Marchenko equation

<h2 id="application-to-scattering-theory">Application to scattering theory</h2>
<p>Suppose that for a potential 
  
    
      
        u
        (
        x
        )
      
    
    {\displaystyle u(x)}
  
 for the <a href="/facts/Schr%C3%B6dinger_operator/3csgSIWF">Schrödinger operator</a> 
  
    
      
        L
        =
        −
        
          
            
              d
              
                2
              
            
            
              d
              
                x
                
                  2
                
              
            
          
        
        +
        u
        (
        x
        )
      
    
    {\displaystyle L=-{\frac {d^{2}}{dx^{2}}}+u(x)}
  
, one has the <a href="/facts/Scattering/nMou7lEH">scattering</a> data 
  
    
      
        (
        r
        (
        k
        )
        ,
        {
        
          χ
          
            1
          
        
        ,
        ⋯
        ,
        
          χ
          
            N
          
        
        }
        )
      
    
    {\displaystyle (r(k),\{\chi _{1},\cdots ,\chi _{N}\})}
  
, where 
  
    
      
        r
        (
        k
        )
      
    
    {\displaystyle r(k)}
  
 are the reflection coefficients from continuous scattering, given as a function 
  
    
      
        r
        :
        
          R
        
        →
        
          C
        
      
    
    {\displaystyle r:\mathbb {R} \rightarrow \mathbb {C} }
  
, and the real parameters 
  
    
      
        
          χ
          
            1
          
        
        ,
        ⋯
        ,
        
          χ
          
            N
          
        
        >
        0
      
    
    {\displaystyle \chi _{1},\cdots ,\chi _{N}>0}
  
 are from the discrete bound spectrum.<a class="footnote-ref" id="fnref:1" href="#fn:1"><sup>1</sup></a>
</p><p>Then defining

F
        (
        x
        )
        =
        
          ∑
          
            n
            =
            1
          
          
            N
          
        
        
          β
          
            n
          
        
        
          e
          
            −
            
              χ
              
                n
              
            
            x
          
        
        +
        
          
            1
            
              2
              π
            
          
        
        
          ∫
          
            
              R
            
          
        
        r
        (
        k
        )
        
          e
          
            i
            k
            x
          
        
        d
        k
        ,
      
    
    {\displaystyle F(x)=\sum _{n=1}^{N}\beta _{n}e^{-\chi _{n}x}+{\frac {1}{2\pi }}\int _{\mathbb {R} }r(k)e^{ikx}dk,}

where the 
  
    
      
        
          β
          
            n
          
        
      
    
    {\displaystyle \beta _{n}}
  
 are non-zero constants, solving the GLM equation

K
        (
        x
        ,
        y
        )
        +
        F
        (
        x
        +
        y
        )
        +
        
          ∫
          
            x
          
          
            ∞
          
        
        K
        (
        x
        ,
        z
        )
        F
        (
        z
        +
        y
        )
        d
        z
        =
        0
      
    
    {\displaystyle K(x,y)+F(x+y)+\int _{x}^{\infty }K(x,z)F(z+y)dz=0}

for 
  
    
      
        K
      
    
    {\displaystyle K}
  
 allows the potential to be recovered using the formula

u
        (
        x
        )
        =
        −
        2
        
          
            d
            
              d
              x
            
          
        
        K
        (
        x
        ,
        x
        )
        .
      
    
    {\displaystyle u(x)=-2{\frac {d}{dx}}K(x,x).}

</p>
<h2 id="see-also">See also</h2>
<ul><li><a href="/facts/Lax_pair/dyzB7Ttb">Lax pair</a></li></ul>
<h2 id="notes">Notes</h2>

<ul><li>Dunajski, Maciej (2009). <i>Solitons, Instantons, and Twistors</i>. Oxford; New York: OUP Oxford. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-19-857063-9. <a href="/facts/OCLC_(identifier)/Yqad8waq">OCLC</a> <a href="https://search.worldcat.org/oclc/320199531">320199531</a>.</li>
<li>Marchenko, V. A. (2011). <i>Sturm–Liouville Operators and Applications</i> (2nd ed.). Providence: <a href="/facts/American_Mathematical_Society/fr5gotCt">American Mathematical Society</a>. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-8218-5316-0. <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=2798059">2798059</a>.</li>
<li>Kay, Irvin W. (1955). <a href="https://archive.org/details/inversescatterin00kayi/page/n3/mode/2up"><i>The inverse scattering problem</i></a>. New York: Courant Institute of Mathematical Sciences, New York University. <a href="/facts/OCLC_(identifier)/Yqad8waq">OCLC</a> <a href="https://search.worldcat.org/oclc/1046812324">1046812324</a>.</li>
<li>Levinson, Norman (1953). "Certain Explicit Relationships between Phase Shift and Scattering Potential". <i>Physical Review</i>. 89 (4): 755–757. <a href="/facts/Bibcode_(identifier)/9HtdQSGB">Bibcode</a>:<a href="https://ui.adsabs.harvard.edu/abs/1953PhRv...89..755L">1953PhRv...89..755L</a>. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1103%2FPhysRev.89.755">10.1103/PhysRev.89.755</a>. <a href="/facts/ISSN_(identifier)/DPAflDvU">ISSN</a> <a href="https://search.worldcat.org/issn/0031-899X">0031-899X</a>.</li></ul>

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

<ol>
<li id="fn:1"><p>Dunajski 2009, pp. 30–31. - Dunajski, Maciej (2009). Solitons, Instantons, and Twistors. Oxford; New York: OUP Oxford. ISBN 978-0-19-857063-9. OCLC 320199531. <a href="https://search.worldcat.org/oclc/320199531" target="_blank">https://search.worldcat.org/oclc/320199531</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li>
</ol>

Marchenko equation open-in-new

Marchenko equation