = μn ψn, where μn tends to zero. The range of T contains H0 so is dense. Hence 0 is not an eigenvalue of T. The resolvent properties of T imply that ψn lies in H0 and that

D
        
          ψ
          
            n
          
        
        =
        
          (
          
            λ
            +
            
              
                1
                
                  μ
                  
                    n
                  
                
              
            
          
          )
        
        
          ψ
          
            n
          
        
      
    
    {\displaystyle D\psi _{n}=\left(\lambda +{\frac {1}{\mu _{n}}}\right)\psi _{n}}

The minimax principle follows because if

λ
        (
        G
        )
        =
        
          min
          
            f
            ⊥
            G
          
        
        
          
            
              (
              D
              f
              ,
              f
              )
            
            
              (
              f
              ,
              f
              )
            
          
        
        ,
      
    
    {\displaystyle \lambda (G)=\min _{f\perp G}{\frac {(Df,f)}{(f,f)}},}

then λ(G) = λk for the <a href="/facts/Linear_span/vPjclEtb">linear span</a> of the first k − 1 eigenfunctions. For any other (k − 1)-dimensional subspace G, some f in the linear span of the first k eigenvectors must be orthogonal to G. Hence λ(G) ≤ (Df,f)/(f,f) ≤ λk.

<h3>Wronskian as a Fredholm determinant</h3>
For simplicity, suppose that m ≤ q(x) ≤ M on [0, π] with Dirichlet boundary conditions. The minimax principle shows that

n
          
            2
          
        
        +
        m
        ≤
        
          λ
          
            n
          
        
        (
        D
        )
        ≤
        
          n
          
            2
          
        
        +
        M
        .
      
    
    {\displaystyle n^{2}+m\leq \lambda _{n}(D)\leq n^{2}+M.}

It follows that the resolvent (D − λ)−1 is a <a href="/facts/Trace-class_operator/2AmWENA3">trace-class operator</a> whenever λ is not an eigenvalue of D and hence that the <a href="/facts/Fredholm_determinant/9TvIBCvI">Fredholm determinant</a> det I − μ(D − λ)−1 is defined.
The Dirichlet boundary conditions imply that

ω
        (
        λ
        )
        =
        
          ϕ
          
            λ
          
        
        (
        b
        )
        .
      
    
    {\displaystyle \omega (\lambda )=\phi _{\lambda }(b).}

Using Picard iteration, Titchmarsh showed that φλ(b), and hence ω(λ), is an <a href="/facts/Entire_function/XNluKzu7">entire function of finite order</a> 1/2:

ω
        (
        λ
        )
        =
        
          
            O
          
        
        
          (
          
            e
            
              
                
                  |
                
                λ
                
                  |
                
              
            
          
          )
        
      
    
    {\displaystyle \omega (\lambda )={\mathcal {O}}\left(e^{\sqrt {|\lambda |}}\right)}

At a zero μ of ω(λ), φμ(b) = 0. Moreover,

ψ
        (
        x
        )
        =
        
          ∂
          
            λ
          
        
        
          φ
          
            λ
          
        
        (
        x
        )
        
          
            |
          
          
            λ
            =
            μ
          
        
      
    
    {\displaystyle \psi (x)=\partial _{\lambda }\varphi _{\lambda }(x)|_{\lambda =\mu }}

satisfies (D − μ)ψ = φμ. Thus

ω
        (
        λ
        )
        =
        (
        λ
        −
        μ
        )
        ψ
        (
        b
        )
        +
        
          
            O
          
        
        (
        (
        λ
        −
        μ
        
          )
          
            2
          
        
        )
      
    
    {\displaystyle \omega (\lambda )=(\lambda -\mu )\psi (b)+{\mathcal {O}}((\lambda -\mu )^{2})}

This implies that<a class="footnote-ref" id="fnref:4" href="#fn:4">4</a>

μ is a simple zero of ω(λ).
For otherwise ψ(b) = 0, so that ψ would have to lie in H0. But then

(
        
          ϕ
          
            μ
          
        
        ,
        
          ϕ
          
            μ
          
        
        )
        =
        (
        (
        D
        −
        μ
        )
        ψ
        ,
        
          ϕ
          
            μ
          
        
        )
        =
        (
        ψ
        ,
        (
        D
        −
        μ
        )
        
          ϕ
          
            μ
          
        
        )
        =
        0
        ,
      
    
    {\displaystyle (\phi _{\mu },\phi _{\mu })=((D-\mu )\psi ,\phi _{\mu })=(\psi ,(D-\mu )\phi _{\mu })=0,}

a contradiction.
On the other hand, the distribution of the zeros of the entire function ω(λ) is already known from the minimax principle.
By the <a href="/facts/Hadamard_factorization_theorem/atdInY5m">Hadamard factorization theorem</a>, it follows that<a class="footnote-ref" id="fnref:5" href="#fn:5">5</a>

ω
        (
        λ
        )
        =
        C
        ∏
        (
        1
        −
        λ
        
          /
        
        
          λ
          
            n
          
        
        )
        ,
      
    
    {\displaystyle \omega (\lambda )=C\prod (1-\lambda /\lambda _{n}),}

for some non-zero constant C.
Hence

det
        (
        I
        −
        μ
        (
        D
        −
        λ
        
          )
          
            −
            1
          
        
        )
        =
        ∏
        
          (
          
            1
            −
            
              
                μ
                
                  
                    λ
                    
                      n
                    
                  
                  −
                  λ
                
              
            
          
          )
        
        =
        ∏
        
          
            
              1
              −
              (
              λ
              +
              μ
              )
              
                /
              
              
                λ
                
                  n
                
              
            
            
              1
              −
              λ
              
                /
              
              
                λ
                
                  n
                
              
            
          
        
        =
        
          
            
              ω
              (
              λ
              +
              μ
              )
            
            
              ω
              (
              λ
              )
            
          
        
        .
      
    
    {\displaystyle \det(I-\mu (D-\lambda )^{-1})=\prod \left(1-{\mu  \over \lambda _{n}-\lambda }\right)=\prod {1-(\lambda +\mu )/\lambda _{n} \over 1-\lambda /\lambda _{n}}={\omega (\lambda +\mu ) \over \omega (\lambda )}.}

In particular if 0 is not an eigenvalue of D

ω
        (
        μ
        )
        =
        ω
        (
        0
        )
        ⋅
        det
        (
        I
        −
        μ
        
          D
          
            −
            1
          
        
        )
        .
      
    
    {\displaystyle \omega (\mu )=\omega (0)\cdot \det(I-\mu D^{-1}).}

<h2 id="tools-from-abstract-spectral-theory">Tools from abstract spectral theory</h2>
<h3>Functions of bounded variation</h3>
See also: <a href="/facts/Bounded_variation/tIjQTpjh">Bounded variation</a>, <a href="/facts/Lebesgue%25E2%2580%2593Stieltjes_integration/W4GD5Tlr">Lebesgue–Stieltjes integration</a>, and <a href="/facts/Riesz_representation_theorem/9CjKDUII">Riesz representation theorem</a>
A function ρ(x) of <a href="/facts/Bounded_variation/tIjQTpjh">bounded variation</a><a class="footnote-ref" id="fnref:6" href="#fn:6">6</a> on a closed interval [a, b] is a complex-valued function such that its <a href="/facts/Total_variation/tyY5fTz9">total variation</a> V(ρ), the <a href="/facts/Supremum/oXCKVopz">supremum</a> of the variations

∑
          
            r
            =
            0
          
          
            k
            −
            1
          
        
        
          |
        
        ρ
        (
        
          x
          
            r
            +
            1
          
        
        )
        −
        ρ
        (
        
          x
          
            r
          
        
        )
        
          |
        
      
    
    {\displaystyle \sum _{r=0}^{k-1}|\rho (x_{r+1})-\rho (x_{r})|}

over all <a href="/facts/Partition_of_an_interval/FKoPazKA">dissections</a>

a
 =
 
 x
 
 0
 
 
 <
 
 x
 
 1
 
 
 <
 ⋯
 <
 
 x
 
 k
 
 
 =
 b
 
 
 {\displaystyle a=x_{0}<x_{1}<\dots <x_{k}=b}

is finite. The real and imaginary parts of ρ are real-valued functions of bounded variation. If ρ is real-valued and normalised so that ρ(a) = 0, it has a canonical decomposition as the difference of two bounded non-decreasing functions:

ρ
        (
        x
        )
        =
        
          ρ
          
            +
          
        
        (
        x
        )
        −
        
          ρ
          
            −
          
        
        (
        x
        )
        ,
      
    
    {\displaystyle \rho (x)=\rho _{+}(x)-\rho _{-}(x),}

where ρ+(x) and ρ–(x) are the total positive and negative variation of ρ over [a, x].
If f is a continuous function on [a, b] its <a href="/facts/Riemann%25E2%2580%2593Stieltjes_integral/C08wyfWn">Riemann–Stieltjes integral</a> with respect to ρ

∫
          
            a
          
          
            b
          
        
        f
        (
        x
        )
        
        d
        ρ
        (
        x
        )
      
    
    {\displaystyle \int _{a}^{b}f(x)\,d\rho (x)}

is defined to be the limit of approximating sums

∑
          
            r
            =
            0
          
          
            k
            −
            1
          
        
        f
        (
        
          x
          
            r
          
        
        )
        (
        ρ
        (
        
          x
          
            r
            +
            1
          
        
        )
        −
        ρ
        (
        
          x
          
            r
          
        
        )
        )
      
    
    {\displaystyle \sum _{r=0}^{k-1}f(x_{r})(\rho (x_{r+1})-\rho (x_{r}))}

as the <a href="/facts/Partition_of_an_interval/FKoPazKA">mesh</a> of the dissection, given by sup |xr+1 − xr|, tends to zero.
This integral satisfies

|
          
            
              ∫
              
                a
              
              
                b
              
            
            f
            (
            x
            )
            
            d
            ρ
            (
            x
            )
          
          |
        
        ≤
        V
        (
        ρ
        )
        ⋅
        ‖
        f
        
          ‖
          
            ∞
          
        
      
    
    {\displaystyle \left|\int _{a}^{b}f(x)\,d\rho (x)\right|\leq V(\rho )\cdot \|f\|_{\infty }}

and thus defines a <a href="/facts/Bounded_linear_functional/LYmDTIqR">bounded linear functional</a> dρ on C[a, b] with <a href="/facts/Norm_(mathematics)/xIbR4uE1">norm</a> ‖dρ‖ = V(ρ).
Every bounded linear functional μ on C[a, b] has an <a href="/facts/Absolute_value/dwJBpEs5">absolute value</a> |μ| defined for non-negative f by<a class="footnote-ref" id="fnref:7" href="#fn:7">7</a>

|
        
        μ
        
          |
        
        (
        f
        )
        =
        
          sup
          
            0
            ≤
            
              |
            
            g
            
              |
            
            ≤
            f
          
        
        
          |
        
        μ
        (
        g
        )
        
          |
        
        .
      
    
    {\displaystyle |\mu |(f)=\sup _{0\leq |g|\leq f}|\mu (g)|.}

The form |μ| extends linearly to a bounded linear form on C[a, b] with norm ‖μ‖ and satisfies the characterizing inequality

|
        
        μ
        (
        f
        )
        
          |
        
        ≤
        
          |
        
        μ
        
          |
        
        (
        
          |
        
        f
        
          |
        
        )
      
    
    {\displaystyle |\mu (f)|\leq |\mu |(|f|)}

for f in C[a, b]. If μ is real, i.e. is real-valued on real-valued functions, then

μ
        =
        
          |
        
        μ
        
          |
        
        −
        (
        
          |
        
        μ
        
          |
        
        −
        μ
        )
        ≡
        
          μ
          
            +
          
        
        −
        
          μ
          
            −
          
        
      
    
    {\displaystyle \mu =|\mu |-(|\mu |-\mu )\equiv \mu _{+}-\mu _{-}}

gives a canonical decomposition as a difference of positive forms, i.e. forms that are non-negative on non-negative functions.
Every positive form μ extends uniquely to the linear span of non-negative bounded lower <a href="/facts/Semicontinuous_function/4unJbWdW">semicontinuous functions</a> g by the formula<a class="footnote-ref" id="fnref:8" href="#fn:8">8</a>

μ
        (
        g
        )
        =
        lim
        μ
        (
        
          f
          
            n
          
        
        )
        ,
      
    
    {\displaystyle \mu (g)=\lim \mu (f_{n}),}

where the non-negative continuous functions fn increase pointwise to g.
The same therefore applies to an arbitrary bounded linear form μ, so that a function ρ of bounded variation may be defined by<a class="footnote-ref" id="fnref:9" href="#fn:9">9</a>

ρ
        (
        x
        )
        =
        μ
        (
        
          χ
          
            [
            a
            ,
            x
            ]
          
        
        )
        ,
      
    
    {\displaystyle \rho (x)=\mu (\chi _{[a,x]}),}

where χA denotes the <a href="/facts/Indicator_function/QEuh04NM">characteristic function</a> of a subset A of [a, b]. Thus μ = dρ and ‖μ‖ = ‖dρ‖.
Moreover μ+ = dρ+ and μ– = dρ–.
This correspondence between functions of bounded variation and bounded linear forms is a special case of the <a href="/facts/Riesz_representation_theorem/9CjKDUII">Riesz representation theorem</a>.
The <a href="/facts/Support_(measure_theory)/mBM9wEAF">support</a> of μ = dρ is the complement of all points x in [a, b] where ρ is constant on some neighborhood of x; by definition it is a closed subset A of [a, b]. Moreover, μ((1 − χA)f) = 0, so that μ(f) = 0 if f vanishes on A.

<h3>Spectral measure</h3>
See also: <a href="/facts/Spectral_theorem/Vqc04B21">Spectral theorem</a> and <a href="/facts/Projection-valued_measure/baNsfTzA">Projection-valued measure</a>
Let H be a Hilbert space and 
 
 
 
 T
 
 
 {\displaystyle T}
 
 a self-adjoint <a href="/facts/Bounded_operator/LYmDTIqR">bounded operator</a> on H with 
 
 
 
 0
 ≤
 T
 ≤
 I
 
 
 {\displaystyle 0\leq T\leq I}
 
, so that the <a href="/facts/Spectrum_(functional_analysis)/cA1evTo6">spectrum</a> 
 
 
 
 σ
 (
 T
 )
 
 
 {\displaystyle \sigma (T)}
 
 of 
 
 
 
 T
 
 
 {\displaystyle T}
 
 is contained in 
 
 
 
 [
 0
 ,
 1
 ]
 
 
 {\displaystyle [0,1]}
 
. If 
 
 
 
 p
 (
 t
 )
 
 
 {\displaystyle p(t)}
 
 is a complex polynomial, then by the <a href="/facts/Continuous_functional_calculus/cJ2Nj9Bo">spectral mapping theorem</a>

σ
        (
        p
        (
        T
        )
        )
        =
        p
        (
        σ
        (
        T
        )
        )
      
    
    {\displaystyle \sigma (p(T))=p(\sigma (T))}

and hence

‖
        p
        (
        T
        )
        ‖
        ≤
        ‖
        p
        
          ‖
          
            ∞
          
        
      
    
    {\displaystyle \|p(T)\|\leq \|p\|_{\infty }}

where 
 
 
 
 ‖
 ⋅
 
 ‖
 
 ∞
 
 
 
 
 {\displaystyle \|\cdot \|_{\infty }}
 
 denotes the <a href="/facts/Uniform_norm/5yIkjNE3">uniform norm</a> on C[0, 1]. By the <a href="/facts/Weierstrass_approximation_theorem/QB07Nwba">Weierstrass approximation theorem</a>, polynomials are uniformly dense in C[0, 1]. It follows that 
 
 
 
 f
 (
 T
 )
 
 
 {\displaystyle f(T)}
 
 can be defined 
 
 
 
 ∀
 f
 ∈
 C
 [
 0
 ,
 1
 ]
 
 
 {\displaystyle \forall f\in C[0,1]}
 
, with

σ
        (
        f
        (
        T
        )
        )
        =
        f
        (
        σ
        (
        T
        )
        )
      
    
    {\displaystyle \sigma (f(T))=f(\sigma (T))}
  
 and 
  
    
      
        ‖
        f
        (
        T
        )
        ‖
        ≤
        ‖
        f
        
          ‖
          
            ∞
          
        
        .
      
    
    {\displaystyle \|f(T)\|\leq \|f\|_{\infty }.}

If 
 
 
 
 0
 ≤
 g
 ≤
 1
 
 
 {\displaystyle 0\leq g\leq 1}
 
 is a lower semicontinuous function on [0, 1], for example the characteristic function 
 
 
 
 
 χ
 
 [
 0
 ,
 α
 ]
 
 
 
 
 {\displaystyle \chi _{[0,\alpha ]}}
 
 of a subinterval of [0, 1], then

g
 
 
 {\displaystyle g}
 
 is a pointwise increasing limit of non-negative 
 
 
 
 
 f
 
 n
 
 
 ∈
 C
 [
 0
 ,
 1
 ]
 
 
 {\displaystyle f_{n}\in C[0,1]}
 
.
If 
 
 
 
 ξ
 
 
 {\displaystyle \xi }
 
 is a vector in H, then the vectors

η
          
            n
          
        
        =
        
          f
          
            n
          
        
        (
        T
        )
        ξ
      
    
    {\displaystyle \eta _{n}=f_{n}(T)\xi }

form a <a href="/facts/Cauchy_sequence/7Q5c8TuF">Cauchy sequence</a> in H, since, for 
 
 
 
 n
 ≥
 m
 
 
 {\displaystyle n\geq m}
 
,

‖
        
          η
          
            n
          
        
        −
        
          η
          
            m
          
        
        
          ‖
          
            2
          
        
        ≤
        (
        
          η
          
            n
          
        
        ,
        ξ
        )
        −
        (
        
          η
          
            m
          
        
        ,
        ξ
        )
        ,
      
    
    {\displaystyle \|\eta _{n}-\eta _{m}\|^{2}\leq (\eta _{n},\xi )-(\eta _{m},\xi ),}

and 
 
 
 
 (
 
 η
 
 n
 
 
 ,
 ξ
 )
 =
 (
 
 f
 
 n
 
 
 (
 T
 )
 ξ
 ,
 ξ
 )
 
 
 {\displaystyle (\eta _{n},\xi )=(f_{n}(T)\xi ,\xi )}
 
 is bounded and increasing, so has a limit.
It follows that 
 
 
 
 g
 (
 T
 )
 
 
 {\displaystyle g(T)}
 
 can be defined by<a class="footnote-ref" id="fnref:10" href="#fn:10">10</a>

g
        (
        T
        )
        ξ
        =
        lim
        
          f
          
            n
          
        
        (
        T
        )
        ξ
        .
      
    
    {\displaystyle g(T)\xi =\lim f_{n}(T)\xi .}

If 
 
 
 
 ξ
 
 
 {\displaystyle \xi }
 
 and η are vectors in H, then

μ
          
            ξ
            ,
            η
          
        
        (
        f
        )
        =
        (
        f
        (
        T
        )
        ξ
        ,
        η
        )
      
    
    {\displaystyle \mu _{\xi ,\eta }(f)=(f(T)\xi ,\eta )}

defines a bounded linear form 
 
 
 
 
 μ
 
 ξ
 ,
 η
 
 
 
 
 {\displaystyle \mu _{\xi ,\eta }}
 
 on H. By the Riesz representation theorem

μ
          
            ξ
            ,
            η
          
        
        =
        d
        
          ρ
          
            ξ
            ,
            η
          
        
      
    
    {\displaystyle \mu _{\xi ,\eta }=d\rho _{\xi ,\eta }}

for a unique normalised function 
 
 
 
 
 ρ
 
 ξ
 ,
 η
 
 
 
 
 {\displaystyle \rho _{\xi ,\eta }}
 
 of bounded variation on [0, 1].

 
 
 
 d
 
 ρ
 
 ξ
 ,
 η
 
 
 
 
 {\displaystyle d\rho _{\xi ,\eta }}
 
 (or sometimes slightly incorrectly 
 
 
 
 
 ρ
 
 ξ
 ,
 η
 
 
 
 
 {\displaystyle \rho _{\xi ,\eta }}
 
 itself) is called the spectral measure determined by 
 
 
 
 ξ
 
 
 {\displaystyle \xi }
 
 and η.
The operator 
 
 
 
 g
 (
 T
 )
 
 
 {\displaystyle g(T)}
 
 is accordingly uniquely characterised by the equation

(
        g
        (
        T
        )
        ξ
        ,
        η
        )
        =
        
          μ
          
            ξ
            ,
            η
          
        
        (
        g
        )
        =
        
          ∫
          
            0
          
          
            1
          
        
        g
        (
        λ
        )
        
        d
        
          ρ
          
            ξ
            ,
            η
          
        
        (
        λ
        )
        .
      
    
    {\displaystyle (g(T)\xi ,\eta )=\mu _{\xi ,\eta }(g)=\int _{0}^{1}g(\lambda )\,d\rho _{\xi ,\eta }(\lambda ).}

The <a href="/facts/Spectral_theorem/Vqc04B21">spectral projection</a> 
 
 
 
 E
 (
 λ
 )
 
 
 {\displaystyle E(\lambda )}
 
 is defined by

E
        (
        λ
        )
        =
        
          χ
          
            [
            0
            ,
            λ
            ]
          
        
        (
        T
        )
        ,
      
    
    {\displaystyle E(\lambda )=\chi _{[0,\lambda ]}(T),}

so that

ρ
          
            ξ
            ,
            η
          
        
        (
        λ
        )
        =
        (
        E
        (
        λ
        )
        ξ
        ,
        η
        )
        .
      
    
    {\displaystyle \rho _{\xi ,\eta }(\lambda )=(E(\lambda )\xi ,\eta ).}

It follows that

g
        (
        T
        )
        =
        
          ∫
          
            0
          
          
            1
          
        
        g
        (
        λ
        )
        
        d
        E
        (
        λ
        )
        ,
      
    
    {\displaystyle g(T)=\int _{0}^{1}g(\lambda )\,dE(\lambda ),}

which is understood in the sense that for any vectors 
  
    
      
        ξ
      
    
    {\displaystyle \xi }
  
 and 
  
    
      
        η
      
    
    {\displaystyle \eta }
  
,

(
        g
        (
        T
        )
        ξ
        ,
        η
        )
        =
        
          ∫
          
            0
          
          
            1
          
        
        g
        (
        λ
        )
        
        d
        (
        E
        (
        λ
        )
        ξ
        ,
        η
        )
        =
        
          ∫
          
            0
          
          
            1
          
        
        g
        (
        λ
        )
        
        d
        
          ρ
          
            ξ
            ,
            η
          
        
        (
        λ
        )
        .
      
    
    {\displaystyle (g(T)\xi ,\eta )=\int _{0}^{1}g(\lambda )\,d(E(\lambda )\xi ,\eta )=\int _{0}^{1}g(\lambda )\,d\rho _{\xi ,\eta }(\lambda ).}

For a single vector 
 
 
 
 ξ
 ,
 
 
 μ
 
 ξ
 
 
 =
 
 μ
 
 ξ
 ,
 ξ
 
 
 
 
 {\displaystyle \xi ,\,\mu _{\xi }=\mu _{\xi ,\xi }}
 
 is a positive form on [0, 1] 
(in other words proportional to a <a href="/facts/Probability_measure/8BjhgoPR">probability measure</a> on [0, 1]) and 
 
 
 
 
 ρ
 
 ξ
 
 
 =
 
 ρ
 
 ξ
 ,
 ξ
 
 
 
 
 {\displaystyle \rho _{\xi }=\rho _{\xi ,\xi }}
 
 is non-negative and non-decreasing. Polarisation shows that all the forms 
 
 
 
 
 μ
 
 ξ
 ,
 η
 
 
 
 
 {\displaystyle \mu _{\xi ,\eta }}
 
 can naturally be expressed in terms of such positive forms, since

μ
          
            ξ
            ,
            η
          
        
        =
        
          
            1
            4
          
        
        
          (
          
            
              μ
              
                ξ
                +
                η
              
            
            +
            i
            
              μ
              
                ξ
                +
                i
                η
              
            
            −
            
              μ
              
                ξ
                −
                η
              
            
            −
            i
            
              μ
              
                ξ
                −
                i
                η
              
            
          
          )
        
      
    
    {\displaystyle \mu _{\xi ,\eta }={\frac {1}{4}}\left(\mu _{\xi +\eta }+i\mu _{\xi +i\eta }-\mu _{\xi -\eta }-i\mu _{\xi -i\eta }\right)}

If the vector 
 
 
 
 ξ
 
 
 {\displaystyle \xi }
 
 is such that the <a href="/facts/Linear_span/vPjclEtb">linear span</a> of the vectors 
 
 
 
 (
 
 T
 
 n
 
 
 ξ
 )
 
 
 {\displaystyle (T^{n}\xi )}
 
 is dense in H, i.e. 
 
 
 
 ξ
 
 
 {\displaystyle \xi }
 
 is a cyclic vector for 
 
 
 
 T
 
 
 {\displaystyle T}
 
, then the map 
 
 
 
 U
 
 
 {\displaystyle U}
 
 defined by

U
        (
        f
        )
        =
        f
        (
        T
        )
        ξ
        ,
        
        C
        [
        0
        ,
        1
        ]
        →
        H
      
    
    {\displaystyle U(f)=f(T)\xi ,\,C[0,1]\rightarrow H}

satisfies

(
        U
        
          f
          
            1
          
        
        ,
        U
        
          f
          
            2
          
        
        )
        =
        
          ∫
          
            0
          
          
            1
          
        
        
          f
          
            1
          
        
        (
        λ
        )
        
          
            
              
                f
                
                  2
                
              
              (
              λ
              )
            
            ¯
          
        
        
        d
        
          ρ
          
            ξ
          
        
        (
        λ
        )
        .
      
    
    {\displaystyle (Uf_{1},Uf_{2})=\int _{0}^{1}f_{1}(\lambda ){\overline {f_{2}(\lambda )}}\,d\rho _{\xi }(\lambda ).}

Let 
 
 
 
 
 L
 
 2
 
 
 (
 [
 0
 ,
 1
 ]
 ,
 d
 
 ρ
 
 ξ
 
 
 )
 
 
 {\displaystyle L_{2}([0,1],d\rho _{\xi })}
 
 denote the Hilbert space completion of 
 
 
 
 C
 [
 0
 ,
 1
 ]
 
 
 {\displaystyle C[0,1]}
 
 associated with the possibly <a href="/facts/Inner_product/HoyjElEy">degenerate inner product</a> on the right hand side.<a class="footnote-ref" id="fnref:11" href="#fn:11">11</a> 
Thus 
 
 
 
 U
 
 
 {\displaystyle U}
 
 extends to a <a href="/facts/Unitary_transformation/DyvU0nms">unitary transformation</a> of 
 
 
 
 
 L
 
 2
 
 
 (
 [
 0
 ,
 1
 ]
 ,
 
 ρ
 
 ξ
 
 
 )
 
 
 {\displaystyle L_{2}([0,1],\rho _{\xi })}
 
 onto H. 
 
 
 
 U
 T
 
 U
 
 ∗
 
 
 
 
 {\displaystyle UTU^{\ast }}
 
 is then just multiplication by 
 
 
 
 λ
 
 
 {\displaystyle \lambda }
 
 on 
 
 
 
 
 L
 
 2
 
 
 (
 [
 0
 ,
 1
 ]
 ,
 d
 
 ρ
 
 ξ
 
 
 )
 
 
 {\displaystyle L_{2}([0,1],d\rho _{\xi })}
 
; and more generally 
 
 
 
 U
 f
 (
 T
 )
 
 U
 
 ∗
 
 
 
 
 {\displaystyle Uf(T)U^{\ast }}
 
 is multiplication by 
 
 
 
 f
 (
 λ
 )
 
 
 {\displaystyle f(\lambda )}
 
. In this case, the support of 
 
 
 
 d
 
 ρ
 
 ξ
 
 
 
 
 {\displaystyle d\rho _{\xi }}
 
 is exactly 
 
 
 
 σ
 (
 T
 )
 
 
 {\displaystyle \sigma (T)}
 
, so that

the self-adjoint operator becomes a multiplication operator on the space of functions on its spectrum with inner product given by the spectral measure.
<h2 id="weyltitchmarshkodaira-theory">Weyl–Titchmarsh–Kodaira theory</h2>
The eigenfunction expansion associated with singular differential operators of the form

D
        f
        =
        −
        (
        p
        
          f
          ′
        
        
          )
          ′
        
        +
        q
        f
      
    
    {\displaystyle Df=-(pf')'+qf}

on an open interval (a, b) requires an initial analysis of the behaviour of the fundamental eigenfunctions near the endpoints a and b to determine possible <a href="/facts/Boundary_condition/QEfmUqmP">boundary conditions</a> there. Unlike the regular Sturm–Liouville case, in some circumstances <a href="/facts/Spectral_theory/8sAEEIMM">spectral values</a> of D can have <a href="/facts/Multiplicity_(mathematics)/dndUyetA">multiplicity</a> 2. In the development outlined below standard assumptions will be imposed on p and q that guarantee that the spectrum of D has multiplicity one everywhere and is bounded below. This includes almost all important applications; modifications required for the more general case will be discussed later.
Having chosen the boundary conditions, as in the classical theory the resolvent of D, (D + R)−1 for R large and positive, is given by an operator T corresponding to a Green's function constructed from two fundamental eigenfunctions. In the classical case T was a compact self-adjoint operator; in this case T is just a self-adjoint bounded operator with 0 ≤ T ≤ I. The abstract theory of spectral measure can therefore be applied to T to give the eigenfunction expansion for D.
The central idea in the proof of Weyl and Kodaira can be explained informally as follows. Assume that the spectrum of D lies in [1, ∞) and that T = D−1 and let

E
        (
        λ
        )
        =
        
          χ
          
            [
            
              λ
              
                −
                1
              
            
            ,
            1
            ]
          
        
        (
        T
        )
      
    
    {\displaystyle E(\lambda )=\chi _{[\lambda ^{-1},1]}(T)}

be the spectral projection of D corresponding to the interval [1, λ]. For an arbitrary function f define

f
        (
        x
        ,
        λ
        )
        =
        (
        E
        (
        λ
        )
        f
        )
        (
        x
        )
        .
      
    
    {\displaystyle f(x,\lambda )=(E(\lambda )f)(x).}

f(x, λ) may be regarded as a differentiable map into the space of functions of bounded variation ρ; or equivalently as a differentiable map

x
        ↦
        (
        
          d
          
            λ
          
        
        f
        )
        (
        x
        )
      
    
    {\displaystyle x\mapsto (d_{\lambda }f)(x)}

into the Banach space E of bounded linear functionals dρ on C[α,β] whenever [α, β] is a compact subinterval of [1, ∞).
Weyl's fundamental observation was that dλ f satisfies a second order ordinary differential equation taking values in E:

D
        (
        
          d
          
            λ
          
        
        f
        )
        =
        λ
        ⋅
        
          d
          
            λ
          
        
        f
        .
      
    
    {\displaystyle D(d_{\lambda }f)=\lambda \cdot d_{\lambda }f.}

After imposing initial conditions on the first two derivatives at a fixed point c, this equation can be solved explicitly in terms of the two fundamental eigenfunctions and the "initial value" functionals

(
        
          d
          
            λ
          
        
        f
        )
        (
        c
        )
        =
        
          d
          
            λ
          
        
        f
        (
        c
        ,
        ⋅
        )
        ,
        
        (
        
          d
          
            λ
          
        
        f
        
          )
          
            ′
          
        
        (
        c
        )
        =
        
          d
          
            λ
          
        
        
          f
          
            x
          
        
        (
        c
        ,
        ⋅
        )
        .
      
    
    {\displaystyle (d_{\lambda }f)(c)=d_{\lambda }f(c,\cdot ),\quad (d_{\lambda }f)^{\prime }(c)=d_{\lambda }f_{x}(c,\cdot ).}

This point of view may now be turned on its head: f(c, λ) and fx(c, λ) may be written as

f
        (
        c
        ,
        λ
        )
        =
        (
        f
        ,
        
          ξ
          
            1
          
        
        (
        λ
        )
        )
        ,
        
        
          f
          
            x
          
        
        (
        c
        ,
        λ
        )
        =
        (
        f
        ,
        
          ξ
          
            2
          
        
        (
        λ
        )
        )
        ,
      
    
    {\displaystyle f(c,\lambda )=(f,\xi _{1}(\lambda )),\quad f_{x}(c,\lambda )=(f,\xi _{2}(\lambda )),}

where ξ1(λ) and ξ2(λ) are given purely in terms of the fundamental eigenfunctions.
The functions of bounded variation

σ
          
            i
            j
          
        
        (
        λ
        )
        =
        (
        
          ξ
          
            i
          
        
        (
        λ
        )
        ,
        
          ξ
          
            j
          
        
        (
        λ
        )
        )
      
    
    {\displaystyle \sigma _{ij}(\lambda )=(\xi _{i}(\lambda ),\xi _{j}(\lambda ))}

determine a spectral measure on the spectrum of D and can be computed explicitly from the behaviour of the fundamental eigenfunctions (the Titchmarsh–Kodaira formula).

<h3>Limit circle and limit point for singular equations</h3>
Let q(x) be a continuous real-valued function on (0, ∞) and let D be the second order differential operator

D
        f
        =
        −
        
          f
          ″
        
        +
        q
        f
      
    
    {\displaystyle Df=-f''+qf}

on (0, ∞). Fix a point c in (0, ∞) and, for complex λ, let 
 
 
 
 
 φ
 
 λ
 
 
 ,
 
 θ
 
 λ
 
 
 
 
 {\displaystyle \varphi _{\lambda },\theta _{\lambda }}
 
 be the unique fundamental eigenfunctions of D on (0, ∞) satisfying

(
        D
        −
        λ
        )
        
          φ
          
            λ
          
        
        =
        0
        ,
        
        (
        D
        −
        λ
        )
        
          θ
          
            λ
          
        
        =
        0
      
    
    {\displaystyle (D-\lambda )\varphi _{\lambda }=0,\quad (D-\lambda )\theta _{\lambda }=0}

together with the initial conditions at c

φ
          
            λ
          
        
        (
        c
        )
        =
        1
        ,
        
        
          φ
          
            λ
          
          ′
        
        (
        c
        )
        =
        0
        ,
        
        
          θ
          
            λ
          
        
        (
        c
        )
        =
        0
        ,
        
        
          θ
          
            λ
          
          ′
        
        (
        c
        )
        =
        1.
      
    
    {\displaystyle \varphi _{\lambda }(c)=1,\,\varphi _{\lambda }'(c)=0,\,\theta _{\lambda }(c)=0,\,\theta _{\lambda }'(c)=1.}

Then their Wronskian satisfies

W
        (
        
          φ
          
            λ
          
        
        ,
        
          θ
          
            λ
          
        
        )
        =
        
          φ
          
            λ
          
        
        
          θ
          
            λ
          
          ′
        
        −
        
          θ
          
            λ
          
        
        
          φ
          
            λ
          
          ′
        
        ≡
        1
        ,
      
    
    {\displaystyle W(\varphi _{\lambda },\theta _{\lambda })=\varphi _{\lambda }\theta _{\lambda }'-\theta _{\lambda }\varphi _{\lambda }'\equiv 1,}

since it is constant and equal to 1 at c.
Let λ be non-real and 0 < x < ∞. If the complex number 
 
 
 
 μ
 
 
 {\displaystyle \mu }
 
 is such that 
 
 
 
 f
 =
 φ
 +
 μ
 θ
 
 
 {\displaystyle f=\varphi +\mu \theta }
 
 satisfies the boundary condition 
 
 
 
 cos
 ⁡
 β
 
 f
 (
 x
 )
 −
 sin
 ⁡
 β
 
 
 f
 ′
 
 (
 x
 )
 =
 0
 
 
 {\displaystyle \cos \beta \,f(x)-\sin \beta \,f'(x)=0}
 
 for some 
 
 
 
 β
 
 
 {\displaystyle \beta }
 
 (or, equivalently, 
 
 
 
 
 f
 ′
 
 (
 x
 )
 
 /
 
 f
 (
 x
 )
 
 
 {\displaystyle f'(x)/f(x)}
 
 is real) then, using integration by parts, one obtains

Im
        ⁡
        (
        λ
        )
        
          ∫
          
            c
          
          
            x
          
        
        
          |
        
        φ
        +
        μ
        θ
        
          
            |
          
          
            2
          
        
        =
        Im
        ⁡
        (
        μ
        )
        .
      
    
    {\displaystyle \operatorname {Im} (\lambda )\int _{c}^{x}|\varphi +\mu \theta |^{2}=\operatorname {Im} (\mu ).}

Therefore, the set of μ satisfying this equation is not empty. This set is a <a href="/facts/Circle/9fy5ggKn">circle</a> in the complex μ-plane. Points μ in its interior are characterized by

∫
 
 c
 
 
 x
 
 
 
 |
 
 φ
 +
 μ
 θ
 
 
 |
 
 
 2
 
 
 <
 
 
 
 Im
 ⁡
 (
 μ
 )
 
 
 Im
 ⁡
 (
 λ
 )
 
 
 
 
 
 {\displaystyle \int _{c}^{x}|\varphi +\mu \theta |^{2}<{\operatorname {Im} (\mu ) \over \operatorname {Im} (\lambda )}}

if x > c and by

∫
 
 x
 
 
 c
 
 
 
 |
 
 φ
 +
 μ
 θ
 
 
 |
 
 
 2
 
 
 <
 
 
 
 Im
 ⁡
 (
 μ
 )
 
 
 Im
 ⁡
 (
 λ
 )
 
 
 
 
 
 {\displaystyle \int _{x}^{c}|\varphi +\mu \theta |^{2}<{\operatorname {Im} (\mu ) \over \operatorname {Im} (\lambda )}}

if x < c.
Let Dx be the closed disc enclosed by the circle. By definition these closed discs are nested and decrease as x approaches 0 or ∞. So in the limit, the circles tend either to a limit circle or a limit point at each end. If 
 
 
 
 μ
 
 
 {\displaystyle \mu }
 
 is a limit point or a point on the limit circle at 0 or ∞, then 
 
 
 
 f
 =
 φ
 +
 μ
 θ
 
 
 {\displaystyle f=\varphi +\mu \theta }
 
 is <a href="/facts/Square_integrable/IASblcJJ">square integrable</a> (L2) near 0 or ∞, since 
 
 
 
 μ
 
 
 {\displaystyle \mu }
 
 lies in Dx for all x > c (in the ∞ case) and so 
 
 
 
 
 ∫
 
 c
 
 
 x
 
 
 
 |
 
 φ
 +
 μ
 θ
 
 
 |
 
 
 2
 
 
 <
 
 
 
 Im
 ⁡
 (
 μ
 )
 
 
 Im
 ⁡
 (
 λ
 )
 
 
 
 
 
 {\displaystyle \int _{c}^{x}|\varphi +\mu \theta |^{2}<{\operatorname {Im} (\mu ) \over \operatorname {Im} (\lambda )}}
 
 is bounded independent of x. In particular:<a class="footnote-ref" id="fnref:12" href="#fn:12">12</a>

<ul><li>there are always non-zero solutions of Df = λf which are square integrable near 0 resp. ∞;</li>
<li>in the limit circle case all solutions of Df = λf are square integrable near 0 resp. ∞.</li></ul>
The radius of the disc Dx can be calculated to be

|
          
            
              1
              
                2
                Im
                ⁡
                (
                λ
                )
                
                  ∫
                  
                    c
                  
                  
                    x
                  
                
                
                  |
                
                θ
                
                  
                    |
                  
                  
                    2
                  
                
              
            
          
          |
        
      
    
    {\displaystyle \left|{1 \over {2\operatorname {Im} (\lambda )\int _{c}^{x}|\theta |^{2}}}\right|}

and this implies that in the limit point case 
 
 
 
 θ
 
 
 {\displaystyle \theta }
 
 cannot be square integrable near 0 resp. ∞. Therefore, we have a converse to the second statement above:

<ul><li>in the limit point case there is exactly one non-zero solution (up to scalar multiples) of Df = λf which is square integrable near 0 resp. ∞.</li></ul>
On the other hand, if Dg = λ′ g for another value λ′, then

h
        (
        x
        )
        =
        g
        (
        x
        )
        −
        (
        
          λ
          
            ′
          
        
        −
        λ
        )
        
          ∫
          
            c
          
          
            x
          
        
        (
        
          φ
          
            λ
          
        
        (
        x
        )
        
          θ
          
            λ
          
        
        (
        y
        )
        −
        
          θ
          
            λ
          
        
        (
        x
        )
        
          φ
          
            λ
          
        
        (
        y
        )
        )
        g
        (
        y
        )
        
        d
        y
      
    
    {\displaystyle h(x)=g(x)-(\lambda ^{\prime }-\lambda )\int _{c}^{x}(\varphi _{\lambda }(x)\theta _{\lambda }(y)-\theta _{\lambda }(x)\varphi _{\lambda }(y))g(y)\,dy}

satisfies Dh = λh, so that

g
        (
        x
        )
        =
        
          c
          
            1
          
        
        
          φ
          
            λ
          
        
        +
        
          c
          
            2
          
        
        
          θ
          
            λ
          
        
        +
        (
        
          λ
          
            ′
          
        
        −
        λ
        )
        
          ∫
          
            c
          
          
            x
          
        
        (
        
          φ
          
            λ
          
        
        (
        x
        )
        
          θ
          
            λ
          
        
        (
        y
        )
        −
        
          θ
          
            λ
          
        
        (
        x
        )
        
          φ
          
            λ
          
        
        (
        y
        )
        )
        g
        (
        y
        )
        
        d
        y
        .
      
    
    {\displaystyle g(x)=c_{1}\varphi _{\lambda }+c_{2}\theta _{\lambda }+(\lambda ^{\prime }-\lambda )\int _{c}^{x}(\varphi _{\lambda }(x)\theta _{\lambda }(y)-\theta _{\lambda }(x)\varphi _{\lambda }(y))g(y)\,dy.}

This formula may also be obtained directly by the variation of constant method from (D − λ)g = (λ′ − λ)g.
Using this to estimate g, it follows that<a class="footnote-ref" id="fnref:13" href="#fn:13">13</a>

<ul><li>the limit point/limit circle behaviour at 0 or ∞ is independent of the choice of λ.</li></ul>
More generally if Dg = (λ – r) g for some function r(x), then<a class="footnote-ref" id="fnref:14" href="#fn:14">14</a>

g
        (
        x
        )
        =
        
          c
          
            1
          
        
        
          φ
          
            λ
          
        
        +
        
          c
          
            2
          
        
        
          θ
          
            λ
          
        
        −
        
          ∫
          
            c
          
          
            x
          
        
        (
        
          φ
          
            λ
          
        
        (
        x
        )
        
          θ
          
            λ
          
        
        (
        y
        )
        −
        
          θ
          
            λ
          
        
        (
        x
        )
        
          φ
          
            λ
          
        
        (
        y
        )
        )
        r
        (
        y
        )
        g
        (
        y
        )
        
        d
        y
        .
      
    
    {\displaystyle g(x)=c_{1}\varphi _{\lambda }+c_{2}\theta _{\lambda }-\int _{c}^{x}(\varphi _{\lambda }(x)\theta _{\lambda }(y)-\theta _{\lambda }(x)\varphi _{\lambda }(y))r(y)g(y)\,dy.}

From this it follows that<a class="footnote-ref" id="fnref:15" href="#fn:15">15</a>

<ul><li>if r is continuous at 0, then D + r is limit point or limit circle at 0 precisely when D is,</li></ul>
so that in particular<a class="footnote-ref" id="fnref:16" href="#fn:16">16</a>

<ul><li>if q(x) − a/x2 is continuous at 0, then D is limit point at 0 if and only if a ≥ ⁠3/4⁠.</li></ul>
Similarly

<ul><li>if r has a finite limit at ∞, then D + r is limit point or limit circle at ∞ precisely when D is,</li></ul>
so that in particular<a class="footnote-ref" id="fnref:17" href="#fn:17">17</a>

<ul><li>if q has a finite limit at ∞, then D is limit point at ∞.</li></ul>
Many more elaborate criteria to be limit point or limit circle can be found in the mathematical literature.

<h3>Green's function (singular case)</h3>
Consider the differential operator

D
          
            0
          
        
        f
        =
        −
        (
        
          p
          
            0
          
        
        
          f
          ′
        
        
          )
          ′
        
        +
        
          q
          
            0
          
        
        f
      
    
    {\displaystyle D_{0}f=-(p_{0}f')'+q_{0}f}

on (0, ∞) with q0 positive and continuous on (0, ∞) and p0 continuously differentiable in [0, ∞), positive in (0, ∞) and p0(0) = 0.
Moreover, assume that after reduction to standard form D0 becomes the equivalent operator

D
        f
        =
        −
        
          f
          ″
        
        +
        q
        f
      
    
    {\displaystyle Df=-f''+qf}

on (0, ∞) where q has a finite limit at ∞. Thus

<ul><li>D is limit point at ∞.</li></ul>
At 0, D may be either limit circle or limit point. In either case there is an eigenfunction Φ0 with DΦ0 = 0 and Φ0 square integrable near 0. In the limit circle case, Φ0 determines a <a href="/facts/Boundary_condition/QEfmUqmP">boundary condition</a> at 0:

W
        (
        f
        ,
        
          Φ
          
            0
          
        
        )
        (
        0
        )
        =
        0.
      
    
    {\displaystyle W(f,\Phi _{0})(0)=0.}

For complex λ, let Φλ and Χλ satisfy

<ul><li>(D – λ)Φλ = 0, (D – λ)Χλ = 0</li>
<li>Χλ square integrable near infinity</li>
<li>Φλ square integrable at 0 if 0 is limit point</li>
<li>Φλ satisfies the boundary condition above if 0 is limit circle.</li></ul>
Let

ω
        (
        λ
        )
        =
        W
        (
        
          Φ
          
            λ
          
        
        ,
        
          
            X
          
          
            λ
          
        
        )
        ,
      
    
    {\displaystyle \omega (\lambda )=W(\Phi _{\lambda },\mathrm {X} _{\lambda }),}

a constant which vanishes precisely when Φλ and Χλ are proportional, i.e. λ is an <a href="/facts/Eigenvalue/8TjEoT8u">eigenvalue</a> of D for these boundary conditions.
On the other hand, this cannot occur if Im λ ≠ 0 or if λ is negative.<a class="footnote-ref" id="fnref:18" href="#fn:18">18</a>
Indeed, if D f = λf with q0 – λ ≥ δ > 0, then by Green's formula (Df,f) = (f,Df), since W(f,f*) is constant. So λ must be real. If f is taken to be real-valued in the D0 realization, then for 0 < x < y

[
        
          p
          
            0
          
        
        f
        
          f
          ′
        
        
          ]
          
            x
          
          
            y
          
        
        =
        
          ∫
          
            x
          
          
            y
          
        
        (
        
          q
          
            0
          
        
        −
        λ
        )
        
          |
        
        f
        
          
            |
          
          
            2
          
        
        +
        
          p
          
            0
          
        
        (
        
          f
          ′
        
        
          )
          
            2
          
        
        .
      
    
    {\displaystyle [p_{0}ff']_{x}^{y}=\int _{x}^{y}(q_{0}-\lambda )|f|^{2}+p_{0}(f')^{2}.}

Since p0(0) = 0 and f is integrable near 0, p0f f′ must vanish at 0. Setting x = 0, it follows that f(y) f′(y) > 0, so that f2 is increasing, contradicting the square integrability of f near ∞.
Thus, adding a positive scalar to q, it may be assumed that

ω
        (
        λ
        )
        ≠
        0
         
         
        
           if 
        
        λ
        ∉
        [
        1
        ,
        ∞
        )
        .
      
    
    {\displaystyle \omega (\lambda )\neq 0~~{\text{ if }}\lambda \notin [1,\infty ).}

If ω(λ) ≠ 0, the <a href="/facts/Green%2527s_function/rKxKKPcp">Green's function</a> Gλ(x,y) at λ is defined by

G
          
            λ
          
        
        (
        x
        ,
        y
        )
        =
        
          
            {
            
              
                
                  
                    Φ
                    
                      λ
                    
                  
                  (
                  x
                  )
                  
                    
                      X
                    
                    
                      λ
                    
                  
                  (
                  y
                  )
                  
                    /
                  
                  ω
                  (
                  λ
                  )
                
                
                  (
                  x
                  ≤
                  y
                  )
                  ,
                
              
              
                
                  
                    
                      X
                    
                    
                      λ
                    
                  
                  (
                  x
                  )
                  
                    Φ
                    
                      λ
                    
                  
                  (
                  y
                  )
                  
                    /
                  
                  ω
                  (
                  λ
                  )
                
                
                  (
                  x
                  ≥
                  y
                  )
                  .
                
              
            
            
          
        
      
    
    {\displaystyle G_{\lambda }(x,y)={\begin{cases}\Phi _{\lambda }(x)\mathrm {X} _{\lambda }(y)/\omega (\lambda )&(x\leq y),\\[1ex]\mathrm {X} _{\lambda }(x)\Phi _{\lambda }(y)/\omega (\lambda )&(x\geq y).\end{cases}}}

and is independent of the choice of Φλ and Χλ.
In the examples there will be a third "bad" eigenfunction Ψλ defined and holomorphic for λ not in [1, ∞) such that Ψλ satisfies the boundary conditions at neither 0 nor ∞. This means that for λ not in [1, ∞)

<ul><li>W(Φλ,Ψλ) is nowhere vanishing;</li>
<li>W(Χλ,Ψλ) is nowhere vanishing.</li></ul>
In this case Χλ is proportional to Φλ + m(λ) Ψλ, where

m
        (
        λ
        )
        =
        −
        W
        (
        
          Φ
          
            λ
          
        
        ,
        
          
            X
          
          
            λ
          
        
        )
        
          /
        
        W
        (
        
          Ψ
          
            λ
          
        
        ,
        
          
            X
          
          
            λ
          
        
        )
        .
      
    
    {\displaystyle m(\lambda )=-W(\Phi _{\lambda },\mathrm {X} _{\lambda })/W(\Psi _{\lambda },\mathrm {X} _{\lambda }).}

Let H1 be the space of square integrable continuous functions on (0, ∞) and let H0 be

<ul><li>the space of C2 functions f on (0, ∞) of <a href="/facts/Compact_support/BTuMGbsb">compact support</a> if D is limit point at 0</li>
<li>the space of C2 functions f on (0, ∞) with W(f, Φ0) = 0 at 0 and with f = 0 near ∞ if D is limit circle at 0.</li></ul>
Define T = G0 by

(
        T
        f
        )
        (
        x
        )
        =
        
          ∫
          
            0
          
          
            ∞
          
        
        
          G
          
            0
          
        
        (
        x
        ,
        y
        )
        f
        (
        y
        )
        
        d
        y
        .
      
    
    {\displaystyle (Tf)(x)=\int _{0}^{\infty }G_{0}(x,y)f(y)\,dy.}

Then T D = I on H0, D T = I on H1 and the operator D is bounded below on H0:

(
        D
        f
        ,
        f
        )
        ≥
        (
        f
        ,
        f
        )
        .
      
    
    {\displaystyle (Df,f)\geq (f,f).}

Thus T is a self-adjoint bounded operator with 0 ≤ T ≤ I.
Formally T = D−1. The corresponding operators Gλ defined for λ not in [1, ∞) can be formally identified with

(
        D
        −
        λ
        
          )
          
            −
            1
          
        
        =
        T
        (
        I
        −
        λ
        T
        
          )
          
            −
            1
          
        
      
    
    {\displaystyle (D-\lambda )^{-1}=T(I-\lambda T)^{-1}}

and satisfy Gλ (D – λ) = I on H0, (D – λ)Gλ = I on H1.

<h3>Spectral theorem and Titchmarsh–Kodaira formula</h3>

Theorem.<a class="footnote-ref" id="fnref:19" href="#fn:19">19</a><a class="footnote-ref" id="fnref:20" href="#fn:20">20</a><a class="footnote-ref" id="fnref:21" href="#fn:21">21</a>—For every real number λ let ρ(λ) be defined by the Titchmarsh–Kodaira formula:

ρ
        (
        λ
        )
        =
        
          lim
          
            δ
            ↓
            0
          
        
        
          lim
          
            ε
            ↓
            0
          
        
        
          
            1
            π
          
        
        
          ∫
          
            δ
          
          
            λ
            +
            δ
          
        
        Im
        ⁡
        m
        (
        t
        +
        i
        ε
        )
        
        d
        t
        .
      
    
    {\displaystyle \rho (\lambda )=\lim _{\delta \downarrow 0}\lim _{\varepsilon \downarrow 0}{\frac {1}{\pi }}\int _{\delta }^{\lambda +\delta }\operatorname {Im} m(t+i\varepsilon )\,dt.}

Then ρ(λ) is a lower semicontinuous non-decreasing function of λ and if

(
        U
        f
        )
        (
        λ
        )
        =
        
          ∫
          
            0
          
          
            ∞
          
        
        f
        (
        x
        )
        Φ
        (
        x
        ,
        λ
        )
        
        d
        x
        ,
      
    
    {\displaystyle (Uf)(\lambda )=\int _{0}^{\infty }f(x)\Phi (x,\lambda )\,dx,}

then U defines a unitary transformation of L2(0, ∞) onto L2([1,∞), dρ) such that UDU−1 corresponds to multiplication by λ. 
The inverse transformation U−1 is given by

(
        
          U
          
            −
            1
          
        
        g
        )
        (
        x
        )
        =
        
          ∫
          
            1
          
          
            ∞
          
        
        g
        (
        λ
        )
        Φ
        (
        x
        ,
        λ
        )
        
        d
        ρ
        (
        λ
        )
        .
      
    
    {\displaystyle (U^{-1}g)(x)=\int _{1}^{\infty }g(\lambda )\Phi (x,\lambda )\,d\rho (\lambda ).}

The spectrum of D equals the support of dρ.

Kodaira gave a streamlined version<a class="footnote-ref" id="fnref:22" href="#fn:22">22</a><a class="footnote-ref" id="fnref:23" href="#fn:23">23</a> of Weyl's original proof.<a class="footnote-ref" id="fnref:24" href="#fn:24">24</a> (<a href="/facts/Marshall_Harvey_Stone/TFzbE4GH">M.H. Stone</a> had previously shown<a class="footnote-ref" id="fnref:25" href="#fn:25">25</a> how part of Weyl's work could be simplified using von Neumann's spectral theorem.)
In fact for T =D−1 with 0 ≤ T ≤ I, the spectral projection E(λ) of T is defined by

It is also the spectral projection of D corresponding to the interval [1, λ].
For f in H1 define

f(x, λ) may be regarded as a differentiable map into the space of functions ρ of bounded variation; or equivalently as a differentiable map

into the Banach space E of bounded linear functionals dρ on [C[α, β]] for any compact subinterval [α, β] of [1, ∞).
The functionals (or measures) dλ f(x) satisfies the following E-valued second order ordinary differential equation:

D
        (
        
          d
          
            λ
          
        
        f
        )
        =
        λ
        ⋅
        
          d
          
            λ
          
        
        f
        ,
      
    
    {\displaystyle D(d_{\lambda }f)=\lambda \cdot d_{\lambda }f,}

with initial conditions at c in (0, ∞)

(
        
          d
          
            λ
          
        
        f
        )
        (
        c
        )
        =
        
          d
          
            λ
          
        
        f
        (
        c
        ,
        ⋅
        )
        =
        
          μ
          
            (
            0
            )
          
        
        ,
        
        (
        
          d
          
            λ
          
        
        f
        
          )
          
            ′
          
        
        (
        c
        )
        =
        
          d
          
            λ
          
        
        
          f
          
            x
          
        
        (
        c
        ,
        ⋅
        )
        =
        
          μ
          
            (
            1
            )
          
        
        .
      
    
    {\displaystyle (d_{\lambda }f)(c)=d_{\lambda }f(c,\cdot )=\mu ^{(0)},\quad (d_{\lambda }f)^{\prime }(c)=d_{\lambda }f_{x}(c,\cdot )=\mu ^{(1)}.}

If φλ and χλ are the special eigenfunctions adapted to c, then

d
          
            λ
          
        
        f
        (
        x
        )
        =
        
          φ
          
            λ
          
        
        (
        x
        )
        
          μ
          
            (
            0
            )
          
        
        +
        
          χ
          
            λ
          
        
        (
        x
        )
        
          μ
          
            (
            1
            )
          
        
        .
      
    
    {\displaystyle d_{\lambda }f(x)=\varphi _{\lambda }(x)\mu ^{(0)}+\chi _{\lambda }(x)\mu ^{(1)}.}

Moreover,

μ
          
            (
            k
            )
          
        
        =
        
          d
          
            λ
          
        
        (
        f
        ,
        
          ξ
          
            λ
          
          
            (
            k
            )
          
        
        )
        ,
      
    
    {\displaystyle \mu ^{(k)}=d_{\lambda }(f,\xi _{\lambda }^{(k)}),}

where

ξ
          
            λ
          
          
            (
            k
            )
          
        
        =
        D
        E
        (
        λ
        )
        
          η
          
            (
            k
            )
          
        
        ,
      
    
    {\displaystyle \xi _{\lambda }^{(k)}=DE(\lambda )\eta ^{(k)},}

with

η
          
            z
          
          
            (
            0
            )
          
        
        (
        y
        )
        =
        
          G
          
            z
          
        
        (
        c
        ,
        y
        )
        ,
        
        
        
        
        
          η
          
            z
          
          
            (
            1
            )
          
        
        (
        x
        )
        =
        
          ∂
          
            x
          
        
        
          G
          
            z
          
        
        (
        c
        ,
        y
        )
        ,
        
        
        
        
        (
        z
        ∉
        [
        1
        ,
        ∞
        )
        )
        .
      
    
    {\displaystyle \eta _{z}^{(0)}(y)=G_{z}(c,y),\,\,\,\,\eta _{z}^{(1)}(x)=\partial _{x}G_{z}(c,y),\,\,\,\,(z\notin [1,\infty )).}

(As the notation suggests, ξλ(0) and ξλ(1) do not depend on the choice of z.)
Setting

σ
          
            i
            j
          
        
        (
        λ
        )
        =
        (
        
          ξ
          
            λ
          
          
            (
            i
            )
          
        
        ,
        
          ξ
          
            λ
          
          
            (
            j
            )
          
        
        )
        ,
      
    
    {\displaystyle \sigma _{ij}(\lambda )=(\xi _{\lambda }^{(i)},\xi _{\lambda }^{(j)}),}

it follows that

d
          
            λ
          
        
        (
        E
        (
        λ
        )
        
          η
          
            z
          
          
            (
            i
            )
          
        
        ,
        
          η
          
            z
          
          
            (
            j
            )
          
        
        )
        =
        
          |
        
        λ
        −
        z
        
          
            |
          
          
            −
            2
          
        
        ⋅
        
          d
          
            λ
          
        
        
          σ
          
            i
            j
          
        
        (
        λ
        )
        .
      
    
    {\displaystyle d_{\lambda }(E(\lambda )\eta _{z}^{(i)},\eta _{z}^{(j)})=|\lambda -z|^{-2}\cdot d_{\lambda }\sigma _{ij}(\lambda ).}

On the other hand, there are holomorphic functions a(λ), b(λ) such that

<ul><li>φλ + a(λ) χλ is proportional to Φλ;</li>
<li>φλ + b(λ) χλ is proportional to Χλ.</li></ul>
Since W(φλ, χλ) = 1, the Green's function is given by

G
          
            λ
          
        
        (
        x
        ,
        y
        )
        =
        
          
            {
            
              
                
                  
                    
                      
                        
                          (
                          
                            φ
                            
                              λ
                            
                          
                          (
                          x
                          )
                          +
                          a
                          (
                          λ
                          )
                          
                            χ
                            
                              λ
                            
                          
                          (
                          x
                          )
                          )
                          (
                          
                            φ
                            
                              λ
                            
                          
                          (
                          y
                          )
                          +
                          b
                          (
                          λ
                          )
                          
                            χ
                            
                              λ
                            
                          
                          (
                          y
                          )
                          )
                        
                        
                          b
                          (
                          λ
                          )
                          −
                          a
                          (
                          λ
                          )
                        
                      
                    
                  
                
                
                  (
                  x
                  ≤
                  y
                  )
                  ,
                
              
              
                
                  
                    
                      
                        
                          (
                          
                            φ
                            
                              λ
                            
                          
                          (
                          x
                          )
                          +
                          b
                          (
                          λ
                          )
                          
                            χ
                            
                              λ
                            
                          
                          (
                          x
                          )
                          )
                          (
                          
                            φ
                            
                              λ
                            
                          
                          (
                          y
                          )
                          +
                          a
                          (
                          λ
                          )
                          
                            χ
                            
                              λ
                            
                          
                          (
                          y
                          )
                          )
                        
                        
                          b
                          (
                          λ
                          )
                          −
                          a
                          (
                          λ
                          )
                        
                      
                    
                  
                
                
                  (
                  y
                  ≤
                  x
                  )
                  .
                
              
            
            
          
        
      
    
    {\displaystyle G_{\lambda }(x,y)={\begin{cases}{\dfrac {(\varphi _{\lambda }(x)+a(\lambda )\chi _{\lambda }(x))(\varphi _{\lambda }(y)+b(\lambda )\chi _{\lambda }(y))}{b(\lambda )-a(\lambda )}}&(x\leq y),\\[1ex]{\dfrac {(\varphi _{\lambda }(x)+b(\lambda )\chi _{\lambda }(x))(\varphi _{\lambda }(y)+a(\lambda )\chi _{\lambda }(y))}{b(\lambda )-a(\lambda )}}&(y\leq x).\end{cases}}}

Direct calculation<a class="footnote-ref" id="fnref:26" href="#fn:26">26</a> shows that

(
        
          η
          
            z
          
          
            (
            i
            )
          
        
        ,
        
          η
          
            z
          
          
            (
            j
            )
          
        
        )
        =
        Im
        ⁡
        
          M
          
            i
            j
          
        
        (
        z
        )
        
          /
        
        Im
        ⁡
        z
        ,
      
    
    {\displaystyle (\eta _{z}^{(i)},\eta _{z}^{(j)})=\operatorname {Im} M_{ij}(z)/\operatorname {Im} z,}

where the so-called characteristic matrix Mij(z) is given by

M
          
            00
          
        
        (
        z
        )
        =
        
          
            
              a
              (
              z
              )
              b
              (
              z
              )
            
            
              a
              (
              z
              )
              −
              b
              (
              z
              )
            
          
        
        ,
        
        
        
          M
          
            01
          
        
        (
        z
        )
        =
        
          M
          
            10
          
        
        (
        z
        )
        =
        
          
            
              a
              (
              z
              )
              +
              b
              (
              z
              )
            
            
              2
              (
              a
              (
              z
              )
              −
              b
              (
              z
              )
              )
            
          
        
        ,
        
        
        
          M
          
            11
          
        
        (
        z
        )
        =
        
          
            1
            
              a
              (
              z
              )
              −
              b
              (
              z
              )
            
          
        
        .
      
    
    {\displaystyle M_{00}(z)={\frac {a(z)b(z)}{a(z)-b(z)}},\,\,M_{01}(z)=M_{10}(z)={\frac {a(z)+b(z)}{2(a(z)-b(z))}},\,\,M_{11}(z)={\frac {1}{a(z)-b(z)}}.}

Hence

∫
          
            −
            ∞
          
          
            ∞
          
        
        (
        Im
        ⁡
        z
        )
        ⋅
        
          |
        
        λ
        −
        z
        
          
            |
          
          
            −
            2
          
        
        
        d
        
          σ
          
            i
            j
          
        
        (
        λ
        )
        =
        Im
        ⁡
        
          M
          
            i
            j
          
        
        (
        z
        )
        ,
      
    
    {\displaystyle \int _{-\infty }^{\infty }(\operatorname {Im} z)\cdot |\lambda -z|^{-2}\,d\sigma _{ij}(\lambda )=\operatorname {Im} M_{ij}(z),}

which immediately implies

σ
          
            i
            j
          
        
        (
        λ
        )
        =
        
          lim
          
            δ
            ↓
            0
          
        
        
          lim
          
            ε
            ↓
            0
          
        
        
          ∫
          
            δ
          
          
            λ
            +
            δ
          
        
        Im
        ⁡
        
          M
          
            i
            j
          
        
        (
        t
        +
        i
        ε
        )
        
        d
        t
        .
      
    
    {\displaystyle \sigma _{ij}(\lambda )=\lim _{\delta \downarrow 0}\lim _{\varepsilon \downarrow 0}\int _{\delta }^{\lambda +\delta }\operatorname {Im} M_{ij}(t+i\varepsilon )\,dt.}

(This is a special case of the <a href="/facts/Stieltjes_transformation/afjPeS41">"Stieltjes inversion formula"</a>.)
Setting ψλ(0) = φλ and ψλ(1) = χλ, it follows that

(
        E
        (
        μ
        )
        f
        )
        (
        x
        )
        =
        
          ∑
          
            i
            .
            j
          
        
        
          ∫
          
            0
          
          
            μ
          
        
        
          ∫
          
            0
          
          
            ∞
          
        
        
          ψ
          
            λ
          
          
            (
            i
            )
          
        
        (
        x
        )
        
          ψ
          
            λ
          
          
            (
            j
            )
          
        
        (
        y
        )
        f
        (
        y
        )
        
        d
        y
        
        d
        
          σ
          
            i
            j
          
        
        (
        λ
        )
        =
        
          ∫
          
            0
          
          
            μ
          
        
        
          ∫
          
            0
          
          
            ∞
          
        
        
          Φ
          
            λ
          
        
        (
        x
        )
        
          Φ
          
            λ
          
        
        (
        y
        )
        f
        (
        y
        )
        
        d
        y
        
        d
        ρ
        (
        λ
        )
        .
      
    
    {\displaystyle (E(\mu )f)(x)=\sum _{i.j}\int _{0}^{\mu }\int _{0}^{\infty }\psi _{\lambda }^{(i)}(x)\psi _{\lambda }^{(j)}(y)f(y)\,dy\,d\sigma _{ij}(\lambda )=\int _{0}^{\mu }\int _{0}^{\infty }\Phi _{\lambda }(x)\Phi _{\lambda }(y)f(y)\,dy\,d\rho (\lambda ).}

This identity is equivalent to the spectral theorem and Titchmarsh–Kodaira formula.

<h2 id="application-to-the-hypergeometric-equation">Application to the hypergeometric equation</h2>
See also: <a href="/facts/Legendre_equation/4C60rXPc">Legendre equation</a> and <a href="/facts/Hypergeometric_equation/yTEC6NBJ">Hypergeometric equation</a>
The Mehler–Fock transform<a class="footnote-ref" id="fnref:27" href="#fn:27">27</a><a class="footnote-ref" id="fnref:28" href="#fn:28">28</a><a class="footnote-ref" id="fnref:29" href="#fn:29">29</a> concerns the eigenfunction expansion associated with the <a href="/facts/Legendre_function/sDo3ra2G">Legendre differential operator</a> D

D
        f
        =
        −
        (
        (
        
          x
          
            2
          
        
        −
        1
        )
        
          f
          ′
        
        
          )
          ′
        
        =
        −
        (
        
          x
          
            2
          
        
        −
        1
        )
        
          f
          ″
        
        −
        2
        x
        
          f
          ′
        
      
    
    {\displaystyle Df=-((x^{2}-1)f')'=-(x^{2}-1)f''-2xf'}

on (1, ∞). The eigenfunctions are the <a href="/facts/Legendre_function/sDo3ra2G">Legendre functions</a><a class="footnote-ref" id="fnref:30" href="#fn:30">30</a>

P
          
            −
            1
            
              /
            
            2
            +
            i
            
              
                λ
              
            
          
        
        (
        cosh
        ⁡
        r
        )
        =
        
          
            1
            
              2
              π
            
          
        
        
          ∫
          
            0
          
          
            2
            π
          
        
        
          
            (
            
              
                
                  sin
                  ⁡
                  θ
                  +
                  i
                  
                    e
                    
                      −
                      r
                    
                  
                  cos
                  ⁡
                  θ
                
                
                  cos
                  ⁡
                  θ
                  −
                  i
                  
                    e
                    
                      −
                      r
                    
                  
                  sin
                  ⁡
                  θ
                
              
            
            )
          
          
            
              
                1
                2
              
            
            +
            i
            
              
                λ
              
            
          
        
        
        d
        θ
      
    
    {\displaystyle P_{-1/2+i{\sqrt {\lambda }}}(\cosh r)={1 \over 2\pi }\int _{0}^{2\pi }\left({\sin \theta +ie^{-r}\cos \theta  \over \cos \theta -ie^{-r}\sin \theta }\right)^{{1 \over 2}+i{\sqrt {\lambda }}}\,d\theta }

with eigenvalue λ ≥ 0. The two Mehler–Fock transformations are<a class="footnote-ref" id="fnref:31" href="#fn:31">31</a>

U
        f
        (
        λ
        )
        =
        
          ∫
          
            1
          
          
            ∞
          
        
        f
        (
        x
        )
        
        
          P
          
            −
            1
            
              /
            
            2
            +
            i
            
              
                λ
              
            
          
        
        (
        x
        )
        
        d
        x
      
    
    {\displaystyle Uf(\lambda )=\int _{1}^{\infty }f(x)\,P_{-1/2+i{\sqrt {\lambda }}}(x)\,dx}

and

U
          
            −
            1
          
        
        g
        (
        x
        )
        =
        
          ∫
          
            0
          
          
            ∞
          
        
        g
        (
        λ
        )
        
        
          
            1
            2
          
        
        tanh
        ⁡
        π
        
          
            λ
          
        
        
        d
        λ
        .
      
    
    {\displaystyle U^{-1}g(x)=\int _{0}^{\infty }g(\lambda )\,{1 \over 2}\tanh \pi {\sqrt {\lambda }}\,d\lambda .}

(Often this is written in terms of the variable τ = √λ.)
Mehler and Fock studied this differential operator because it arose as the radial component of the Laplacian on 2-dimensional hyperbolic space. 
More generally,<a class="footnote-ref" id="fnref:32" href="#fn:32">32</a> consider the group G = <a href="/facts/SL(2%2cR)/xjKM9JLJ">SU(1,1)</a> consisting of complex matrices of the form

[
            
              
                
                  α
                
                
                  β
                
              
              
                
                  
                    
                      β
                      ¯
                    
                  
                
                
                  
                    
                      α
                      ¯
                    
                  
                
              
            
            ]
          
        
      
    
    {\displaystyle {\begin{bmatrix}\alpha &\beta \\{\overline {\beta }}&{\overline {\alpha }}\end{bmatrix}}}

with determinant |α|2 − |β|2 = 1.

<h2 id="application-to-the-hydrogen-atom">Application to the hydrogen atom</h2>
See also: <a href="/facts/Hydrogen_atom/ncVpPbuE">Hydrogen atom</a>
<h2 id="generalisations-and-alternative-approaches">Generalisations and alternative approaches</h2>
A Weyl function can be defined at a singular endpoint a giving rise to a singular version of Weyl–Titchmarsh–Kodaira theory.<a class="footnote-ref" id="fnref:33" href="#fn:33">33</a> this applies for example to the case of radial Schrödinger operators

D
        f
        =
        −
        
          f
          ″
        
        +
        
          
            
              ℓ
              (
              ℓ
              +
              1
              )
            
            
              x
              
                2
              
            
          
        
        f
        +
        V
        (
        x
        )
        f
        ,
        
        x
        ∈
        (
        0
        ,
        ∞
        )
      
    
    {\displaystyle Df=-f''+{\frac {\ell (\ell +1)}{x^{2}}}f+V(x)f,\qquad x\in (0,\infty )}

The whole theory can also be extended to the case where the coefficients are allowed to be measures.<a class="footnote-ref" id="fnref:34" href="#fn:34">34</a>

<h2 id="gelfandlevitan-theory">Gelfand–Levitan theory</h2>
See also: <a href="/facts/Inverse_scattering_theory/CJ5dfzXJ">Inverse scattering theory</a>
<h2 id="notes">Notes</h2>

<h3>Citations</h3>

<h3>Bibliography</h3>

<ul><li><a href="/facts/Naum_Akhiezer/tFszwj15">Akhiezer, Naum Ilich</a>; Glazman, Izrael Markovich (1993), Theory of Linear Operators in Hilbert Space, Dover, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-486-67748-4</li>
<li>Bellman, Richard (1969), <a href="https://archive.org/details/stabilitytheoryo0000bell">Stability Theory of Differential Equations</a>, Dover, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-486-62210-1</li>
<li>Burkill, J.C. (1951), The Lebesgue Integral, Cambridge Tracts in Mathematics and Mathematical Physics, vol. 40, Cambridge University Press, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-521-04382-3 {{citation}}: ISBN / Date incompatibility (help)</li>
<li>Coddington, Earl A.; Levinson, Norman (1955), Theory of Ordinary Differential equations, McGraw-Hill, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-07-011542-2 {{citation}}: ISBN / Date incompatibility (help)</li>
<li><a href="/facts/Richard_Courant/QDToNfWl">Courant, Richard</a>; <a href="/facts/David_Hilbert/1vhlHn4b">Hilbert, David</a> (1989), Method of Mathematical Physics, Vol. I, Wiley-Interscience, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-471-50447-4</li>
<li><a href="/facts/Jean_Dieudonn%25C3%25A9/gWagzMdu">Dieudonné, Jean</a> (1969), Treatise on Analysis, Vol. I [Foundations of Modern Analysis], Academic Press, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-1-4067-2791-3</li>
<li><a href="/facts/Jean_Dieudonn%25C3%25A9/gWagzMdu">Dieudonné, Jean</a> (1988), Treatise on Analysis, Vol. VIII, Academic Press, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-12-215507-9</li>
<li><a href="/facts/Nelson_Dunford/Q72PJMF7">Dunford, Nelson</a>; <a href="/facts/Jacob_T._Schwartz/UAvyPwCr">Schwartz, Jacob T.</a> (1963), Linear Operators, Part II Spectral Theory. Self Adjoint Operators in Hilbert space, Wiley Interscience, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-471-60847-9 {{citation}}: ISBN / Date incompatibility (help)</li>
<li>Fock, V.A. (1943), "On the representation of an arbitrary function by an integral involving Legendre's functions with a complex index", C. R. Acad. Sci. URSS, 39</li>
<li><a href="/facts/Einar_Hille/56qNTKUA">Hille, Einar</a> (1969), Lectures on Ordinary Differential Equations, Addison-Wesley, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-201-53083-4</li>
<li><a href="/facts/Kunihiko_Kodaira/u73B5W0D">Kodaira, Kunihiko</a> (1949), "The eigenvalue problem for ordinary differential equations of the second order and Heisenberg's theory of S-matrices", <a href="/facts/American_Journal_of_Mathematics/lkgftcD7">American Journal of Mathematics</a>, 71 (4): 921–945, <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.2307%2F2372377">10.2307/2372377</a>, <a href="/facts/JSTOR_(identifier)/YTeVmaJ7">JSTOR</a> <a href="https://www.jstor.org/stable/2372377">2372377</a></li>
<li><a href="/facts/Kunihiko_Kodaira/u73B5W0D">Kodaira, Kunihiko</a> (1950), "On ordinary differential equations of any even order and the corresponding eigenfunction expansions", <a href="/facts/American_Journal_of_Mathematics/lkgftcD7">American Journal of Mathematics</a>, 72 (3): 502–544, <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.2307%2F2372051">10.2307/2372051</a>, <a href="/facts/JSTOR_(identifier)/YTeVmaJ7">JSTOR</a> <a href="https://www.jstor.org/stable/2372051">2372051</a></li>
<li>Kolmogorov, A.N.; Fomin, S.V. (1975). <a href="https://archive.org/details/introductoryreal00kolm_0/page/374">Introductory Real Analysis</a>. Dover. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-486-61226-3.</li>
<li>Kostenko, Aleksey; Sakhnovich, Alexander; <a href="/facts/Gerald_Teschl/tEBmvGQA">Teschl, Gerald</a> (2012), "Weyl–Titchmarsh Theory for Schrödinger Operators with Strongly Singular Potentials", Int Math Res Notices, 2012, <a href="/facts/ArXiv_(identifier)/H6EtgnBe">arXiv</a>:<a href="https://arxiv.org/abs/1007.0136">1007.0136</a>, <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1093%2Fimrn%2Frnr065">10.1093/imrn/rnr065</a></li>
<li>Lebedev, N.N. (1972), Special Functions and Their Applications, Dover, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-486-60624-8</li>
<li>Loomis, Lynn H. (1953), An Introduction to Abstract Harmonic Analysis, van Nostrand</li>
<li>Mehler, F.G. (1881), <a href="http://gdz.sub.uni-goettingen.de/index.php?id=11&PPN=PPN235181684_0018&DMDID=DMDLOG_0021&L=1">"Ueber mit der Kugel- und Cylinderfunctionen verwandte Function und ihre Anwendung in der Theorie der Elektricitätsverteilung"</a>, <a href="/facts/Mathematische_Annalen/GPhGaonc">Mathematische Annalen</a>, 18 (2): 161–194, <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1007%2FBF01445847">10.1007/BF01445847</a>, <a href="/facts/S2CID_(identifier)/ldJsHa2Y">S2CID</a> <a href="https://api.semanticscholar.org/CorpusID:122590188">122590188</a></li>
<li><a href="/facts/Michael_C._Reed/PjUGTNwk">Reed, Michael</a>; <a href="/facts/Barry_Simon/BImmcE2I">Simon, Barry</a> (1975), Methods of Modern Mathematical Physics II, Fourier Analysis, Self-Adjointness, Academic Press, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-12-585002-5</li>
<li>Riesz, Frigyes; Szőkefalvi-Nagy, Béla (1990). Functional Analysis. Dover Publications. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-486-66289-6.</li>
<li><a href="/facts/Marshall_Harvey_Stone/TFzbE4GH">Stone, Marshall Harvey</a> (1932), Linear transformations in Hilbert space and Their Applications to Analysis, AMS Colloquium Publications, vol. 16, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-8218-1015-6 {{citation}}: ISBN / Date incompatibility (help)</li>
<li><a href="/facts/Audrey_Terras/pmVYDCq9">Terras, Audrey</a> (1984), "Non-Euclidean harmonic analysis, the central limit theorem, and long transmission lines with random inhomogeneities", J. Multivariate Anal., 15 (2): 261–276, <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1016%2F0047-259X%2884%2990031-9">10.1016/0047-259X(84)90031-9</a></li>
<li><a href="/facts/Gerald_Teschl/tEBmvGQA">Teschl, Gerald</a> (2009). <a href="https://www.mat.univie.ac.at/~gerald/ftp/book-schroe/">Mathematical Methods in Quantum Mechanics; With Applications to Schrödinger Operators</a>. AMS Graduate Studies in Mathematics. Vol. 99. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-8218-4660-5.</li>
<li><a href="/facts/Gerald_Teschl/tEBmvGQA">Teschl, Gerald</a> (2012). <a href="https://www.mat.univie.ac.at/~gerald/ftp/book-ode/">Ordinary Differential Equations and Dynamical Systems</a>. AMS Graduate Studies in Mathematics. Vol. 140. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-8218-8328-0.</li>
<li>Titchmarsh, Edward Charles (1939), Theory of Functions, Oxford University Press</li>
<li><a href="/facts/Edward_Charles_Titchmarsh/607FjJwB">Titchmarsh, Edward Charles</a> (1946), Eigenfunction expansions associated with second order differential equations, Vol. I (1st ed.), Oxford University Press</li>
<li><a href="/facts/Edward_Charles_Titchmarsh/607FjJwB">Titchmarsh, Edward Charles</a> (1962), Eigenfunction expansions associated with second order differential equations, Vol. I (2nd ed.), Oxford University Press, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-608-08254-7 {{citation}}: ISBN / Date incompatibility (help)</li>
<li>Eckhardt, Jonathan; <a href="/facts/Gerald_Teschl/tEBmvGQA">Teschl, Gerald</a> (2013), "Sturm–Liouville operators with measure-valued coefficients", <a href="/facts/Journal_d%2527Analyse_Math%25C3%25A9matique/dQD4alPo">Journal d'Analyse Mathématique</a>, 120: 151–224, <a href="/facts/ArXiv_(identifier)/H6EtgnBe">arXiv</a>:<a href="https://arxiv.org/abs/1105.3755">1105.3755</a>, <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1007%2Fs11854-013-0018-x">10.1007/s11854-013-0018-x</a></li>
<li>Vilenkin, Naoum Iakovlevitch (1968). Special Functions and the Theory of Group Representations. Translations of Mathematical Monographs. Vol. 22. <a href="/facts/American_Mathematical_Society/fr5gotCt">American Mathematical Society</a>. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-8218-1572-4.</li>
<li>Weidmann, Joachim (1987). Spectral Theory of Ordinary Differential Operators. Lecture Notes in Mathematics. Vol. 1258. Springer-Verlag. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-387-17902-5.</li>
<li><a href="/facts/Hermann_Weyl/4C5HhyGU">Weyl, Hermann</a> (1910a), <a href="http://gdz.sub.uni-goettingen.de/index.php?id=11&PPN=PPN235181684_0068&DMDID=DMDLOG_0023&L=1">"Über gewöhnliche Differentialgleichungen mit Singularitäten und die zugehörigen Entwicklungen willkürlicher Functionen"</a>, <a href="/facts/Mathematische_Annalen/GPhGaonc">Mathematische Annalen</a>, 68 (2): 220–269, <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1007%2FBF01474161">10.1007/BF01474161</a>, <a href="/facts/S2CID_(identifier)/ldJsHa2Y">S2CID</a> <a href="https://api.semanticscholar.org/CorpusID:119727984">119727984</a></li>
<li><a href="/facts/Hermann_Weyl/4C5HhyGU">Weyl, Hermann</a> (1910b), "Über gewöhnliche Differentialgleichungen mit Singulären Stellen und ihre Eigenfunktionen", Nachr. Akad. Wiss. Göttingen. Math.-Phys.: 442–446</li>
<li><a href="/facts/Hermann_Weyl/4C5HhyGU">Weyl, Hermann</a> (1935), "Über das Pick-Nevanlinnasche Interpolationsproblem und sein infinitesimales Analogen", <a href="/facts/Annals_of_Mathematics/CUYktUqD">Annals of Mathematics</a>, 36 (1): 230–254, <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.2307%2F1968677">10.2307/1968677</a>, <a href="/facts/JSTOR_(identifier)/YTeVmaJ7">JSTOR</a> <a href="https://www.jstor.org/stable/1968677">1968677</a></li></ul>

<h2 id="references">References</h2>

<ol>
<li id="fn:1">Titchmarsh 1962, p. 22 - Titchmarsh, Edward Charles (1962), Eigenfunction expansions associated with second order differential equations, Vol. I (2nd ed.), Oxford University Press, ISBN 978-0-608-08254-7 <a href="#fnref:1" class="footnote-back-ref">↩</a></li>
<li id="fn:2">Dieudonné 1969, Chapter X - Dieudonné, Jean (1969), Treatise on Analysis, Vol. I [Foundations of Modern Analysis], Academic Press, ISBN 978-1-4067-2791-3 <a href="#fnref:2" class="footnote-back-ref">↩</a></li>
<li id="fn:3">Courant & Hilbert 1989 - Courant, Richard; Hilbert, David (1989), Method of Mathematical Physics, Vol. I, Wiley-Interscience, ISBN 978-0-471-50447-4 <a href="#fnref:3" class="footnote-back-ref">↩</a></li>
<li id="fn:4">Titchmarsh 1962 - Titchmarsh, Edward Charles (1962), Eigenfunction expansions associated with second order differential equations, Vol. I (2nd ed.), Oxford University Press, ISBN 978-0-608-08254-7 <a href="#fnref:4" class="footnote-back-ref">↩</a></li>
<li id="fn:5">Titchmarsh 1939, §8.2 - Titchmarsh, Edward Charles (1939), Theory of Functions, Oxford University Press <a href="#fnref:5" class="footnote-back-ref">↩</a></li>
<li id="fn:6">Burkill 1951, pp. 50–52 - Burkill, J.C. (1951), The Lebesgue Integral, Cambridge Tracts in Mathematics and Mathematical Physics, vol. 40, Cambridge University Press, ISBN 978-0-521-04382-3 <a href="#fnref:6" class="footnote-back-ref">↩</a></li>
<li id="fn:7">Loomis 1953, p. page 40 - Loomis, Lynn H. (1953), An Introduction to Abstract Harmonic Analysis, van Nostrand <a href="#fnref:7" class="footnote-back-ref">↩</a></li>
<li id="fn:8">Loomis 1953, pp. 30–31 - Loomis, Lynn H. (1953), An Introduction to Abstract Harmonic Analysis, van Nostrand <a href="#fnref:8" class="footnote-back-ref">↩</a></li>
<li id="fn:9">Kolmogorov & Fomin 1975, pp. 374–376 - Kolmogorov, A.N.; Fomin, S.V. (1975). Introductory Real Analysis. Dover. ISBN 978-0-486-61226-3. <a href="https://archive.org/details/introductoryreal00kolm_0/page/374" target="_blank">https://archive.org/details/introductoryreal00kolm_0/page/374</a> <a href="#fnref:9" class="footnote-back-ref">↩</a></li>
<li id="fn:10">This is a limit in the strong operator topology. <a href="/wiki/Strong_operator_topology" target="_blank">/wiki/Strong_operator_topology</a> <a href="#fnref:10" class="footnote-back-ref">↩</a></li>
<li id="fn:11">A bona fide inner product is defined on the quotient by the subspace of null functions 
 
 
 
 f
 
 
 {\displaystyle f}
 
, i.e. those with 
 
 
 
 
 μ
 
 ξ
 
 
 (
 
 |
 
 f
 
 
 |
 
 
 2
 
 
 )
 =
 0
 
 
 {\displaystyle \mu _{\xi }(|f|_{2})=0}
 
. Alternatively in this case the support of the measure is 
 
 
 
 σ
 (
 T
 )
 
 
 {\displaystyle \sigma (T)}
 
, so the right hand side defines a (non-degenerate) inner product on 
 
 
 
 C
 (
 σ
 (
 T
 )
 )
 
 
 {\displaystyle C(\sigma (T))}
 
. <a href="#fnref:11" class="footnote-back-ref">↩</a></li>
<li id="fn:12">Weyl 1910 harvnb error: no target: CITEREFWeyl1910 (help).[specify] <a href="#fnref:12" class="footnote-back-ref">↩</a></li>
<li id="fn:13">Weyl 1910 harvnb error: no target: CITEREFWeyl1910 (help).[specify] <a href="#fnref:13" class="footnote-back-ref">↩</a></li>
<li id="fn:14">Bellman 1969, p. 116 - Bellman, Richard (1969), Stability Theory of Differential Equations, Dover, ISBN 978-0-486-62210-1 <a href="https://archive.org/details/stabilitytheoryo0000bell" target="_blank">https://archive.org/details/stabilitytheoryo0000bell</a> <a href="#fnref:14" class="footnote-back-ref">↩</a></li>
<li id="fn:15">Bellman 1969, p. 116 - Bellman, Richard (1969), Stability Theory of Differential Equations, Dover, ISBN 978-0-486-62210-1 <a href="https://archive.org/details/stabilitytheoryo0000bell" target="_blank">https://archive.org/details/stabilitytheoryo0000bell</a> <a href="#fnref:15" class="footnote-back-ref">↩</a></li>
<li id="fn:16">Reed & Simon 1975, p. 159 - Reed, Michael; Simon, Barry (1975), Methods of Modern Mathematical Physics II, Fourier Analysis, Self-Adjointness, Academic Press, ISBN 978-0-12-585002-5 <a href="#fnref:16" class="footnote-back-ref">↩</a></li>
<li id="fn:17">Reed & Simon 1975, p. 154 - Reed, Michael; Simon, Barry (1975), Methods of Modern Mathematical Physics II, Fourier Analysis, Self-Adjointness, Academic Press, ISBN 978-0-12-585002-5 <a href="#fnref:17" class="footnote-back-ref">↩</a></li>
<li id="fn:18">Weyl 1910 harvnb error: no target: CITEREFWeyl1910 (help).[specify] <a href="#fnref:18" class="footnote-back-ref">↩</a></li>
<li id="fn:19">Weyl 1910 harvnb error: no target: CITEREFWeyl1910 (help).[specify] <a href="#fnref:19" class="footnote-back-ref">↩</a></li>
<li id="fn:20">Titchmarsh 1946, Chapter III - Titchmarsh, Edward Charles (1946), Eigenfunction expansions associated with second order differential equations, Vol. I (1st ed.), Oxford University Press <a href="#fnref:20" class="footnote-back-ref">↩</a></li>
<li id="fn:21">Kodaira 1949, pp. 935–936 - Kodaira, Kunihiko (1949), "The eigenvalue problem for ordinary differential equations of the second order and Heisenberg's theory of S-matrices", American Journal of Mathematics, 71 (4): 921–945, doi:10.2307/2372377, JSTOR 2372377 <a href="https://doi.org/10.2307%2F2372377" target="_blank">https://doi.org/10.2307%2F2372377</a> <a href="#fnref:21" class="footnote-back-ref">↩</a></li>
<li id="fn:22">Kodaira 1949, pp. 929–932; for omitted details, see Kodaira 1950, pp. 529–536 - Kodaira, Kunihiko (1949), "The eigenvalue problem for ordinary differential equations of the second order and Heisenberg's theory of S-matrices", American Journal of Mathematics, 71 (4): 921–945, doi:10.2307/2372377, JSTOR 2372377 <a href="https://doi.org/10.2307%2F2372377" target="_blank">https://doi.org/10.2307%2F2372377</a> <a href="#fnref:22" class="footnote-back-ref">↩</a></li>
<li id="fn:23">Dieudonné 1988 - Dieudonné, Jean (1988), Treatise on Analysis, Vol. VIII, Academic Press, ISBN 978-0-12-215507-9 <a href="#fnref:23" class="footnote-back-ref">↩</a></li>
<li id="fn:24">Weyl 1910 harvnb error: no target: CITEREFWeyl1910 (help).[specify] <a href="#fnref:24" class="footnote-back-ref">↩</a></li>
<li id="fn:25">Stone 1932, Chapter X - Stone, Marshall Harvey (1932), Linear transformations in Hilbert space and Their Applications to Analysis, AMS Colloquium Publications, vol. 16, ISBN 978-0-8218-1015-6 <a href="#fnref:25" class="footnote-back-ref">↩</a></li>
<li id="fn:26">Kodaira 1950, pp. 534–535 - Kodaira, Kunihiko (1950), "On ordinary differential equations of any even order and the corresponding eigenfunction expansions", American Journal of Mathematics, 72 (3): 502–544, doi:10.2307/2372051, JSTOR 2372051 <a href="https://doi.org/10.2307%2F2372051" target="_blank">https://doi.org/10.2307%2F2372051</a> <a href="#fnref:26" class="footnote-back-ref">↩</a></li>
<li id="fn:27">Mehler 1881. - Mehler, F.G. (1881), "Ueber mit der Kugel- und Cylinderfunctionen verwandte Function und ihre Anwendung in der Theorie der Elektricitätsverteilung", Mathematische Annalen, 18 (2): 161–194, doi:10.1007/BF01445847, S2CID 122590188 <a href="http://gdz.sub.uni-goettingen.de/index.php?id=11&PPN=PPN235181684_0018&DMDID=DMDLOG_0021&L=1" target="_blank">http://gdz.sub.uni-goettingen.de/index.php?id=11&PPN=PPN235181684_0018&DMDID=DMDLOG_0021&L=1</a> <a href="#fnref:27" class="footnote-back-ref">↩</a></li>
<li id="fn:28">Fock 1943, pp. 253–256 - Fock, V.A. (1943), "On the representation of an arbitrary function by an integral involving Legendre's functions with a complex index", C. R. Acad. Sci. URSS, 39 <a href="#fnref:28" class="footnote-back-ref">↩</a></li>
<li id="fn:29">Vilenkin 1968 - Vilenkin, Naoum Iakovlevitch (1968). Special Functions and the Theory of Group Representations. Translations of Mathematical Monographs. Vol. 22. American Mathematical Society. ISBN 978-0-8218-1572-4. <a href="#fnref:29" class="footnote-back-ref">↩</a></li>
<li id="fn:30">Terras 1984, pp. 261–276 - Terras, Audrey (1984), "Non-Euclidean harmonic analysis, the central limit theorem, and long transmission lines with random inhomogeneities", J. Multivariate Anal., 15 (2): 261–276, doi:10.1016/0047-259X(84)90031-9 <a href="https://doi.org/10.1016%2F0047-259X%2884%2990031-9" target="_blank">https://doi.org/10.1016%2F0047-259X%2884%2990031-9</a> <a href="#fnref:30" class="footnote-back-ref">↩</a></li>
<li id="fn:31">Lebedev 1972 - Lebedev, N.N. (1972), Special Functions and Their Applications, Dover, ISBN 978-0-486-60624-8 <a href="#fnref:31" class="footnote-back-ref">↩</a></li>
<li id="fn:32">Vilenkin 1968, Chapter VI - Vilenkin, Naoum Iakovlevitch (1968). Special Functions and the Theory of Group Representations. Translations of Mathematical Monographs. Vol. 22. American Mathematical Society. ISBN 978-0-8218-1572-4. <a href="#fnref:32" class="footnote-back-ref">↩</a></li>
<li id="fn:33">Kostenko, Sakhnovich & Teschl 2012, pp. 1699–1747 - Kostenko, Aleksey; Sakhnovich, Alexander; Teschl, Gerald (2012), "Weyl–Titchmarsh Theory for Schrödinger Operators with Strongly Singular Potentials", Int Math Res Notices, 2012, arXiv:1007.0136, doi:10.1093/imrn/rnr065 <a href="https://arxiv.org/abs/1007.0136" target="_blank">https://arxiv.org/abs/1007.0136</a> <a href="#fnref:33" class="footnote-back-ref">↩</a></li>
<li id="fn:34">Eckhardt & Teschl 2013, pp. 151–224 - Eckhardt, Jonathan; Teschl, Gerald (2013), "Sturm–Liouville operators with measure-valued coefficients", Journal d'Analyse Mathématique, 120: 151–224, arXiv:1105.3755, doi:10.1007/s11854-013-0018-x <a href="https://arxiv.org/abs/1105.3755" target="_blank">https://arxiv.org/abs/1105.3755</a> <a href="#fnref:34" class="footnote-back-ref">↩</a></li>
</ol>

Spectral theory of ordinary differential equations open-in-new

Spectral theory of ordinary differential equations