Sturm separation theorem

<h2 id="sturm-separation-theorem">Sturm separation theorem</h2>
If u(x) and v(x) are two non-trivial continuous linearly independent solutions to a homogeneous second order linear differential equation with x0 and x1 being successive roots of u(x), then v(x) has exactly one root in the open interval (x0, x1). It is a special case of the <a href="/facts/Sturm-Picone_comparison_theorem/ABukvpTe">Sturm-Picone comparison theorem</a>.

<h2 id="proof">Proof</h2>
Since 
 
 
 
 
 u
 
 
 
 {\displaystyle \displaystyle u}
 
 and 
 
 
 
 
 v
 
 
 
 {\displaystyle \displaystyle v}
 
 are linearly independent it follows that the <a href="/facts/Wronskian/sJNmIF5C">Wronskian</a> 
 
 
 
 
 W
 [
 u
 ,
 v
 ]
 
 
 
 {\displaystyle \displaystyle W[u,v]}
 
 must satisfy 
 
 
 
 W
 [
 u
 ,
 v
 ]
 (
 x
 )
 ≡
 W
 (
 x
 )
 ≠
 0
 
 
 {\displaystyle W[u,v](x)\equiv W(x)\neq 0}
 
 for all 
 
 
 
 
 x
 
 
 
 {\displaystyle \displaystyle x}
 
 where the differential equation is defined, say 
 
 
 
 
 I
 
 
 
 {\displaystyle \displaystyle I}
 
. Without loss of generality, suppose that 
 
 
 
 W
 (
 x
 )
 <
 0
 
 
  
 
 
 ∀
 
 
  
 
 
 x
 ∈
 I
 
 
 {\displaystyle W(x)<0{\mbox{ }}\forall {\mbox{ }}x\in I}
 
. Then

u
        (
        x
        )
        
          v
          ′
        
        (
        x
        )
        −
        
          u
          ′
        
        (
        x
        )
        v
        (
        x
        )
        ≠
        0.
      
    
    {\displaystyle u(x)v'(x)-u'(x)v(x)\neq 0.}

So at 
 
 
 
 
 x
 =
 
 x
 
 0
 
 
 
 
 
 {\displaystyle \displaystyle x=x_{0}}

W
        (
        
          x
          
            0
          
        
        )
        =
        −
        
          u
          ′
        
        
          (
          
            x
            
              0
            
          
          )
        
        v
        
          (
          
            x
            
              0
            
          
          )
        
      
    
    {\displaystyle W(x_{0})=-u'\left(x_{0}\right)v\left(x_{0}\right)}

and either 
 
 
 
 
 u
 ′
 
 
 (
 
 x
 
 0
 
 
 )
 
 
 
 {\displaystyle u'\left(x_{0}\right)}
 
 and 
 
 
 
 v
 
 (
 
 x
 
 0
 
 
 )
 
 
 
 {\displaystyle v\left(x_{0}\right)}
 
 are both positive or both negative. Without loss of generality, suppose that they are both positive. Now, at 
 
 
 
 
 x
 =
 
 x
 
 1
 
 
 
 
 
 {\displaystyle \displaystyle x=x_{1}}

W
        (
        
          x
          
            1
          
        
        )
        =
        −
        
          u
          ′
        
        
          (
          
            x
            
              1
            
          
          )
        
        v
        
          (
          
            x
            
              1
            
          
          )
        
      
    
    {\displaystyle W(x_{1})=-u'\left(x_{1}\right)v\left(x_{1}\right)}

and since 
 
 
 
 
 x
 =
 
 x
 
 0
 
 
 
 
 
 {\displaystyle \displaystyle x=x_{0}}
 
 and 
 
 
 
 
 x
 =
 
 x
 
 1
 
 
 
 
 
 {\displaystyle \displaystyle x=x_{1}}
 
 are successive zeros of 
 
 
 
 
 u
 (
 x
 )
 
 
 
 {\displaystyle \displaystyle u(x)}
 
 it causes 
 
 
 
 
 u
 ′
 
 
 (
 
 x
 
 1
 
 
 )
 
 <
 0
 
 
 {\displaystyle u'\left(x_{1}\right)<0}
 
. Thus, to keep 
 
 
 
 
 W
 (
 x
 )
 <
 0
 
 
 
 {\displaystyle \displaystyle W(x)<0}
 
 we must have 
 
 
 
 v
 
 (
 
 x
 
 1
 
 
 )
 
 <
 0
 
 
 {\displaystyle v\left(x_{1}\right)<0}
 
. We see this by observing that if 
 
 
 
 
 
 u
 ′
 
 (
 x
 )
 >
 0
 
 
  
 
 
 ∀
 
 
  
 
 
 x
 ∈
 
 (
 
 
 x
 
 0
 
 
 ,
 
 x
 
 1
 
 
 
 ]
 
 
 
 
 {\displaystyle \displaystyle u'(x)>0{\mbox{ }}\forall {\mbox{ }}x\in \left(x_{0},x_{1}\right]}
 
 then 
 
 
 
 
 u
 (
 x
 )
 
 
 
 {\displaystyle \displaystyle u(x)}
 
 would be increasing (away from the 
 
 
 
 
 x
 
 
 
 {\displaystyle \displaystyle x}
 
-axis), which would never lead to a zero at 
 
 
 
 
 x
 =
 
 x
 
 1
 
 
 
 
 
 {\displaystyle \displaystyle x=x_{1}}
 
. So for a zero to occur at 
 
 
 
 
 x
 =
 
 x
 
 1
 
 
 
 
 
 {\displaystyle \displaystyle x=x_{1}}
 
 at most 
 
 
 
 
 u
 ′
 
 
 (
 
 x
 
 1
 
 
 )
 
 =
 0
 
 
 {\displaystyle u'\left(x_{1}\right)=0}
 
 (i.e., 
 
 
 
 
 u
 ′
 
 
 (
 
 x
 
 1
 
 
 )
 
 ≤
 0
 
 
 {\displaystyle u'\left(x_{1}\right)\leq 0}
 
 and it turns out, by our result from the <a href="/facts/Wronskian/sJNmIF5C">Wronskian</a> that 
 
 
 
 
 u
 ′
 
 
 (
 
 x
 
 1
 
 
 )
 
 ≤
 0
 
 
 {\displaystyle u'\left(x_{1}\right)\leq 0}
 
). So somewhere in the interval 
 
 
 
 
 (
 
 
 x
 
 0
 
 
 ,
 
 x
 
 1
 
 
 
 )
 
 
 
 {\displaystyle \left(x_{0},x_{1}\right)}
 
 the sign of 
 
 
 
 
 v
 (
 x
 )
 
 
 
 {\displaystyle \displaystyle v(x)}
 
 changed. By the <a href="/facts/Intermediate_Value_Theorem/SaFdYE7W">Intermediate Value Theorem</a> there exists 
 
 
 
 
 x
 
 ∗
 
 
 ∈
 
 (
 
 
 x
 
 0
 
 
 ,
 
 x
 
 1
 
 
 
 )
 
 
 
 {\displaystyle x^{*}\in \left(x_{0},x_{1}\right)}
 
 such that 
 
 
 
 v
 
 (
 
 x
 
 ∗
 
 
 )
 
 =
 0
 
 
 {\displaystyle v\left(x^{*}\right)=0}
 
.
On the other hand, there can be only one zero in 
 
 
 
 
 (
 
 
 x
 
 0
 
 
 ,
 
 x
 
 1
 
 
 
 )
 
 
 
 {\displaystyle \left(x_{0},x_{1}\right)}
 
, because otherwise 
 
 
 
 v
 
 
 {\displaystyle v}
 
 would have two zeros and there would be no zeros of 
 
 
 
 u
 
 
 {\displaystyle u}
 
 in between, and it was just proved that this is impossible.

<ul><li><a href="/facts/Gerald_Teschl/tEBmvGQA">Teschl, G.</a> (2012). <a href="https://www.mat.univie.ac.at/~gerald/ftp/book-ode/">Ordinary Differential Equations and Dynamical Systems</a>. <a href="/facts/Providence,_Rhode_Island/77kh9zJT">Providence</a>: <a href="/facts/American_Mathematical_Society/fr5gotCt">American Mathematical Society</a>. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-8218-8328-0.</li></ul>

Sturm separation theorem open-in-new

Sturm separation theorem