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A problem in which it is required to reconstruct a function (a potential) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090780/s0907801.png" /> from some spectral characteristics of the operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090780/s0907802.png" /> generated by the differential expression <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090780/s0907803.png" /> and some boundary conditions in the Hilbert space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090780/s0907804.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090780/s0907805.png" /> varies in a finite or infinite interval <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090780/s0907806.png" />. Moreover, one should also reconstruct the boundary conditions corresponding to the operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090780/s0907807.png" />.
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When studying inverse problems, the following natural questions arise: 1) to find out which spectral characteristics determine the operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090780/s0907808.png" /> uniquely; 2) to give a method of reconstructing the operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090780/s0907809.png" /> from these spectral characteristics; and 3) to find particular properties of the spectral characteristics considered. Depending on the choice of the spectral characteristics, different statements of inverse problems are possible (often arising in applications).
+
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 +
{{TEX|done}}
  
The first result concerning inverse problems (see [[#References|[10]]]), which gave a start to the whole theory, is: Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090780/s09078010.png" /> be the eigenvalues for the problem
+
A problem in which it is required to reconstruct a function (a potential)  $  q $
 +
from some spectral characteristics of the operator  $  A $
 +
generated by the differential expression  $  l[ y] = - y  ^ {\prime\prime} + q( x) y $
 +
and some boundary conditions in the Hilbert space  $  L _ {2} ( a, b) $,  
 +
where  $  x $
 +
varies in a finite or infinite interval  $  ( a, b) $.  
 +
Moreover, one should also reconstruct the boundary conditions corresponding to the operator  $  A $.
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090780/s09078011.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</td></tr></table>
+
When studying inverse problems, the following natural questions arise: 1) to find out which spectral characteristics determine the operator  $  A $
 +
uniquely; 2) to give a method of reconstructing the operator  $  A $
 +
from these spectral characteristics; and 3) to find particular properties of the spectral characteristics considered. Depending on the choice of the spectral characteristics, different statements of inverse problems are possible (often arising in applications).
  
and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090780/s09078012.png" /> be a real-valued continuous function on the interval <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090780/s09078013.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090780/s09078014.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090780/s09078015.png" /> then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090780/s09078016.png" />.
+
The first result concerning inverse problems (see [[#References|[10]]]), which gave a start to the whole theory, is: Let  $  \lambda _ {0} , \lambda _ {1} \dots $
 +
be the eigenvalues for the problem
  
A profound study of inverse problems started in the 1940's (see [[#References|[11]]], [[#References|[12]]]). Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090780/s09078017.png" /> be the eigenvalues for the equation (1) under the boundary conditions
+
$$ \tag{1 }
 +
\left .
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090780/s09078018.png" /></td> <td valign="top" style="width:5%;text-align:right;">(2)</td></tr></table>
+
\begin{array}{c}
 +
- y  ^ {\prime\prime} + q( x) y  = \lambda y ,\  0 \leq  x \leq  \pi ,  \\
 +
y  ^  \prime  ( 0= y  ^  \prime  ( \pi ) =  0,  \\
 +
\end{array}
  
(<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090780/s09078019.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090780/s09078020.png" /> are real numbers), and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090780/s09078021.png" /> be the eigenvalues of (1) under the boundary conditions
+
\right \}
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090780/s09078022.png" /></td> </tr></table>
+
and let  $  q $
 +
be a real-valued continuous function on the interval  $  [ 0, \pi ] $.
 +
If  $  \lambda _ {n} = n  ^ {2} $,
 +
$  n = 0, 1 \dots $
 +
then  $  q( x) \equiv 0 $.
  
Then the sequences <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090780/s09078023.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090780/s09078024.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090780/s09078025.png" /> determine the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090780/s09078026.png" /> and the numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090780/s09078027.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090780/s09078028.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090780/s09078029.png" /> uniquely. Moreover, if at least one eigenvalue in these problems has not been determined, then all the others do not determine the equation (1) uniquely. In particular, generally speaking, one spectrum does not determine the equation uniquely (the above-mentioned result is an exception to the general rule).
+
A profound study of inverse problems started in the 1940's (see [[#References|[11]]], [[#References|[12]]]). Let  $  \lambda _ {0} , \lambda _ {1} \dots $
 +
be the eigenvalues for the equation (1) under the boundary conditions
  
If the equation (1) is studied on the half-line <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090780/s09078030.png" /> and the potential function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090780/s09078031.png" /> satisfies the requirement
+
$$ \tag{2 }
 +
y  ^  \prime  ( 0) - hy( 0)  = 0,\ \
 +
y  ^  \prime  ( \pi ) + Hy( \pi )  = 0
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090780/s09078032.png" /></td> </tr></table>
+
( $  h $
 +
and  $  H $
 +
are real numbers), and let  $  \mu _ {0} , \mu _ {1} \dots $
 +
be the eigenvalues of (1) under the boundary conditions
  
then the solution <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090780/s09078033.png" /> of the problem <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090780/s09078034.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090780/s09078035.png" />, has an asymptotic representation for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090780/s09078036.png" />,
+
$$
 +
y  ^  \prime  ( 0) - h _ {1} y( 0= 0,\ \
 +
y  ^  \prime  ( \pi ) + Hy( \pi )  = 0 ,\ \
 +
h _ {1} \neq h.
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090780/s09078037.png" /></td> </tr></table>
+
Then the sequences  $  \{ \lambda _ {n} \} $
 +
and  $  \{ \mu _ {n} \} $,
 +
$  n = 0, 1 \dots $
 +
determine the function  $  q $
 +
and the numbers  $  h $,
 +
$  h _ {1} $
 +
and  $  H $
 +
uniquely. Moreover, if at least one eigenvalue in these problems has not been determined, then all the others do not determine the equation (1) uniquely. In particular, generally speaking, one spectrum does not determine the equation uniquely (the above-mentioned result is an exception to the general rule).
  
The function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090780/s09078038.png" /> is called the scattering phase. The main result is that if a problem (considered in the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090780/s09078039.png" />) does not have discrete eigenvalues, then the scattering phase determines the potential function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090780/s09078040.png" /> uniquely.
+
If the equation (1) is studied on the half-line  $  ( 0, \infty ) $
 +
and the potential function $  q $
 +
satisfies the requirement
 +
 
 +
$$
 +
\int\limits _ { 0 } ^  \infty  x | q( x) |  dx  < \infty ,
 +
$$
 +
 
 +
then the solution  $  \phi ( x, \lambda ) $
 +
of the problem  $  - y  ^ {\prime\prime} + q( x) y = \lambda  ^ {2} y $,
 +
$  y( 0) = 0 $,
 +
has an asymptotic representation for  $  x \rightarrow \infty $,
 +
 
 +
$$
 +
\phi ( x, \lambda )  = M( \lambda )  \sin [ \lambda x + \delta ( \lambda )] + o( 1).
 +
$$
 +
 
 +
The function  $  \delta ( \lambda ) $
 +
is called the scattering phase. The main result is that if a problem (considered in the space $  L _ {2} ( 0, \infty ) $)  
 +
does not have discrete eigenvalues, then the scattering phase determines the potential function $  q $
 +
uniquely.
  
 
A decisive step in the further development of the theory of inverse problems was the application of so-called operator-transform techniques (see [[Sturm–Liouville equation|Sturm–Liouville equation]]), which was naturally developed in the framework of the theory of generalized shift operators (see [[#References|[4]]]).
 
A decisive step in the further development of the theory of inverse problems was the application of so-called operator-transform techniques (see [[Sturm–Liouville equation|Sturm–Liouville equation]]), which was naturally developed in the framework of the theory of generalized shift operators (see [[#References|[4]]]).
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In principle, all inverse problems can be reduced to the inverse problem of reconstructing an operator from its spectral function. However, this way is not always the simplest; moreover, when following it, one often encounters difficulties in finding necessary and sufficient conditions for the spectral characteristics that are used to reconstruct the operator.
 
In principle, all inverse problems can be reduced to the inverse problem of reconstructing an operator from its spectral function. However, this way is not always the simplest; moreover, when following it, one often encounters difficulties in finding necessary and sufficient conditions for the spectral characteristics that are used to reconstruct the operator.
  
The significance of inverse problems became greater after discovering a possibility to apply them to solve some non-linear evolution equations of mathematical physics. In particular, a relationship (see [[#References|[25]]]) between inverse problems for some Sturm–Liouville operators with a finite number of gaps in the spectrum and the [[Jacobi inversion problem|Jacobi inversion problem]] for Abelian integrals was established. Recent development of these ideas made it possible to obtain explicit formulas, which express finite gap potentials by Riemann <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090780/s09078041.png" />-functions (see [[#References|[1]]], [[#References|[5]]]).
+
The significance of inverse problems became greater after discovering a possibility to apply them to solve some non-linear evolution equations of mathematical physics. In particular, a relationship (see [[#References|[25]]]) between inverse problems for some Sturm–Liouville operators with a finite number of gaps in the spectrum and the [[Jacobi inversion problem|Jacobi inversion problem]] for Abelian integrals was established. Recent development of these ideas made it possible to obtain explicit formulas, which express finite gap potentials by Riemann $  \theta $-
 +
functions (see [[#References|[1]]], [[#References|[5]]]).
  
 
Below two versions of the statement and solution of inverse problems will be considered.
 
Below two versions of the statement and solution of inverse problems will be considered.
  
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090780/s09078042.png" />. Given a known spectral function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090780/s09078043.png" />, find a differential equation in the form
+
$  1 $.  
 +
Given a known spectral function $  \rho ( \lambda ) $,  
 +
find a differential equation in the form
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090780/s09078044.png" /></td> </tr></table>
+
$$
 +
l[ y]  = - y  ^ {\prime\prime} + q( x) y
 +
$$
  
with a real locally-summable potential <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090780/s09078045.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090780/s09078046.png" />, and a number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090780/s09078047.png" /> appearing in the boundary condition
+
with a real locally-summable potential $  q( x) $,
 +
0 \leq  x < \infty $,  
 +
and a number $  h $
 +
appearing in the boundary condition
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090780/s09078048.png" /></td> <td valign="top" style="width:5%;text-align:right;">(3)</td></tr></table>
+
$$ \tag{3 }
 +
y  ^  \prime  ( 0) - hy( 0)  = 0.
 +
$$
  
 
To solve this problem it is assumed that
 
To solve this problem it is assumed that
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090780/s09078049.png" /></td> <td valign="top" style="width:5%;text-align:right;">(4)</td></tr></table>
+
$$ \tag{4 }
 +
\Phi ( x)  = \int\limits _ {- \infty } ^ { {+ }  \infty }
 +
\frac{1 - \cos  \sqrt \lambda x } \lambda
 +
 
 +
d \rho ( \lambda ),\ \
 +
0 < x < \infty ,
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090780/s09078050.png" /></td> </tr></table>
+
$$
 +
f( x, y)  =
 +
\frac{1}{2}
 +
\{ \Phi  ^ {\prime\prime} ( x+ y) + \Phi  ^ {\prime\prime} ( | x- y | ) \} .
 +
$$
  
 
It turns out that the integral equation
 
It turns out that the integral equation
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090780/s09078051.png" /></td> <td valign="top" style="width:5%;text-align:right;">(5)</td></tr></table>
+
$$ \tag{5 }
 +
f( x, y) + K( x, y) + \int\limits _ { 0 } ^ { x }  K( x, t) f( t, y)  dt  = 0,\ \
 +
0 \leq  y \leq  x,
 +
$$
 +
 
 +
has a unique solution  $  K( x, y) $
 +
for each fixed  $  x $.
 +
The potential  $  q $
 +
is defined by the formula
  
has a unique solution <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090780/s09078052.png" /> for each fixed <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090780/s09078053.png" />. The potential <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090780/s09078054.png" /> is defined by the formula
+
$$
 +
q( x)  = 2
 +
\frac{dK( x, x) }{dx}
 +
,
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090780/s09078055.png" /></td> </tr></table>
+
and the number  $  h $
 +
in (3) is given by the formula  $  h = K( 0, 0) $(
 +
see [[#References|[14]]]). The solution  $  \phi ( x, y) $
 +
of the equation  $  l [ y] = \lambda y $
 +
and satisfying the boundary conditions  $  \phi ( 0, \lambda ) = 1 $
 +
and  $  \phi  ^  \prime  ( 0, \lambda ) = h $
 +
can be found by the formula
  
and the number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090780/s09078056.png" /> in (3) is given by the formula <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090780/s09078057.png" /> (see [[#References|[14]]]). The solution <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090780/s09078058.png" /> of the equation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090780/s09078059.png" /> and satisfying the boundary conditions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090780/s09078060.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090780/s09078061.png" /> can be found by the formula
+
$$
 +
\phi ( x, \lambda ) = \cos  \sqrt \lambda x + \int\limits _ { 0 } ^ { x }  K( x, t) \
 +
\cos  \sqrt \lambda t  dt.
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090780/s09078062.png" /></td> </tr></table>
+
Further, a non-decreasing function  $  \rho ( \lambda ) $,
 +
$  - \infty < \lambda < \infty $,
 +
is the spectral function for some problem  $  - y  ^ {\prime\prime} + q( x) y = \lambda y $,
 +
$  0 \leq  x < \infty $,
 +
$  y  ^  \prime  ( 0) - hy( 0) = 0 $,
 +
where  $  q $
 +
is a real-valued function with  $  m $
 +
locally-summable derivatives and  $  h $
 +
is a real number, if and only if the function  $  \Phi ( x) $
 +
constructed from  $  \rho ( \lambda ) $
 +
by formula (4) has  $  m+ 3 $
 +
locally-summable derivatives and  $  \Phi  ^ {\prime\prime} (+ 0) = - h $(
 +
see [[#References|[14]]], [[#References|[17]]], [[#References|[9]]]). In a number of particular cases,  $  q $
 +
and  $  h $
 +
can be found effectively from the function  $  \rho ( \lambda ) $.  
 +
For example, let
  
Further, a non-decreasing function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090780/s09078063.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090780/s09078064.png" />, is the spectral function for some problem <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090780/s09078065.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090780/s09078066.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090780/s09078067.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090780/s09078068.png" /> is a real-valued function with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090780/s09078069.png" /> locally-summable derivatives and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090780/s09078070.png" /> is a real number, if and only if the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090780/s09078071.png" /> constructed from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090780/s09078072.png" /> by formula (4) has <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090780/s09078073.png" /> locally-summable derivatives and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090780/s09078074.png" /> (see [[#References|[14]]], [[#References|[17]]], [[#References|[9]]]). In a number of particular cases, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090780/s09078075.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090780/s09078076.png" /> can be found effectively from the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090780/s09078077.png" />. For example, let
+
$$
 +
\rho ( \lambda ) = \left \{
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090780/s09078078.png" /></td> </tr></table>
+
\begin{array}{lll}
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090780/s09078079.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090780/s09078080.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090780/s09078081.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090780/s09078082.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090780/s09078083.png" /> is a positive number. In this case the integral equation (5) is an equation with degenerate kernel <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090780/s09078084.png" /> and its solution is
+
\frac{2} \pi
 +
\sqrt \lambda + \alpha \chi ( \lambda - \lambda _ {0} )  & \textrm{ for }  &\lambda > 0,  \\
 +
\alpha \chi ( \lambda - \lambda _ {0} ) & \textrm{ for }  &\lambda < 0,  \\
 +
\end{array}
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090780/s09078085.png" /></td> </tr></table>
+
\right .$$
  
Now the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090780/s09078086.png" /> and the number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090780/s09078087.png" /> are determined by the formulas
+
where  $  \chi ( \lambda ) = 0 $
 +
for  $  \lambda < 0 $
 +
and  $  \chi ( \lambda ) = 1 $
 +
for  $  \lambda > 0 $
 +
and $  \alpha $
 +
is a positive number. In this case the integral equation (5) is an equation with degenerate kernel  $  f( x, y) = \alpha  \cos  \sqrt {\lambda _ {0} } x  \cos  \sqrt {\lambda _ {0} } y $
 +
and its solution is
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090780/s09078088.png" /></td> </tr></table>
+
$$
 +
K( x, y)  = -  
 +
\frac{\alpha  \cos  \sqrt {\lambda _ {0} } x  \cos  \sqrt {\lambda _ {0}  } y }{1 + \alpha \int\limits _ { 0 } ^ { x }  \cos  ^ {2}  \sqrt {\lambda _ {0} } t  dt }
 +
.
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090780/s09078089.png" /></td> </tr></table>
+
Now the function  $  q $
 +
and the number  $  h $
 +
are determined by the formulas
  
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090780/s09078090.png" />. Let a real-valued function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090780/s09078091.png" /> satisfy the inequality
+
$$
 +
q( x)  = 2
 +
\frac{dK( x, x) }{dx}
 +
  = - 2
 +
\frac{d}{dx}
 +
\left (
 +
\frac{\alpha \
 +
\cos  ^ {2}  \sqrt {
 +
\lambda _ {0} } x }{1 + \alpha \int\limits _ { 0 } ^ { x }  \cos  ^ {2}  \sqrt {\lambda _ {0} } t  dt }
 +
\right ) ,
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090780/s09078092.png" /></td> <td valign="top" style="width:5%;text-align:right;">(6)</td></tr></table>
+
$$
 +
= K( 0, 0)  = - \alpha .
 +
$$
 +
 
 +
$  2 $.  
 +
Let a real-valued function  $  q $
 +
satisfy the inequality
 +
 
 +
$$ \tag{6 }
 +
\int\limits _ { 0 } ^  \infty  x | q( x) |  dx  < \infty .
 +
$$
  
 
Then the boundary value problem
 
Then the boundary value problem
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090780/s09078093.png" /></td> <td valign="top" style="width:5%;text-align:right;">(7)</td></tr></table>
+
$$ \tag{7 }
 +
- y  ^ {\prime\prime} + q( x) y  = \lambda  ^ {2} y,\ \
 +
0 < x < \infty ,
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090780/s09078094.png" /></td> <td valign="top" style="width:5%;text-align:right;">(7prm)</td></tr></table>
+
$$ \tag{7'}
 +
y( 0)  = 0,
 +
$$
  
has bounded solutions for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090780/s09078095.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090780/s09078096.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090780/s09078097.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090780/s09078098.png" />. Moreover, these solutions satisfy for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090780/s09078099.png" /> the asymptotic formulas
+
has bounded solutions for $  \lambda  ^ {2} > 0 $
 +
and $  \lambda = i \lambda _ {k} $,
 +
$  \lambda _ {k} > 0 $,  
 +
$  k = 1 \dots n $.  
 +
Moreover, these solutions satisfy for $  x \rightarrow \infty $
 +
the asymptotic formulas
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090780/s090780100.png" /></td> </tr></table>
+
$$
 +
y( x, \lambda )  = e ^ {- i \lambda x } - S( \lambda ) e ^ {i \lambda x } +
 +
o( 1),\ \
 +
0 < \lambda  ^ {2} < \infty ,
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090780/s090780101.png" /></td> </tr></table>
+
$$
 +
y( x, i \lambda _ {k} )  = m _ {k} e ^ {- \lambda _ {k} x } [ 1+ o( 1)] ,\  m _ {k} > 0,\  k = 1 \dots n.
 +
$$
  
Here <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090780/s090780102.png" /> is a normalization factor: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090780/s090780103.png" />. The set of values <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090780/s090780104.png" /> is called the scattering data for the boundary value problem (7), (7prm). It is required to reconstruct the potential <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090780/s090780105.png" /> from the scattering data.
+
Here $  m _ {k} $
 +
is a normalization factor: $  \int _ {0}  ^  \infty  | y( x , i \lambda _ {k} ) |  ^ {2}  dx = 1 $.  
 +
The set of values $  \{ {S(\lambda) } : {-\infty<\lambda<\infty;  \lambda _ {k} , m _ {k} ;  k= 1\dots n } \} $
 +
is called the scattering data for the boundary value problem (7), (7'}). It is required to reconstruct the potential $  q $
 +
from the scattering data.
  
To solve this problem, a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090780/s090780106.png" /> is constructed by the formula
+
To solve this problem, a function $  F $
 +
is constructed by the formula
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090780/s090780107.png" /></td> </tr></table>
+
$$
 +
F( x)  = \sum _ { k= } 1 ^ { n }  m _ {k}  ^ {2} e ^ {- \lambda _ {k} x } +
 +
\frac{1}{2 \pi }
 +
\int\limits _ {- \infty } ^  \infty  [ 1- S( \lambda )] e ^ {i \lambda x }  d \lambda
 +
$$
  
 
and one considers the following equation:
 
and one considers the following equation:
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090780/s090780108.png" /></td> <td valign="top" style="width:5%;text-align:right;">(8)</td></tr></table>
+
$$ \tag{8 }
 +
F( x+ y) + K( x, y) + \int\limits _ { x } ^  \infty  K( x, t) F( t+ y)  dt  = 0.
 +
$$
  
This equation has a unique solution <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090780/s090780109.png" /> for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090780/s090780110.png" />. After this equation has been solved, the potential <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090780/s090780111.png" /> is determined by the formula
+
This equation has a unique solution $  K( x, y) $
 +
for any $  x \geq  0 $.  
 +
After this equation has been solved, the potential $  q $
 +
is determined by the formula
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090780/s090780112.png" /></td> </tr></table>
+
$$
 +
q( x)  = - 2
 +
\frac{dK( x, x) }{dx}
 +
.
 +
$$
  
For a set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090780/s090780113.png" /> to be the scattering data for some boundary value problem of the form (7), (7prm) with a real potential <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090780/s090780114.png" /> which satisfies the condition (6), it is necessary and sufficient that the following conditions are satisfied (see [[#References|[1]]]):
+
For a set $  \{ {S(\lambda) } : {-\infty<\lambda<\infty;  \lambda _ {k} , m _ {k} ;  \lambda _ {k} > 0,  m _ {k} > 0,  k= 1\dots n } \} $
 +
to be the scattering data for some boundary value problem of the form (7), (7'}) with a real potential $  q $
 +
which satisfies the condition (6), it is necessary and sufficient that the following conditions are satisfied (see [[#References|[1]]]):
  
a) the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090780/s090780115.png" /> is continuous on the whole line, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090780/s090780116.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090780/s090780117.png" /> tends to zero as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090780/s090780118.png" /> and is the Fourier transform of the function
+
a) the function $  S( \lambda ) $
 +
is continuous on the whole line, $  \overline{ {S( \lambda ) }}\; = S(- \lambda ) = [ S( \lambda )]  ^ {-} 1 $,  
 +
$  1- S( \lambda ) $
 +
tends to zero as $  | \lambda | \rightarrow \infty $
 +
and is the Fourier transform of the function
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090780/s090780119.png" /></td> </tr></table>
+
$$
 +
F _ {S} ( x)  =
 +
\frac{1}{2 \pi }
 +
\int\limits _ {- \infty } ^  \infty  [ 1- S( \lambda )] e ^ {i
 +
\lambda x }  d \lambda ,
 +
$$
  
which is representable as a sum of two functions, one of which belongs to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090780/s090780120.png" />, while the second is bounded and belongs to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090780/s090780121.png" />. On the half-line <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090780/s090780122.png" /> the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090780/s090780123.png" /> has a derivative <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090780/s090780124.png" /> satisfying the condition <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090780/s090780125.png" />;
+
which is representable as a sum of two functions, one of which belongs to $  L _ {1} (- \infty , \infty ) $,  
 +
while the second is bounded and belongs to $  L _ {2} (- \infty , \infty ) $.  
 +
On the half-line $  0 < x < \infty $
 +
the function $  F _ {S} ( x) $
 +
has a derivative $  F _ {S} ^ { \prime } ( x) $
 +
satisfying the condition $  \int _ {0}  ^  \infty  x | F _ {S} ^ { \prime } ( x) |  dx < \infty $;
  
b) the increment of the argument in the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090780/s090780126.png" /> is connected with the number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090780/s090780127.png" /> of negative eigenvalues (i.e. of the numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090780/s090780128.png" />) of the boundary value problem (7), (7prm) by the formula
+
b) the increment of the argument in the function $  S( \lambda ) $
 +
is connected with the number $  n $
 +
of negative eigenvalues (i.e. of the numbers $  - \lambda _ {1}  ^ {2} \dots - \lambda _ {n}  ^ {2} $)  
 +
of the boundary value problem (7), (7'}) by the formula
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090780/s090780129.png" /></td> </tr></table>
+
$$
 +
=
 +
\frac{ \mathop{\rm ln}  S(+ 0) -  \mathop{\rm ln}  S(+ \infty ) }{2 \pi i }
 +
- 1- S(
 +
\frac{0)}{4}
 +
.
 +
$$
  
The integral equation (8) for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090780/s090780130.png" /> has an explicit solution if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090780/s090780131.png" /> is a rational function. Solutions of the equation (7) and the potential <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090780/s090780132.png" /> are obtained in this case as rational functions in the trigonometric and hyperbolic functions. For example, if
+
The integral equation (8) for $  K( x, y) $
 +
has an explicit solution if $  S( \lambda ) $
 +
is a rational function. Solutions of the equation (7) and the potential $  q $
 +
are obtained in this case as rational functions in the trigonometric and hyperbolic functions. For example, if
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090780/s090780133.png" /></td> </tr></table>
+
$$
 +
S ( \lambda )  =
 +
\frac{( \lambda + i)( \lambda + 2i) }{( \lambda - i)( \lambda
 +
- 2i) }
 +
,\ \
 +
\lambda _ {1}  = 1,\ \
 +
m _ {1}  = \sqrt 6 ,
 +
$$
  
 
then the corresponding potential has the form
 
then the corresponding potential has the form
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090780/s090780134.png" /></td> </tr></table>
+
$$
 +
q( x)  = -
 +
\frac{24}{( 2  \cosh  2x- \sinh  2x)  ^ {2} }
 +
.
 +
$$
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  V.A. Marchenko,  "Sturm–Liouville operators and applications" , Birkhäuser  (1986)  (Translated from Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  Z.S. Agranovich,  V.A. Marchenko,  "The inverse problem in scattering theory" , Khar'kov  (1960)  (In Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  C. Shadan,  P. Sabatier,  "Inverse problems in quantum scattering theory" , Springer  (1989)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  B.M. Levitan,  "Generalized translation operators and some of their applications" , Israel Program Sci. Transl.  (1964)  (Translated from Russian)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top"> , ''The theory of solitons: methods of the inverse problem'' , Moscow  (1980)  (In Russian)</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top">  Yu.M. [Yu.M. Berezanskii] Berezanskiy,  "Expansion in eigenfunctions of selfadjoint operators" , Amer. Math. Soc.  (1968)  (Translated from Russian)</TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top">  L.D. Faddeev,  "The inverse problem in quantum scattering theory"  ''Uspekhi Mat. Nauk'' , '''14''' :  4  (1959)  pp. 57–119  (In Russian)</TD></TR><TR><TD valign="top">[8]</TD> <TD valign="top">  L.D. Faddeev,  "Inverse problem of quantum scattering theory II"  ''J. Soviet. Math.'' , '''5''' :  3  (1976)  pp. 334–450  ''Itogi Nauk. i Tekhn. Sovremen. Probl. Mat.'' , '''3'''  (1974)  pp. 93–180</TD></TR><TR><TD valign="top">[9]</TD> <TD valign="top">  B.M. Levitan,  M.G. Gasymov,  "Determination of a differential equation by two of its spectra"  ''Russian Math. Surveys'' , '''19''' :  2  (1964)  pp. 1–64  ''Uspkehi Mat. Nauk'' , '''19''' :  2  (1964)  pp. 3–63</TD></TR><TR><TD valign="top">[10]</TD> <TD valign="top">  V. Ambarzumian,  ''Z. Phys.'' , '''53'''  (1929)  pp. 690–695</TD></TR><TR><TD valign="top">[11]</TD> <TD valign="top">  G. Borg,  "Eine Umkehrung der Sturm–Liouvilleschen Eigenwertaufgabe, Bestimmung der Differentialgleichung durch die Eigenwerte"  ''Acta Math.'' , '''78'''  (1946)  pp. 1–96</TD></TR><TR><TD valign="top">[12]</TD> <TD valign="top">  N. Levinson,  "On the uniqueness of the potential in a Schrödinger equation for a given asymptotic phase"  ''Danske Vid. Selsk. Mat. -Fys. Medd.'' , '''25''' :  9  (1949)  pp. 1–29</TD></TR><TR><TD valign="top">[13]</TD> <TD valign="top">  V.A. Marchenko,  "On the theory of a second-order differential operator"  ''Dokl. Akad. Nauk SSSR'' , '''72''' :  3  (1950)  pp. 457–460  (In Russian)</TD></TR><TR><TD valign="top">[14]</TD> <TD valign="top">  I.M. Gel'fand,  B.M. Levitan,  "On the determination of a differential equation from its spectral function"  ''Izv. Akad. Nauk SSSR Ser. Mat.'' , '''15''' :  4  (1951)  pp. 309–360  (In Russian)</TD></TR><TR><TD valign="top">[15a]</TD> <TD valign="top">  M.G. Krein,  "Solution of the inverse Sturm–Liouville problem"  ''Dokl. Akad. Nauk SSSR'' , '''76''' :  1  (1951)  pp. 21–24  (In Russian)</TD></TR><TR><TD valign="top">[15b]</TD> <TD valign="top">  M.G. Krein,  "On inverse problems for a nonhomogeneous chord"  ''Dokl. Akad. Nauk SSSR'' , '''82''' :  5  (1952)  pp. 669–672  (In Russian)</TD></TR><TR><TD valign="top">[15c]</TD> <TD valign="top">  M.G. Krein,  "On the transfer function of a one-dimensional second-order boundary problem"  ''Dokl. Akad. Nauk SSSR'' , '''88''' :  3  (1953)  pp. 405–408  (In Russian)</TD></TR><TR><TD valign="top">[16]</TD> <TD valign="top">  V.A. Marchenko,  "On reconstructing the potential energy from phases of scattered waves"  ''Dokl. Akad. Nauk SSSR'' , '''104''' :  5  (1955)  pp. 695–698  (In Russian)</TD></TR><TR><TD valign="top">[17]</TD> <TD valign="top">  B.Ya. Levin,  "Transformations of Fourier and Laplace types by means of solutions of second-order differential equations"  ''Dokl. Akad. Nauk SSSR'' , '''106''' :  2  (1956)  pp. 187–190  (In Russian)</TD></TR><TR><TD valign="top">[18]</TD> <TD valign="top">  R. Newton,  R. Jost,  "The construction of potentials from the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090780/s090780135.png" />-matrix for systems of differential equations"  ''Nuovo Cimento'' , '''1''' :  4  (1955)  pp. 590–622</TD></TR><TR><TD valign="top">[19]</TD> <TD valign="top">  M.G. Gasymov,  B.M. Levitan,  "Determination of Dirac's system from the scattering phase"  ''Soviet Math. Dokl.'' , '''7''' :  2  (1966)  pp. 543–547  ''Dokl. Akad. Nauk SSSR'' , '''167''' :  6  (1966)  pp. 1219–1222</TD></TR><TR><TD valign="top">[20]</TD> <TD valign="top">  M.G. Gasymov,  "The inverse problem of scattering theory for a system of Dirac equations of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090780/s090780136.png" />"  ''Trudy Moskov. Mat. Obshch.'' , '''19'''  (1968)  pp. 41–112  (In Russian)</TD></TR><TR><TD valign="top">[21]</TD> <TD valign="top">  F.S. Rofe-Beketov,  "The spectral matrix and the inverse Sturm–Liouville problem on the axis <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090780/s090780137.png" />"  ''Teor. Funktsii, Funktional. Analiz. i Prilozhen.'' :  4  (1967)  pp. 189–197  (In Russian)</TD></TR><TR><TD valign="top">[22]</TD> <TD valign="top">  B.M. Levitan,  "The inverse problem in quantum scattering theory at fixed energy" , ''Probl. Mekh. i Mat. Fiz. (I.G. Petrovskii)'' , Moscow  (1976)  pp. 166–207  (In Russian)</TD></TR><TR><TD valign="top">[23]</TD> <TD valign="top">  N.I. Akhiezer,  "A continuous analogue of orthogonal polynomials on a system of intervals"  ''Soviet Math. Dokl.'' , '''2''' :  6  (1961)  pp. 1409–1412  ''Dokl. Akad. SSSR'' , '''141''' :  2  (1961)  pp. 263–266</TD></TR><TR><TD valign="top">[24]</TD> <TD valign="top">  G.Sh. Guseinov,  "The inverse problem of scattering theory for a second-order difference equation on the whole axis"  ''Soviet Math. Dokl.'' , '''17''' :  6  (1976)  pp. 1684–1688  ''Dokl. Akad. Nauk SSSR'' , '''231''' :  5  (1976)  pp. 1045–1048</TD></TR><TR><TD valign="top">[25]</TD> <TD valign="top">  B.M. Levitan,  "Inverse Sturm–Liouville problems" , VNU  (1987)  (Translated from Russian)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  V.A. Marchenko,  "Sturm–Liouville operators and applications" , Birkhäuser  (1986)  (Translated from Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  Z.S. Agranovich,  V.A. Marchenko,  "The inverse problem in scattering theory" , Khar'kov  (1960)  (In Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  C. Shadan,  P. Sabatier,  "Inverse problems in quantum scattering theory" , Springer  (1989)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  B.M. Levitan,  "Generalized translation operators and some of their applications" , Israel Program Sci. Transl.  (1964)  (Translated from Russian)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top"> , ''The theory of solitons: methods of the inverse problem'' , Moscow  (1980)  (In Russian)</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top">  Yu.M. [Yu.M. Berezanskii] Berezanskiy,  "Expansion in eigenfunctions of selfadjoint operators" , Amer. Math. Soc.  (1968)  (Translated from Russian)</TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top">  L.D. Faddeev,  "The inverse problem in quantum scattering theory"  ''Uspekhi Mat. Nauk'' , '''14''' :  4  (1959)  pp. 57–119  (In Russian)</TD></TR><TR><TD valign="top">[8]</TD> <TD valign="top">  L.D. Faddeev,  "Inverse problem of quantum scattering theory II"  ''J. Soviet. Math.'' , '''5''' :  3  (1976)  pp. 334–450  ''Itogi Nauk. i Tekhn. Sovremen. Probl. Mat.'' , '''3'''  (1974)  pp. 93–180</TD></TR><TR><TD valign="top">[9]</TD> <TD valign="top">  B.M. Levitan,  M.G. Gasymov,  "Determination of a differential equation by two of its spectra"  ''Russian Math. Surveys'' , '''19''' :  2  (1964)  pp. 1–64  ''Uspkehi Mat. Nauk'' , '''19''' :  2  (1964)  pp. 3–63</TD></TR><TR><TD valign="top">[10]</TD> <TD valign="top">  V. Ambarzumian,  ''Z. Phys.'' , '''53'''  (1929)  pp. 690–695</TD></TR><TR><TD valign="top">[11]</TD> <TD valign="top">  G. Borg,  "Eine Umkehrung der Sturm–Liouvilleschen Eigenwertaufgabe, Bestimmung der Differentialgleichung durch die Eigenwerte"  ''Acta Math.'' , '''78'''  (1946)  pp. 1–96</TD></TR><TR><TD valign="top">[12]</TD> <TD valign="top">  N. Levinson,  "On the uniqueness of the potential in a Schrödinger equation for a given asymptotic phase"  ''Danske Vid. Selsk. Mat. -Fys. Medd.'' , '''25''' :  9  (1949)  pp. 1–29</TD></TR><TR><TD valign="top">[13]</TD> <TD valign="top">  V.A. Marchenko,  "On the theory of a second-order differential operator"  ''Dokl. Akad. Nauk SSSR'' , '''72''' :  3  (1950)  pp. 457–460  (In Russian)</TD></TR><TR><TD valign="top">[14]</TD> <TD valign="top">  I.M. Gel'fand,  B.M. Levitan,  "On the determination of a differential equation from its spectral function"  ''Izv. Akad. Nauk SSSR Ser. Mat.'' , '''15''' :  4  (1951)  pp. 309–360  (In Russian)</TD></TR><TR><TD valign="top">[15a]</TD> <TD valign="top">  M.G. Krein,  "Solution of the inverse Sturm–Liouville problem"  ''Dokl. Akad. Nauk SSSR'' , '''76''' :  1  (1951)  pp. 21–24  (In Russian)</TD></TR><TR><TD valign="top">[15b]</TD> <TD valign="top">  M.G. Krein,  "On inverse problems for a nonhomogeneous chord"  ''Dokl. Akad. Nauk SSSR'' , '''82''' :  5  (1952)  pp. 669–672  (In Russian)</TD></TR><TR><TD valign="top">[15c]</TD> <TD valign="top">  M.G. Krein,  "On the transfer function of a one-dimensional second-order boundary problem"  ''Dokl. Akad. Nauk SSSR'' , '''88''' :  3  (1953)  pp. 405–408  (In Russian)</TD></TR><TR><TD valign="top">[16]</TD> <TD valign="top">  V.A. Marchenko,  "On reconstructing the potential energy from phases of scattered waves"  ''Dokl. Akad. Nauk SSSR'' , '''104''' :  5  (1955)  pp. 695–698  (In Russian)</TD></TR><TR><TD valign="top">[17]</TD> <TD valign="top">  B.Ya. Levin,  "Transformations of Fourier and Laplace types by means of solutions of second-order differential equations"  ''Dokl. Akad. Nauk SSSR'' , '''106''' :  2  (1956)  pp. 187–190  (In Russian)</TD></TR><TR><TD valign="top">[18]</TD> <TD valign="top">  R. Newton,  R. Jost,  "The construction of potentials from the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090780/s090780135.png" />-matrix for systems of differential equations"  ''Nuovo Cimento'' , '''1''' :  4  (1955)  pp. 590–622</TD></TR><TR><TD valign="top">[19]</TD> <TD valign="top">  M.G. Gasymov,  B.M. Levitan,  "Determination of Dirac's system from the scattering phase"  ''Soviet Math. Dokl.'' , '''7''' :  2  (1966)  pp. 543–547  ''Dokl. Akad. Nauk SSSR'' , '''167''' :  6  (1966)  pp. 1219–1222</TD></TR><TR><TD valign="top">[20]</TD> <TD valign="top">  M.G. Gasymov,  "The inverse problem of scattering theory for a system of Dirac equations of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090780/s090780136.png" />"  ''Trudy Moskov. Mat. Obshch.'' , '''19'''  (1968)  pp. 41–112  (In Russian)</TD></TR><TR><TD valign="top">[21]</TD> <TD valign="top">  F.S. Rofe-Beketov,  "The spectral matrix and the inverse Sturm–Liouville problem on the axis <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090780/s090780137.png" />"  ''Teor. Funktsii, Funktional. Analiz. i Prilozhen.'' :  4  (1967)  pp. 189–197  (In Russian)</TD></TR><TR><TD valign="top">[22]</TD> <TD valign="top">  B.M. Levitan,  "The inverse problem in quantum scattering theory at fixed energy" , ''Probl. Mekh. i Mat. Fiz. (I.G. Petrovskii)'' , Moscow  (1976)  pp. 166–207  (In Russian)</TD></TR><TR><TD valign="top">[23]</TD> <TD valign="top">  N.I. Akhiezer,  "A continuous analogue of orthogonal polynomials on a system of intervals"  ''Soviet Math. Dokl.'' , '''2''' :  6  (1961)  pp. 1409–1412  ''Dokl. Akad. SSSR'' , '''141''' :  2  (1961)  pp. 263–266</TD></TR><TR><TD valign="top">[24]</TD> <TD valign="top">  G.Sh. Guseinov,  "The inverse problem of scattering theory for a second-order difference equation on the whole axis"  ''Soviet Math. Dokl.'' , '''17''' :  6  (1976)  pp. 1684–1688  ''Dokl. Akad. Nauk SSSR'' , '''231''' :  5  (1976)  pp. 1045–1048</TD></TR><TR><TD valign="top">[25]</TD> <TD valign="top">  B.M. Levitan,  "Inverse Sturm–Liouville problems" , VNU  (1987)  (Translated from Russian)</TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====
The inverse problem for the Schrödinger operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090780/s090780138.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090780/s090780139.png" />, has been investigated in [[#References|[a1]]]. (See also [[Schrödinger equation|Schrödinger equation]].) In order to derive necessary and sufficient conditions in the scattering data to correspond with the equation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090780/s090780140.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090780/s090780141.png" />, it is assumed in [[#References|[a1]]] that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090780/s090780142.png" /> satisfies <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090780/s090780143.png" />, cf. [[#References|[6]]]. The condition on the Fourier transforms <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090780/s090780144.png" />, are analogous to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090780/s090780145.png" />, defined above is also changed: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090780/s090780146.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090780/s090780147.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090780/s090780148.png" />.
+
The inverse problem for the Schrödinger operator $  - ( {d  ^ {2} } / {dx  ^ {2} } ) + q( x) $,  
 +
$  - \infty < x < + \infty $,  
 +
has been investigated in [[#References|[a1]]]. (See also [[Schrödinger equation|Schrödinger equation]].) In order to derive necessary and sufficient conditions in the scattering data to correspond with the equation $  - y  ^ {\prime\prime} + q( x) y= \lambda  ^ {2} y $,  
 +
$  - \infty < x< \infty $,  
 +
it is assumed in [[#References|[a1]]] that $  q $
 +
satisfies $  \int _ {- \infty }  ^ {+ \infty } | q( x) | ( 1+ x  ^ {2} )  dx < \infty $,  
 +
cf. [[#References|[6]]]. The condition on the Fourier transforms $  F _ {i} $,  
 +
are analogous to $  F _ {S} $,  
 +
defined above is also changed: $  \int _ {a}  ^  \infty  | F _ {1} ^ { \prime } ( x) |  dx $,  
 +
$  \int _ {- \infty }  ^ {a} | F _ {2} ^ { \prime } ( x) | ( 1+ x  ^ {2} )  dx < \infty $
 +
for all $  - \infty < a < + \infty $.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  P. Deift,  E. Trubowitz,  "Inverse scattering on the line"  ''Comm. Pure Appl. Math.'' , '''32'''  (1979)  pp. 121–251</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  J. Pöschel,  E. Trubowitz,  "Inverse spectral theory" , Acad. Press  (1987)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  A.G. Ramm,  "Inverse scattering on half-line"  ''J. Math. Anal. Appl.'' , '''133'''  (1988)  pp. 543–572</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  A.G. Ramm,  B.A. Taylor,  "Example of a potential in one-dimensional scattering problem for which there are infinitely many purely imaginary resonances"  ''Phys. Lett.'' , '''124A'''  (1987)  pp. 313–319</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  L.E. Andersson,  "Inverse eigenvalue problems with discontinuous coefficients"  ''Inverse Probl.'' , '''4'''  (1988)  pp. 353–397</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top">  A.G. Ramm,  "Recovery of the potential from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090780/s090780149.png" />-function"  ''Math. Reports Canad. Acad. Sci.'' , '''9'''  (1987)  pp. 177–182</TD></TR><TR><TD valign="top">[a7a]</TD> <TD valign="top">  A.G. Ramm,  "Necessary and sufficient conditions on the scattering data for the potential to be in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090780/s090780150.png" /> for the Schrödinger operator on the half-line"  ''Inverse Probl.'' , '''3'''  (1987)  pp. L71–76</TD></TR><TR><TD valign="top">[a7b]</TD> <TD valign="top">  A.G. Ramm,  "An inverse problem for the Helmholtz equation in a semi-infinite medium"  ''Inverse Probl.'' , '''3'''  (1987)  pp. L19–22</TD></TR><TR><TD valign="top">[a8]</TD> <TD valign="top">  E.L. Isaacson,  E. Trubowitz,  "The inverse Sturm–Liouville problem I"  ''Comm. Pure Appl. Math.'' , '''36'''  (1983)  pp. 767–784</TD></TR><TR><TD valign="top">[a9]</TD> <TD valign="top">  E.L. Isaacson,  H.P. McKean,  E. Trubowitz,  "The inverse Sturm–Liouville problem II"  ''Comm. Pure Appl. Math.'' , '''37'''  (1984)  pp. 1–12</TD></TR><TR><TD valign="top">[a10]</TD> <TD valign="top">  B.E.J. Dahlberg,  E. Trubowitz,  "The inverse Sturm–Liouville problem III"  ''Comm. Pure Appl. Math.'' , '''37'''  (1984)  pp. 255–267</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  P. Deift,  E. Trubowitz,  "Inverse scattering on the line"  ''Comm. Pure Appl. Math.'' , '''32'''  (1979)  pp. 121–251</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  J. Pöschel,  E. Trubowitz,  "Inverse spectral theory" , Acad. Press  (1987)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  A.G. Ramm,  "Inverse scattering on half-line"  ''J. Math. Anal. Appl.'' , '''133'''  (1988)  pp. 543–572</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  A.G. Ramm,  B.A. Taylor,  "Example of a potential in one-dimensional scattering problem for which there are infinitely many purely imaginary resonances"  ''Phys. Lett.'' , '''124A'''  (1987)  pp. 313–319</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  L.E. Andersson,  "Inverse eigenvalue problems with discontinuous coefficients"  ''Inverse Probl.'' , '''4'''  (1988)  pp. 353–397</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top">  A.G. Ramm,  "Recovery of the potential from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090780/s090780149.png" />-function"  ''Math. Reports Canad. Acad. Sci.'' , '''9'''  (1987)  pp. 177–182</TD></TR><TR><TD valign="top">[a7a]</TD> <TD valign="top">  A.G. Ramm,  "Necessary and sufficient conditions on the scattering data for the potential to be in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090780/s090780150.png" /> for the Schrödinger operator on the half-line"  ''Inverse Probl.'' , '''3'''  (1987)  pp. L71–76</TD></TR><TR><TD valign="top">[a7b]</TD> <TD valign="top">  A.G. Ramm,  "An inverse problem for the Helmholtz equation in a semi-infinite medium"  ''Inverse Probl.'' , '''3'''  (1987)  pp. L19–22</TD></TR><TR><TD valign="top">[a8]</TD> <TD valign="top">  E.L. Isaacson,  E. Trubowitz,  "The inverse Sturm–Liouville problem I"  ''Comm. Pure Appl. Math.'' , '''36'''  (1983)  pp. 767–784</TD></TR><TR><TD valign="top">[a9]</TD> <TD valign="top">  E.L. Isaacson,  H.P. McKean,  E. Trubowitz,  "The inverse Sturm–Liouville problem II"  ''Comm. Pure Appl. Math.'' , '''37'''  (1984)  pp. 1–12</TD></TR><TR><TD valign="top">[a10]</TD> <TD valign="top">  B.E.J. Dahlberg,  E. Trubowitz,  "The inverse Sturm–Liouville problem III"  ''Comm. Pure Appl. Math.'' , '''37'''  (1984)  pp. 255–267</TD></TR></table>

Latest revision as of 14:55, 7 June 2020


A problem in which it is required to reconstruct a function (a potential) $ q $ from some spectral characteristics of the operator $ A $ generated by the differential expression $ l[ y] = - y ^ {\prime\prime} + q( x) y $ and some boundary conditions in the Hilbert space $ L _ {2} ( a, b) $, where $ x $ varies in a finite or infinite interval $ ( a, b) $. Moreover, one should also reconstruct the boundary conditions corresponding to the operator $ A $.

When studying inverse problems, the following natural questions arise: 1) to find out which spectral characteristics determine the operator $ A $ uniquely; 2) to give a method of reconstructing the operator $ A $ from these spectral characteristics; and 3) to find particular properties of the spectral characteristics considered. Depending on the choice of the spectral characteristics, different statements of inverse problems are possible (often arising in applications).

The first result concerning inverse problems (see [10]), which gave a start to the whole theory, is: Let $ \lambda _ {0} , \lambda _ {1} \dots $ be the eigenvalues for the problem

$$ \tag{1 } \left . \begin{array}{c} - y ^ {\prime\prime} + q( x) y = \lambda y ,\ 0 \leq x \leq \pi , \\ y ^ \prime ( 0) = y ^ \prime ( \pi ) = 0, \\ \end{array} \right \} $$

and let $ q $ be a real-valued continuous function on the interval $ [ 0, \pi ] $. If $ \lambda _ {n} = n ^ {2} $, $ n = 0, 1 \dots $ then $ q( x) \equiv 0 $.

A profound study of inverse problems started in the 1940's (see [11], [12]). Let $ \lambda _ {0} , \lambda _ {1} \dots $ be the eigenvalues for the equation (1) under the boundary conditions

$$ \tag{2 } y ^ \prime ( 0) - hy( 0) = 0,\ \ y ^ \prime ( \pi ) + Hy( \pi ) = 0 $$

( $ h $ and $ H $ are real numbers), and let $ \mu _ {0} , \mu _ {1} \dots $ be the eigenvalues of (1) under the boundary conditions

$$ y ^ \prime ( 0) - h _ {1} y( 0) = 0,\ \ y ^ \prime ( \pi ) + Hy( \pi ) = 0 ,\ \ h _ {1} \neq h. $$

Then the sequences $ \{ \lambda _ {n} \} $ and $ \{ \mu _ {n} \} $, $ n = 0, 1 \dots $ determine the function $ q $ and the numbers $ h $, $ h _ {1} $ and $ H $ uniquely. Moreover, if at least one eigenvalue in these problems has not been determined, then all the others do not determine the equation (1) uniquely. In particular, generally speaking, one spectrum does not determine the equation uniquely (the above-mentioned result is an exception to the general rule).

If the equation (1) is studied on the half-line $ ( 0, \infty ) $ and the potential function $ q $ satisfies the requirement

$$ \int\limits _ { 0 } ^ \infty x | q( x) | dx < \infty , $$

then the solution $ \phi ( x, \lambda ) $ of the problem $ - y ^ {\prime\prime} + q( x) y = \lambda ^ {2} y $, $ y( 0) = 0 $, has an asymptotic representation for $ x \rightarrow \infty $,

$$ \phi ( x, \lambda ) = M( \lambda ) \sin [ \lambda x + \delta ( \lambda )] + o( 1). $$

The function $ \delta ( \lambda ) $ is called the scattering phase. The main result is that if a problem (considered in the space $ L _ {2} ( 0, \infty ) $) does not have discrete eigenvalues, then the scattering phase determines the potential function $ q $ uniquely.

A decisive step in the further development of the theory of inverse problems was the application of so-called operator-transform techniques (see Sturm–Liouville equation), which was naturally developed in the framework of the theory of generalized shift operators (see [4]).

Applying operator transforms to inverse problems (see [13]) allowed one to generalize the above-mentioned theorems. Namely, it turned out that the most general inverse problem is the problem of reconstructing equation (1) from its spectral function (see Sturm–Liouville problem). It was shown that the spectral function determines this equation uniquely. Moreover, it is of no importance whether the considered interval is finite or infinite.

In principle, all inverse problems can be reduced to the inverse problem of reconstructing an operator from its spectral function. However, this way is not always the simplest; moreover, when following it, one often encounters difficulties in finding necessary and sufficient conditions for the spectral characteristics that are used to reconstruct the operator.

The significance of inverse problems became greater after discovering a possibility to apply them to solve some non-linear evolution equations of mathematical physics. In particular, a relationship (see [25]) between inverse problems for some Sturm–Liouville operators with a finite number of gaps in the spectrum and the Jacobi inversion problem for Abelian integrals was established. Recent development of these ideas made it possible to obtain explicit formulas, which express finite gap potentials by Riemann $ \theta $- functions (see [1], [5]).

Below two versions of the statement and solution of inverse problems will be considered.

$ 1 $. Given a known spectral function $ \rho ( \lambda ) $, find a differential equation in the form

$$ l[ y] = - y ^ {\prime\prime} + q( x) y $$

with a real locally-summable potential $ q( x) $, $ 0 \leq x < \infty $, and a number $ h $ appearing in the boundary condition

$$ \tag{3 } y ^ \prime ( 0) - hy( 0) = 0. $$

To solve this problem it is assumed that

$$ \tag{4 } \Phi ( x) = \int\limits _ {- \infty } ^ { {+ } \infty } \frac{1 - \cos \sqrt \lambda x } \lambda d \rho ( \lambda ),\ \ 0 < x < \infty , $$

$$ f( x, y) = \frac{1}{2} \{ \Phi ^ {\prime\prime} ( x+ y) + \Phi ^ {\prime\prime} ( | x- y | ) \} . $$

It turns out that the integral equation

$$ \tag{5 } f( x, y) + K( x, y) + \int\limits _ { 0 } ^ { x } K( x, t) f( t, y) dt = 0,\ \ 0 \leq y \leq x, $$

has a unique solution $ K( x, y) $ for each fixed $ x $. The potential $ q $ is defined by the formula

$$ q( x) = 2 \frac{dK( x, x) }{dx} , $$

and the number $ h $ in (3) is given by the formula $ h = K( 0, 0) $( see [14]). The solution $ \phi ( x, y) $ of the equation $ l [ y] = \lambda y $ and satisfying the boundary conditions $ \phi ( 0, \lambda ) = 1 $ and $ \phi ^ \prime ( 0, \lambda ) = h $ can be found by the formula

$$ \phi ( x, \lambda ) = \cos \sqrt \lambda x + \int\limits _ { 0 } ^ { x } K( x, t) \ \cos \sqrt \lambda t dt. $$

Further, a non-decreasing function $ \rho ( \lambda ) $, $ - \infty < \lambda < \infty $, is the spectral function for some problem $ - y ^ {\prime\prime} + q( x) y = \lambda y $, $ 0 \leq x < \infty $, $ y ^ \prime ( 0) - hy( 0) = 0 $, where $ q $ is a real-valued function with $ m $ locally-summable derivatives and $ h $ is a real number, if and only if the function $ \Phi ( x) $ constructed from $ \rho ( \lambda ) $ by formula (4) has $ m+ 3 $ locally-summable derivatives and $ \Phi ^ {\prime\prime} (+ 0) = - h $( see [14], [17], [9]). In a number of particular cases, $ q $ and $ h $ can be found effectively from the function $ \rho ( \lambda ) $. For example, let

$$ \rho ( \lambda ) = \left \{ \begin{array}{lll} \frac{2} \pi \sqrt \lambda + \alpha \chi ( \lambda - \lambda _ {0} ) & \textrm{ for } &\lambda > 0, \\ \alpha \chi ( \lambda - \lambda _ {0} ) & \textrm{ for } &\lambda < 0, \\ \end{array} \right .$$

where $ \chi ( \lambda ) = 0 $ for $ \lambda < 0 $ and $ \chi ( \lambda ) = 1 $ for $ \lambda > 0 $ and $ \alpha $ is a positive number. In this case the integral equation (5) is an equation with degenerate kernel $ f( x, y) = \alpha \cos \sqrt {\lambda _ {0} } x \cos \sqrt {\lambda _ {0} } y $ and its solution is

$$ K( x, y) = - \frac{\alpha \cos \sqrt {\lambda _ {0} } x \cos \sqrt {\lambda _ {0} } y }{1 + \alpha \int\limits _ { 0 } ^ { x } \cos ^ {2} \sqrt {\lambda _ {0} } t dt } . $$

Now the function $ q $ and the number $ h $ are determined by the formulas

$$ q( x) = 2 \frac{dK( x, x) }{dx} = - 2 \frac{d}{dx} \left ( \frac{\alpha \ \cos ^ {2} \sqrt { \lambda _ {0} } x }{1 + \alpha \int\limits _ { 0 } ^ { x } \cos ^ {2} \sqrt {\lambda _ {0} } t dt } \right ) , $$

$$ h = K( 0, 0) = - \alpha . $$

$ 2 $. Let a real-valued function $ q $ satisfy the inequality

$$ \tag{6 } \int\limits _ { 0 } ^ \infty x | q( x) | dx < \infty . $$

Then the boundary value problem

$$ \tag{7 } - y ^ {\prime\prime} + q( x) y = \lambda ^ {2} y,\ \ 0 < x < \infty , $$

$$ \tag{7'} y( 0) = 0, $$

has bounded solutions for $ \lambda ^ {2} > 0 $ and $ \lambda = i \lambda _ {k} $, $ \lambda _ {k} > 0 $, $ k = 1 \dots n $. Moreover, these solutions satisfy for $ x \rightarrow \infty $ the asymptotic formulas

$$ y( x, \lambda ) = e ^ {- i \lambda x } - S( \lambda ) e ^ {i \lambda x } + o( 1),\ \ 0 < \lambda ^ {2} < \infty , $$

$$ y( x, i \lambda _ {k} ) = m _ {k} e ^ {- \lambda _ {k} x } [ 1+ o( 1)] ,\ m _ {k} > 0,\ k = 1 \dots n. $$

Here $ m _ {k} $ is a normalization factor: $ \int _ {0} ^ \infty | y( x , i \lambda _ {k} ) | ^ {2} dx = 1 $. The set of values $ \{ {S(\lambda) } : {-\infty<\lambda<\infty; \lambda _ {k} , m _ {k} ; k= 1\dots n } \} $ is called the scattering data for the boundary value problem (7), (7'}). It is required to reconstruct the potential $ q $ from the scattering data.

To solve this problem, a function $ F $ is constructed by the formula

$$ F( x) = \sum _ { k= } 1 ^ { n } m _ {k} ^ {2} e ^ {- \lambda _ {k} x } + \frac{1}{2 \pi } \int\limits _ {- \infty } ^ \infty [ 1- S( \lambda )] e ^ {i \lambda x } d \lambda $$

and one considers the following equation:

$$ \tag{8 } F( x+ y) + K( x, y) + \int\limits _ { x } ^ \infty K( x, t) F( t+ y) dt = 0. $$

This equation has a unique solution $ K( x, y) $ for any $ x \geq 0 $. After this equation has been solved, the potential $ q $ is determined by the formula

$$ q( x) = - 2 \frac{dK( x, x) }{dx} . $$

For a set $ \{ {S(\lambda) } : {-\infty<\lambda<\infty; \lambda _ {k} , m _ {k} ; \lambda _ {k} > 0, m _ {k} > 0, k= 1\dots n } \} $ to be the scattering data for some boundary value problem of the form (7), (7'}) with a real potential $ q $ which satisfies the condition (6), it is necessary and sufficient that the following conditions are satisfied (see [1]):

a) the function $ S( \lambda ) $ is continuous on the whole line, $ \overline{ {S( \lambda ) }}\; = S(- \lambda ) = [ S( \lambda )] ^ {-} 1 $, $ 1- S( \lambda ) $ tends to zero as $ | \lambda | \rightarrow \infty $ and is the Fourier transform of the function

$$ F _ {S} ( x) = \frac{1}{2 \pi } \int\limits _ {- \infty } ^ \infty [ 1- S( \lambda )] e ^ {i \lambda x } d \lambda , $$

which is representable as a sum of two functions, one of which belongs to $ L _ {1} (- \infty , \infty ) $, while the second is bounded and belongs to $ L _ {2} (- \infty , \infty ) $. On the half-line $ 0 < x < \infty $ the function $ F _ {S} ( x) $ has a derivative $ F _ {S} ^ { \prime } ( x) $ satisfying the condition $ \int _ {0} ^ \infty x | F _ {S} ^ { \prime } ( x) | dx < \infty $;

b) the increment of the argument in the function $ S( \lambda ) $ is connected with the number $ n $ of negative eigenvalues (i.e. of the numbers $ - \lambda _ {1} ^ {2} \dots - \lambda _ {n} ^ {2} $) of the boundary value problem (7), (7'}) by the formula

$$ n = \frac{ \mathop{\rm ln} S(+ 0) - \mathop{\rm ln} S(+ \infty ) }{2 \pi i } - 1- S( \frac{0)}{4} . $$

The integral equation (8) for $ K( x, y) $ has an explicit solution if $ S( \lambda ) $ is a rational function. Solutions of the equation (7) and the potential $ q $ are obtained in this case as rational functions in the trigonometric and hyperbolic functions. For example, if

$$ S ( \lambda ) = \frac{( \lambda + i)( \lambda + 2i) }{( \lambda - i)( \lambda - 2i) } ,\ \ \lambda _ {1} = 1,\ \ m _ {1} = \sqrt 6 , $$

then the corresponding potential has the form

$$ q( x) = - \frac{24}{( 2 \cosh 2x- \sinh 2x) ^ {2} } . $$

References

[1] V.A. Marchenko, "Sturm–Liouville operators and applications" , Birkhäuser (1986) (Translated from Russian)
[2] Z.S. Agranovich, V.A. Marchenko, "The inverse problem in scattering theory" , Khar'kov (1960) (In Russian)
[3] C. Shadan, P. Sabatier, "Inverse problems in quantum scattering theory" , Springer (1989)
[4] B.M. Levitan, "Generalized translation operators and some of their applications" , Israel Program Sci. Transl. (1964) (Translated from Russian)
[5] , The theory of solitons: methods of the inverse problem , Moscow (1980) (In Russian)
[6] Yu.M. [Yu.M. Berezanskii] Berezanskiy, "Expansion in eigenfunctions of selfadjoint operators" , Amer. Math. Soc. (1968) (Translated from Russian)
[7] L.D. Faddeev, "The inverse problem in quantum scattering theory" Uspekhi Mat. Nauk , 14 : 4 (1959) pp. 57–119 (In Russian)
[8] L.D. Faddeev, "Inverse problem of quantum scattering theory II" J. Soviet. Math. , 5 : 3 (1976) pp. 334–450 Itogi Nauk. i Tekhn. Sovremen. Probl. Mat. , 3 (1974) pp. 93–180
[9] B.M. Levitan, M.G. Gasymov, "Determination of a differential equation by two of its spectra" Russian Math. Surveys , 19 : 2 (1964) pp. 1–64 Uspkehi Mat. Nauk , 19 : 2 (1964) pp. 3–63
[10] V. Ambarzumian, Z. Phys. , 53 (1929) pp. 690–695
[11] G. Borg, "Eine Umkehrung der Sturm–Liouvilleschen Eigenwertaufgabe, Bestimmung der Differentialgleichung durch die Eigenwerte" Acta Math. , 78 (1946) pp. 1–96
[12] N. Levinson, "On the uniqueness of the potential in a Schrödinger equation for a given asymptotic phase" Danske Vid. Selsk. Mat. -Fys. Medd. , 25 : 9 (1949) pp. 1–29
[13] V.A. Marchenko, "On the theory of a second-order differential operator" Dokl. Akad. Nauk SSSR , 72 : 3 (1950) pp. 457–460 (In Russian)
[14] I.M. Gel'fand, B.M. Levitan, "On the determination of a differential equation from its spectral function" Izv. Akad. Nauk SSSR Ser. Mat. , 15 : 4 (1951) pp. 309–360 (In Russian)
[15a] M.G. Krein, "Solution of the inverse Sturm–Liouville problem" Dokl. Akad. Nauk SSSR , 76 : 1 (1951) pp. 21–24 (In Russian)
[15b] M.G. Krein, "On inverse problems for a nonhomogeneous chord" Dokl. Akad. Nauk SSSR , 82 : 5 (1952) pp. 669–672 (In Russian)
[15c] M.G. Krein, "On the transfer function of a one-dimensional second-order boundary problem" Dokl. Akad. Nauk SSSR , 88 : 3 (1953) pp. 405–408 (In Russian)
[16] V.A. Marchenko, "On reconstructing the potential energy from phases of scattered waves" Dokl. Akad. Nauk SSSR , 104 : 5 (1955) pp. 695–698 (In Russian)
[17] B.Ya. Levin, "Transformations of Fourier and Laplace types by means of solutions of second-order differential equations" Dokl. Akad. Nauk SSSR , 106 : 2 (1956) pp. 187–190 (In Russian)
[18] R. Newton, R. Jost, "The construction of potentials from the -matrix for systems of differential equations" Nuovo Cimento , 1 : 4 (1955) pp. 590–622
[19] M.G. Gasymov, B.M. Levitan, "Determination of Dirac's system from the scattering phase" Soviet Math. Dokl. , 7 : 2 (1966) pp. 543–547 Dokl. Akad. Nauk SSSR , 167 : 6 (1966) pp. 1219–1222
[20] M.G. Gasymov, "The inverse problem of scattering theory for a system of Dirac equations of order " Trudy Moskov. Mat. Obshch. , 19 (1968) pp. 41–112 (In Russian)
[21] F.S. Rofe-Beketov, "The spectral matrix and the inverse Sturm–Liouville problem on the axis " Teor. Funktsii, Funktional. Analiz. i Prilozhen. : 4 (1967) pp. 189–197 (In Russian)
[22] B.M. Levitan, "The inverse problem in quantum scattering theory at fixed energy" , Probl. Mekh. i Mat. Fiz. (I.G. Petrovskii) , Moscow (1976) pp. 166–207 (In Russian)
[23] N.I. Akhiezer, "A continuous analogue of orthogonal polynomials on a system of intervals" Soviet Math. Dokl. , 2 : 6 (1961) pp. 1409–1412 Dokl. Akad. SSSR , 141 : 2 (1961) pp. 263–266
[24] G.Sh. Guseinov, "The inverse problem of scattering theory for a second-order difference equation on the whole axis" Soviet Math. Dokl. , 17 : 6 (1976) pp. 1684–1688 Dokl. Akad. Nauk SSSR , 231 : 5 (1976) pp. 1045–1048
[25] B.M. Levitan, "Inverse Sturm–Liouville problems" , VNU (1987) (Translated from Russian)

Comments

The inverse problem for the Schrödinger operator $ - ( {d ^ {2} } / {dx ^ {2} } ) + q( x) $, $ - \infty < x < + \infty $, has been investigated in [a1]. (See also Schrödinger equation.) In order to derive necessary and sufficient conditions in the scattering data to correspond with the equation $ - y ^ {\prime\prime} + q( x) y= \lambda ^ {2} y $, $ - \infty < x< \infty $, it is assumed in [a1] that $ q $ satisfies $ \int _ {- \infty } ^ {+ \infty } | q( x) | ( 1+ x ^ {2} ) dx < \infty $, cf. [6]. The condition on the Fourier transforms $ F _ {i} $, are analogous to $ F _ {S} $, defined above is also changed: $ \int _ {a} ^ \infty | F _ {1} ^ { \prime } ( x) | dx $, $ \int _ {- \infty } ^ {a} | F _ {2} ^ { \prime } ( x) | ( 1+ x ^ {2} ) dx < \infty $ for all $ - \infty < a < + \infty $.

References

[a1] P. Deift, E. Trubowitz, "Inverse scattering on the line" Comm. Pure Appl. Math. , 32 (1979) pp. 121–251
[a2] J. Pöschel, E. Trubowitz, "Inverse spectral theory" , Acad. Press (1987)
[a3] A.G. Ramm, "Inverse scattering on half-line" J. Math. Anal. Appl. , 133 (1988) pp. 543–572
[a4] A.G. Ramm, B.A. Taylor, "Example of a potential in one-dimensional scattering problem for which there are infinitely many purely imaginary resonances" Phys. Lett. , 124A (1987) pp. 313–319
[a5] L.E. Andersson, "Inverse eigenvalue problems with discontinuous coefficients" Inverse Probl. , 4 (1988) pp. 353–397
[a6] A.G. Ramm, "Recovery of the potential from -function" Math. Reports Canad. Acad. Sci. , 9 (1987) pp. 177–182
[a7a] A.G. Ramm, "Necessary and sufficient conditions on the scattering data for the potential to be in for the Schrödinger operator on the half-line" Inverse Probl. , 3 (1987) pp. L71–76
[a7b] A.G. Ramm, "An inverse problem for the Helmholtz equation in a semi-infinite medium" Inverse Probl. , 3 (1987) pp. L19–22
[a8] E.L. Isaacson, E. Trubowitz, "The inverse Sturm–Liouville problem I" Comm. Pure Appl. Math. , 36 (1983) pp. 767–784
[a9] E.L. Isaacson, H.P. McKean, E. Trubowitz, "The inverse Sturm–Liouville problem II" Comm. Pure Appl. Math. , 37 (1984) pp. 1–12
[a10] B.E.J. Dahlberg, E. Trubowitz, "The inverse Sturm–Liouville problem III" Comm. Pure Appl. Math. , 37 (1984) pp. 255–267
How to Cite This Entry:
Sturm-Liouville problem, inverse. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Sturm-Liouville_problem,_inverse&oldid=12687
This article was adapted from an original article by G.Sh. GuseinovB.M. Levitan (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article