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Let $q ( x ) \in L _ { 1,1 } : = \left\{ q : \int _ { - \infty } ^ { \infty } ( 1 + | x | ) | q ( x ) | d x < \infty , q = \overline { q } \right\}$, where the bar stands for complex conjugation. Consider the (direct) scattering problem:

\begin{equation}
\tag{a1}
\mathrm{l}u_- - k^2 u_- := (-u'' + q(x) - k^2) u_- = 0,
\end{equation}

\begin{equation*} x \in \mathbf{R} : = ( - \infty , \infty ), \end{equation*}

\begin{equation} \tag{a2} u _ { - } = \left\{ \begin{array} { l } { e ^ { - i k x } + r _ { - } ( k ) e ^ { - i k x } } &amp;  {x \xrightarrow{\qquad\qquad\qquad\qquad\qquad\qquad }-\infty ,} \\ { t _{-} ( k ) e ^ { i k x } ,} &amp;  {x \xrightarrow{\qquad\qquad\qquad\qquad\qquad\qquad }+\infty .} \end{array} \right. \end{equation}

The coefficients $r ( k )$ and $t ( k )$ are called the reflection and transmission coefficients. One can prove that $t_- ( k )$ is analytic in $\mathbf{C} _ { + } : = \{ k : \operatorname { Im } k > 0 \}$ except at a finite number of points $i k_j$, $1 \leq j \leq J$, $k_ j > 0$, which are simple poles of $t ( k )$.

Problem (a1)–(a2) describes scattering by a plane wave $e ^ { i k x }$ falling from $- \infty$ and scattered by the potential $q ( x )$.

One can also consider the scattering of the plane wave falling from $+ \infty$:

\begin{equation} \tag{a3} \text{l}u _ { + } -  { k } ^ { 2 } u _ { + } = 0 , x \in \mathbf{R}, \end{equation}

\begin{equation} 
\tag{a4} 
u_{+} = \left\{ 
\begin{array} { ll } 
t_{+}(k) e^{-ikx}, & x \xrightarrow{\qquad\qquad\qquad\qquad\qquad\qquad } +\infty, \\
e^{-ikx} + r_{+}e^{ikx}, & x \xrightarrow{\qquad\qquad\qquad\qquad\qquad\qquad } -\infty.
\end{array} \right. \end{equation}

One proves that $t _ { - } ( k ) = t _ { + } ( k ) : = t ( k )$, $t ( - k ) = \overline { t ( k ) }$, $k \in \mathbf{R}$, where the bar stands for complex conjugation, $r _ { \pm } ( - k ) = \overline { r _ { \pm } ( k ) }$. The matrix

\begin{equation*} 
\left( 
\begin{array} { c c } 
{ t ( k ) } & { r _ { - } ( k ) } \\ 
{ r _ { + } ( k ) } & { t ( k ) } 
\end{array} 
\right) = S ( k ) 
\end{equation*}

is called the $S$-matrix (cf. [[Scattering matrix|Scattering matrix]]). Conservation of energy implies $| t ( k ) | ^ { 2 } + | r ( k ) | ^ { 2 } = 1$.

Let $f ( x , k )$ and $g ( x , k )$ be the solutions to (a1) satisfying the conditions:

\begin{equation*} 
f ( x , k ) = e ^ { i k x } + o ( 1 ) , \quad x \rightarrow \infty , 
\end{equation*}

\begin{equation*} 
g ( x , k ) = e ^ { - i k x } + o ( 1 ) , \quad x \rightarrow - \infty. 
\end{equation*}

Then

\begin{equation*} f ( x , k ) = e ^ { i k x } + \int _ { x } ^ { \infty } A _ { + } ( x , y ) e ^ { i k y } d y, \end{equation*}

\begin{equation*} g ( x , k ) = e ^ { - i k x } + \int _ { - \infty } ^ { x } A _ { - } ( x , y ) e ^ { - i k y } d y, \end{equation*}

where $A _ { \pm } ( x , y )$ are the kernels which define the transformation operators. One has

\begin{equation*} f ( x , k ) = b ( k ) g ( x , k ) + a ( k ) g ( x , - k ), \end{equation*}

\begin{equation*} g ( x , k ) = - b ( - k ) f ( x , k ) + a ( k ) f ( x , - k ), \end{equation*}

where

\begin{equation*} 
a ( - k ) = \overline { a ( k ) } , \quad b ( - k ) = \overline { b ( k ) }, 
\end{equation*}

\begin{equation*} 
| a ( k ) | ^ { 2 } = 1 + | b ( k ) | ^ { 2 } , \quad 
r _ { - } ( k ) = \frac { b ( k ) } { a ( k ) } , \quad 
r _ { + } ( k ) = - \frac { b ( - k ) } { a ( k ) }, 
\end{equation*}

\begin{equation*} t ( k ) = \frac { 1 } { \alpha ( k ) }. \end{equation*}

The function $a ( k )$ is analytic in $\mathbf{C} _ { + } : = \{ k : \operatorname { Im } k > 0 \}$ and has finitely many simple zeros all of which are at the points $i k_j$, $1 \leq j \leq J$, $a ( i k _ { j } ) = 0$, $\dot { a } ( i k _ { j } ) \neq 0$, $\dot { a } : = d a / d  k $.

If $k = i k_j$, then $f ( x , i k_{ j} ) \in L ^ { 2 } ( \mathbf{R} )$,

\begin{equation*} - f ^ { \prime \prime } ( x , i k _ { j } ) + q ( x ) f ( x , i k _ { j } ) + k ^ { 2 _ j } f ( x , i k _ { j } ) = 0, \end{equation*}

\begin{equation*} \int _ { - \infty } ^ { \infty } | f ( x , i k _ { j } ) | ^ { 2 } d x = ( m _ { j } ^ { + } ) ^ { - 2 }, \end{equation*}

\begin{equation*} \int _ { - \infty } ^ { \infty } | g ( x , i k _ { j } ) | ^ { 2 } d x = ( m _ { j } ^ { - } ) ^ { - 2 }. \end{equation*}

The numbers $- k _ { j } ^ { 2 }$ are the eigenvalues of the operator $- d ^ { 2 } / d x ^ { 2 } + q ( x )$ in $L ^ { 2 } ( \mathbf{R} )$. They are called the bound states.

The scattering data are the values

\begin{equation*} S : = \{ r _ { + } ( k ) , i k _ { j } , ( m _ { j } ^ { + } ) ^ { 2 } : \forall k > 0,1 \leq j \leq J \}. \end{equation*}

The inverse scattering problem (ISP) consists of finding $q ( x ) \in L _ { 1,1 }$ from $\mathcal{S}$.

The inverse scattering problem has at most one solution in the class $L _ { 1  , 1}$. This solution can be calculated by the following Marchenko method:
<table border="0" cellpadding="0" cellspacing="0" style="background-color:black;"> <tr><td> <table border="0" cellpadding="4" cellspacing="1" style="background-color:black;">  <tr> <td colname="1" colspan="1" style="background-color:white;">1</td> <td colname="2" colspan="1" style="background-color:white;">Define

\begin{equation} \tag{a5} F + ( x ) = \sum _ { j = 1 } ^ { J } ( m _ { j } ^ { + } ) ^ { 2 } e ^ { - k _ { j } x } + \frac { 1 } { 2 \pi } \int _ { - \infty } ^ { \infty } r _ { + } ( k ) e ^ { i k x } d k \end{equation}

and solve the following Marchenko equatio for $A _ { + } ( x , y )$:

\begin{equation*} A _ { + } ( x , y ) + F _ { + } ( x + y ) + \int _ { x } ^ { \infty } A ( x , t ) F _ { + } ( t , y ) d t = 0, \end{equation*}

\begin{equation*} y \geq x. \end{equation*}

If the data $\left\{ r _ { + } ( k ) , i k _ { j } , ( m _ { j } ^ { + } ) ^ { 2 } : 1 \leq j \leq J \right\}$ correspond to a $q \in L _ { 1 , 1} $, then equation (a5) is uniquely solvable in $L ^ { 1 } ( x , \infty )$ for every $x > - \infty$.</td> </tr> <tr> <td colname="1" colspan="1" style="background-color:white;">2</td> <td colname="2" colspan="1" style="background-color:white;">If $A _ { + } ( x , y )$ is found, then $q ( x ) = - 2 d A _ { + } ( x , x ) / d x$.</td> </tr>  </table>

</td></tr> </table>

The main result [[#References|[a7]]] is the characterization property for the scattering data: In order that $\mathcal{S} : = \left\{ r _ { + } ( k ) , i k _ { j } , ( m _ { j } ^ { + } ) ^ { 2 } : 1 \leq j \leq J , k _ { j } > 0 , m _ { j } ^ { + } > 0 , k > 0 \right\}$ be the scattering data corresponding to a $q ( x ) \in L _ { 1,1 } ( \mathbf{R} )$, it is necessary and sufficient that the following conditions hold:

i) $r ( - k ) = \overline { r ( k ) }$ for $k > 0$, the function $r ( k )$ for $k \neq 0$ is continuous,

\begin{equation*} | r _ { + } ( k ) | \leq 1 - c k ^ { 2 } ( 1 + k ^ { 2 } ) ^ { - 1 }, \end{equation*}

where $c = \text{const} > 0$, and $r_+ ( k ) = O ( 1 / k )$ as $k \rightarrow \pm \infty$.

ii) The function

\begin{equation*} R _ { + } ( x ) : = \frac { 1 } { 2 \pi } \int _ { - \infty } ^ { \infty } r _ { + } ( k ) e ^ { i k x } d k \end{equation*}

is absolutely continuous and

\begin{equation*} \int _ { s } ^ { \infty } | R _ { + } ^ { \prime } ( x ) | ( 1 + | x | ) d x < \infty \end{equation*}

for every $s > - \infty$.

iii) Denote

\begin{equation*} a ( z ) : = \prod _ { j = 1 } ^ { J } \frac { z - i k _ { j } } { z + i k _ { j } } \operatorname { exp } \left\{ - \frac { 1 } { 2 \pi i } \int _ { - \infty } ^ { \infty } \frac { \operatorname { ln } ( 1 - | r _ { + } ( k ) | ^ { 2 } ) } { k - z } d k \right\} . \end{equation*}

The function $a ( z )$ is continuous in $\overline { \mathbf{C} } _ { + }$ and

\begin{equation*} \operatorname { lim } _ { k \rightarrow 0 } k \alpha ( k ) [ r _ { + } ( k ) + 1 ] = 0. \end{equation*}

iv) The function

\begin{equation*} R _ { - } ( x ) : = - \frac { 1 } { 2 \pi } \int _ { - \infty } ^ { \infty } r _ { + } ( - k ) \frac { a ( - k ) } { a ( k ) } e ^ { - i k x } d k \end{equation*}

is absolutely continuous and

\begin{equation*} \int _ { s } ^ { \infty } ( 1 + | x | ) | R _ { - } ^ { \prime } ( x ) | d x < \infty \end{equation*}

for every $s > - \infty$.

A similar result holds for the data

\begin{equation*} \{ r _ { - } ( k ) , i k _ { j } , ( m _ { j } ^ { - } ) ^ { 2 } : 1 \leq j \leq J , \forall k > 0 \} \end{equation*}

and the potential $q ( x )$ can be obtained by the Marchenko method, $q ( x ) = - 2 d A _ { - } ( x , x ) / d x$.

In [[#References|[a2]]] the above theory is generalized to the case when $q ( x )$ tends to a different constants as $x \rightarrow + \infty$ and $x \rightarrow - \infty$.

In [[#References|[a5]]] a different approach to solving the inverse scattering problem is described for

\begin{equation*} q \in L _ { 1,2 } : = \left\{ q : q = \overline { q } , \int _ { - \infty } ^ { \infty } ( 1 + x ^ { 2 } ) | q ( x ) | d x < \infty \right\}. \end{equation*}

The approach in [[#References|[a5]]] is based on a trace formula.

If $q ( x ) = 0$ for $x < x _ { 0 } < \infty$, then the reflection coefficient $\{ r_+ ( k ) : \forall k > 0 \}$ alone, without the knowledge of $i k_j$ and $( m _ { j } ^ { + } ) ^ { 2 }$, determines $q ( x )$ uniquely. A simple proof of this and similar statements, based on property $C$ for ordinary differential equations (cf. [[Ordinary differential equations, property C for|Ordinary differential equations, property $C$ for]]), is given in [[#References|[a10]]].

An inverse scattering problem for an inhomogeneous [[Schrödinger equation|Schrödinger equation]] is studied in [[#References|[a5]]].

The inverse scattering method is a tool for solving many evolution equations (cf. also [[Evolution equation|Evolution equation]]) and is used in, e.g., soliton theory [[#References|[a7]]], [[#References|[a1]]], [[#References|[a3]]], [[#References|[a6]]] (cf. also [[Korteweg–de Vries equation|Korteweg–de Vries equation]]; [[Harry Dym equation|Harry Dym equation]]).

Methods for adding and removing bound states are described in [[#References|[a5]]]. They are based on the Darboux–Crum transformations and commutation formulas.

A large bibliography can be found in [[#References|[a4]]].

====References====
<table><tr><td valign="top">[a1]</td> <td valign="top">  M. Ablowitz,   H. Segur,   "Solutions and inverse scattering transform" , SIAM  (1981)</td></tr><tr><td valign="top">[a2]</td> <td valign="top">  A. Cohen,   T. Kappeler,   "Scattering and inverse scattering for step-like potentials in the Schrödinger equation"  ''Indiana Math. J.'' , '''34'''  (1985)  pp. 127–180</td></tr><tr><td valign="top">[a3]</td> <td valign="top">  F. Calogero,   A. Degasperis,   "Solutions and the spectral transform" , North-Holland  (1982)</td></tr><tr><td valign="top">[a4]</td> <td valign="top">  K. Chadan,   P. Sabatier,   "Inverse problems in quantum scattering" , Springer  (1989)</td></tr><tr><td valign="top">[a5]</td> <td valign="top">  P. Deift,   E. Trubowitz,   "Inverse scattering on the line"  ''Commun. Pure Appl. Math.'' , '''32'''  (1979)  pp. 121–251</td></tr><tr><td valign="top">[a6]</td> <td valign="top">  L. Faddeev,   L. Takhtadjian,   "Hamiltonian methods in the theory of solutions" , Springer  (1986)</td></tr><tr><td valign="top">[a7]</td> <td valign="top">  V. Marchenko,   "Sturm–Liouville operators and applications" , Birkhäuser  (1986)</td></tr><tr><td valign="top">[a8]</td> <td valign="top">  A.G. Ramm,   "Multidimensional inverse scattering problems" , Longman/Wiley  (1992)</td></tr><tr><td valign="top">[a9]</td> <td valign="top">  A.G. Ramm,   "Inverse problem for an inhomogeneous Schrödinger equation"  ''J. Math. Phys.'' , '''40''' :  8  (1999)  pp. 3876–3880</td></tr><tr><td valign="top">[a10]</td> <td valign="top">  A.G. Ramm,   "Property C for ODE and applications to inverse problems"  A.G. Ramm (ed.)  P.N. Shivakumar (ed.)  A.V. Strauss (ed.) , ''Operator Theory and Applications'' , ''Fields Inst. Commun.'' , '''25''' , Amer. Math. Soc.  (2000)  pp. 15–75</td></tr></table>