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There are many multi-dimensional inverse scattering problems. Below, inverse potential scattering and inverse geophysical scattering are briefly discussed; see [[Obstacle scattering|Obstacle scattering]] for inverse obstacle scattering problems.
 
There are many multi-dimensional inverse scattering problems. Below, inverse potential scattering and inverse geophysical scattering are briefly discussed; see [[Obstacle scattering|Obstacle scattering]] for inverse obstacle scattering problems.
  
Line 4: Line 12:
 
To formulate the inverse potential scattering problem, consider first the direct scattering problem (see [[#References|[a1]]], [[#References|[a2]]], [[#References|[a4]]], [[#References|[a5]]], [[#References|[a6]]], Appendix):
 
To formulate the inverse potential scattering problem, consider first the direct scattering problem (see [[#References|[a1]]], [[#References|[a2]]], [[#References|[a4]]], [[#References|[a5]]], [[#References|[a6]]], Appendix):
  
<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/i/i130/i130070/i1300701.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a1)</td></tr></table>
+
\begin{equation} \tag{a1} [ - \nabla ^ { 2 } + q ( x ) - k ^ { 2 } ] u = 0\, \operatorname { in } \mathbf{R} ^ { 3 } ,\, k = \text{const} &gt; 0, \end{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/i/i130/i130070/i1300702.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a2)</td></tr></table>
+
\begin{equation} \tag{a2} u = e ^ { i k \alpha x } + v , \alpha \in S ^ { 2 }, \end{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/i/i130/i130070/i1300703.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a3)</td></tr></table>
+
\begin{equation} \tag{a3} \operatorname { lim } _ { r \rightarrow \infty } \int _ { |x| = r } \left| \frac { \partial v } { \partial r } - i k v \right| ^ { 2 } d s = 0, \end{equation}
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130070/i1300704.png" /> is given, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130070/i1300705.png" /> is the unit sphere, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130070/i1300706.png" /> is the scattered field, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130070/i1300707.png" /> is the scattering solution, condition (a3) is called the (outgoing) radiation condition, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130070/i1300708.png" /> is the incident plane wave, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130070/i1300709.png" /> is a real-valued function, called a potential,
+
where $\alpha$ is given, $S ^ { 2 }$ is the unit sphere, $v$ is the scattered field, $u$ is the scattering solution, condition (a3) is called the (outgoing) radiation condition, $e ^ { i k  \alpha x}$ is the incident plane wave, and $q ( x )$ is a real-valued function, called a potential,
  
<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/i/i130/i130070/i13007010.png" /></td> </tr></table>
+
\begin{equation*} q ( x ) \in L  ^ { 2  }_\text { loc }  ( \mathbf{R} ^ { 3 } ), \end{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/i/i130/i130070/i13007011.png" /></td> </tr></table>
+
\begin{equation*} | q ( x ) | \leq c ( 1 + | x | ) ^ { - b } , b &gt; 2,\text{ for large }|x|. \end{equation*}
  
The existence and uniqueness of the solution to (a1)–(a3) has been proved under less restrictive assumptions on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130070/i13007012.png" /> [[#References|[a2]]]. The function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130070/i13007013.png" /> has the form
+
The existence and uniqueness of the solution to (a1)–(a3) has been proved under less restrictive assumptions on $q ( x )$ [[#References|[a2]]]. The function $v$ 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/i/i130/i130070/i13007014.png" /></td> </tr></table>
+
\begin{equation*} v ( x , \alpha , k ) = A ( \alpha ^ { \prime } , \alpha , k ) \frac { e ^ { i k r} } { r } + o \left( \frac { 1 } { r } \right), \end{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/i/i130/i130070/i13007015.png" /></td> </tr></table>
+
\begin{equation*} r \rightarrow \infty , \frac { x } { r } = \alpha ^ { \prime }, \end{equation*}
  
where the coefficient <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130070/i13007016.png" /> is called the scattering amplitude.
+
where the coefficient $A ( \alpha ^ { \prime } , \alpha , k )$ is called the scattering amplitude.
  
The inverse potential scattering problem consists of finding <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130070/i13007017.png" /> given <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130070/i13007018.png" /> on some subsets of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130070/i13007019.png" />.
+
The inverse potential scattering problem consists of finding $q ( x )$ given $A ( \alpha ^ { \prime } , \alpha , k )$ on some subsets of $S ^ { 2 } \times S ^ { 2 } \times \mathbf{R} _ { + }$.
  
The first result is simple: If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130070/i13007020.png" /> is known for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130070/i13007021.png" /> and all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130070/i13007022.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130070/i13007023.png" /> is uniquely determined.
+
The first result is simple: If $A ( \alpha ^ { \prime } , \alpha , k )$ is known for all $\alpha ^ { \prime } , \alpha \in S ^ { 2 }$ and all $k &gt; 0$, then $q ( x )$ is uniquely determined.
  
 
If
 
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/i/i130/i130070/i13007024.png" /></td> </tr></table>
+
\begin{equation*} q \in Q _ { m } : = \left\{ \begin{array} { c } { q = \overline { q } }, \\ { q : | q ( x ) | + | \nabla ^ { m } q | \leq c ( 1 + | x | ) ^ { - b } }, \\ { b &gt; 3 } \end{array} \right\}, \end{equation*}
  
 
then it is known (e.g. [[#References|[a6]]], p. 233, see also [[#References|[a4]]]) that
 
then it is known (e.g. [[#References|[a6]]], p. 233, see also [[#References|[a4]]]) 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/i/i130/i130070/i13007025.png" /></td> </tr></table>
+
\begin{equation*} A ( \alpha ^ { \prime } , \alpha , k ) = - \frac { 1 } { 4 \pi } \int _ { \mathbf{R} ^ { 3 } } e ^ { i k (\alpha - \alpha ^ { \prime } ) x } q ( x ) d x + O \left( \frac { 1 } { k } \right), \end{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/i/i130/i130070/i13007026.png" /></td> </tr></table>
+
\begin{equation*} k \rightarrow \infty, \end{equation*}
  
so that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130070/i13007027.png" /> can be found:
+
so that $\tilde{q} ( \xi ) : = \int _ { \mathbf{R} ^ { 3 } } e ^ { - i \xi x } q ( x ) d x$ can be found:
  
<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/i/i130/i130070/i13007028.png" /></td> </tr></table>
+
<table class="eq" style="width:100%;"> <tr><td style="width:94%;text-align:center;" valign="top"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130070/i13007028.png"/></td> </tr></table>
  
 
The second result is much more difficult.
 
The second result is much more difficult.
  
For decades it was not known if the data <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130070/i13007029.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130070/i13007030.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130070/i13007031.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130070/i13007032.png" /> fixed, determine <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130070/i13007033.png" /> uniquely. In 1987 the uniqueness result has been established by A.G. Ramm (see [[#References|[a7]]], [[#References|[a8]]]) under the assumptions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130070/i13007034.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130070/i13007035.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130070/i13007036.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130070/i13007037.png" /> is an arbitrary large fixed number, and in 1988 inversion procedures were published; see [[#References|[a7]]]. One of them, proposed by Ramm, is based on the formula
+
For decades it was not known if the data $A ( \alpha ^ { \prime } , \alpha ) : = A ( \alpha ^ { \prime } , k _ { 0 } )$, $\forall \alpha ^ { \prime }$, $\alpha \in S ^ { 2 }$ and $k _ { 0 } &gt; 0$ fixed, determine $q ( x )$ uniquely. In 1987 the uniqueness result has been established by A.G. Ramm (see [[#References|[a7]]], [[#References|[a8]]]) under the assumptions $q ( x ) \in L ^ { 2 } ( \mathbf{R} ^ { 3 } )$, $q ( x ) = 0$ for $| x | &gt; a$, where $a &gt; 0$ is an arbitrary large fixed number, and in 1988 inversion procedures were published; see [[#References|[a7]]]. One of them, proposed by Ramm, is based on 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/i/i130/i130070/i13007038.png" /></td> </tr></table>
+
<table class="eq" style="width:100%;"> <tr><td style="width:94%;text-align:center;" valign="top"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130070/i13007038.png"/></td> </tr></table>
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130070/i13007039.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130070/i13007040.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130070/i13007041.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130070/i13007042.png" /> is an arbitrary point.
+
where $M : = \left\{ \theta : \theta \in \mathbf{C} ^ { 3 } , \theta . \theta = k ^ { 2_0 }  \right\}$, $\theta . w : = \sum _ { j = 1 } ^ { 3 } \theta _ { j } .w _ { j }$, $v ( \alpha , \theta ) \in L ^ { 2 } ( S ^ { 2 } )$, and $\xi \in \mathbf{R} ^ { 3 }$ is an arbitrary point.
  
Another inversion procedure ([[#References|[a3]]], [[#References|[a7]]]) is based on the reconstruction of the Dirichlet-to-Neumann mapping and then finding <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130070/i13007043.png" />.
+
Another inversion procedure ([[#References|[a3]]], [[#References|[a7]]]) is based on the reconstruction of the Dirichlet-to-Neumann mapping and then finding $q ( x )$.
  
Error estimates for Ramm's inversion procedure in the case of noisy data and an algorithm for calculating the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130070/i13007044.png" /> in the inversion formula are obtained in [[#References|[a9]]].
+
Error estimates for Ramm's inversion procedure in the case of noisy data and an algorithm for calculating the function $v ( \alpha , \theta )$ in the inversion formula are obtained in [[#References|[a9]]].
  
The uniqueness problem for inverse potential scattering with the data <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130070/i13007045.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130070/i13007046.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130070/i13007047.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130070/i13007048.png" />, fixed, is still open (as of 2000).
+
The uniqueness problem for inverse potential scattering with the data $A ( \alpha ^ { \prime } , \alpha_0 , k )$, $\forall \alpha ^ { \prime } \in S ^ { 2 }$, $\forall k &gt; 0$, $\alpha _ { 0 } \in S ^ { 2 }$, fixed, is still open (as of 2000).
  
The same is true for the uniqueness problem for inverse potential scattering with the (backscattering) data <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130070/i13007049.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130070/i13007050.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130070/i13007051.png" />, although for this problem a uniqueness theorem for small <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130070/i13007052.png" /> holds.
+
The same is true for the uniqueness problem for inverse potential scattering with the (backscattering) data $A ( - \alpha , \alpha , k )$, $\forall \alpha \in S ^ { 2 }$, $\forall k &gt; 0$, although for this problem a uniqueness theorem for small $q ( x )$ holds.
  
 
==Inverse geophysical scattering.==
 
==Inverse geophysical scattering.==
The inverse geophysical scattering problem consists of finding the unknown coefficient <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130070/i13007053.png" /> in the equation
+
The inverse geophysical scattering problem consists of finding the unknown coefficient $v ( x )$ in the 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/i/i130/i130070/i13007054.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a4)</td></tr></table>
+
\begin{equation} \tag{a4} ( \nabla ^ { 2 } +  k  ^ { 2_0 }  + k  ^ { 2_0 }v ( x ) ) u ( x , y , k _ { 0 } ) = - \delta ( x - y ) \text { in } \mathbf{R}  ^ { 3 }, \end{equation}
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130070/i13007055.png" /> satisfies the outgoing radiation condition (a3), <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130070/i13007056.png" /> is fixed, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130070/i13007057.png" /> is a real-valued <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130070/i13007058.png" /> function with compact support in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130070/i13007059.png" />.
+
where $u : = u ( x , y ) : = u ( x , y , k _ { 0 } )$ satisfies the outgoing radiation condition (a3), $k _ { 0 } = \text { const } &gt; 0$ is fixed, and $v ( x )$ is a real-valued $L _ { \text{loc} } ^ { 2 }$ function with compact support in $\mathbf{R} _ { - } ^ { 3 } : = \{ x : x _ { 3 } &lt; 0 \}$.
  
The scattering data are the values <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130070/i13007060.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130070/i13007061.png" />, that is, the values of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130070/i13007062.png" /> on the surface of the Earth. The function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130070/i13007063.png" /> describes an inhomogeneity in the velocity profile (in the refraction coefficient), <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130070/i13007064.png" /> can be an acoustic pressure. Uniqueness of the solution to inverse geophysical scattering problem was proved in 1987 [[#References|[a8]]], [[#References|[a7]]].
+
The scattering data are the values $u ( x , y )$, $\forall x , y \in P : = \{ x : x_ {3} = 0 \}$, that is, the values of $u$ on the surface of the Earth. The function $v ( x )$ describes an inhomogeneity in the velocity profile (in the refraction coefficient), $u$ can be an acoustic pressure. Uniqueness of the solution to inverse geophysical scattering problem was proved in 1987 [[#References|[a8]]], [[#References|[a7]]].
  
The uniqueness problem for inverse geophysical scattering with data <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130070/i13007065.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130070/i13007066.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130070/i13007067.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130070/i13007068.png" /> fixed, is open (as of 2000).
+
The uniqueness problem for inverse geophysical scattering with data $u ( x , y_{0} , k )$, $\forall x \in P$, $\forall k &gt; 0$, and $y _ { 0 } \in P$ fixed, is open (as of 2000).
  
A reduction of the inverse geophysical scattering problem with the data <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130070/i13007069.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130070/i13007070.png" />, to the inverse potential scattering problem with the data <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130070/i13007071.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130070/i13007072.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130070/i13007073.png" /> fixed, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130070/i13007074.png" />, with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130070/i13007075.png" /> the unit vector along <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130070/i13007076.png" />-axis, is done in [[#References|[a7]]].
+
A reduction of the inverse geophysical scattering problem with the data $u ( x , y , k _ { 0 } )$, $\forall x , y \in P$, to the inverse potential scattering problem with the data $A ( \alpha ^ { \prime } , \alpha , k _ { 0 } )$, $\forall \alpha , \alpha ^ { \prime } \in S _ { + } ^ { 2 }$, $k _ { 0 } &gt; 0$ fixed, $S _ { + } ^ { 2 } : = \left\{ \alpha : \alpha \in S ^ { 2 } , \alpha \cdot e _ { 3 } &gt; 0 \right\}$, with $e_3$ the unit vector along $x _ { 3 }$-axis, is done in [[#References|[a7]]].
  
 
==Inverse potential scattering: Open problem.==
 
==Inverse potential scattering: Open problem.==
An interesting open problem (as of 2000) in inverse potential scattering is the problem of finding discontinuities of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130070/i13007077.png" /> and the number of bound states of the Schrödinger operator generated by the expression <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130070/i13007078.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130070/i13007079.png" /> from the knowledge of fixed energy scattering data <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130070/i13007080.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130070/i13007081.png" />.
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An interesting open problem (as of 2000) in inverse potential scattering is the problem of finding discontinuities of $q ( x )$ and the number of bound states of the Schrödinger operator generated by the expression $- \nabla ^ { 2 } + q ( x )$ in $L ^ { 2 } ( \mathbf{R} _ { 3 } )$ from the knowledge of fixed energy scattering data $A ( \alpha ^ { \prime } , \alpha , k _ { 0 } )$, $\forall \alpha ^ { \prime } , \alpha \in S ^ { 2 }$.
  
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130070/i13007082.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130070/i13007083.png" /> is an [[Analytic function|analytic function]] of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130070/i13007084.png" />. Therefore, knowledge of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130070/i13007085.png" /> on an open set in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130070/i13007086.png" />, however small, allows one to recover <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130070/i13007087.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130070/i13007088.png" />.
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If $q \in L ^ { 2_0 }  (\mathbf{ R} ^ { 3 } )$, then $A ( \alpha ^ { \prime } , \alpha )$ is an [[Analytic function|analytic function]] of $\alpha ^ { \prime } , \alpha \in M$. Therefore, knowledge of $A ( \alpha ^ { \prime } , \alpha )$ on an open set in $S ^ { 2 } \times S ^ { 2 }$, however small, allows one to recover $A ( \alpha ^ { \prime } , \alpha )$ on $M \times M$.
  
The assumption concerning compactness of the support of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130070/i13007089.png" /> is natural in inverse potential scattering because the scattering data are always noisy and it is not possible in principle to recover the tail of a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130070/i13007090.png" /> (that is, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130070/i13007091.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130070/i13007092.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130070/i13007093.png" /> is sufficiently large) from knowledge of noisy data <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130070/i13007094.png" />,
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The assumption concerning compactness of the support of $q ( x )$ is natural in inverse potential scattering because the scattering data are always noisy and it is not possible in principle to recover the tail of a $q ( x ) \in Q$ (that is, $q ( x )$ for $| x | &gt; R$, where $R &gt; 0$ is sufficiently large) from knowledge of noisy data $A _ { \delta } ( \alpha ^ { \prime } , \alpha )$,
  
<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/i/i130/i130070/i13007095.png" /></td> </tr></table>
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\begin{equation*} \operatorname { sup } _ { \alpha , \alpha ^ { \prime } \in S ^ { 2 } } | A _ { \delta } ( \alpha ^ { \prime } , \alpha ) - A ( \alpha ^ { \prime } , \alpha ) | &lt; \delta \end{equation*}
  
 
(see [[#References|[a7]]] for a proof).
 
(see [[#References|[a7]]] for a proof).
  
 
====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  H. Cycon,  R. Froese,  W. Kirsch,  B. Simon,  "Schrödinger operators" , Springer  (1986)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  L. Hörmander,  "Analysis of linear partial differential operators" , '''IV''' , Springer  (1985)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  A. Nachman,  "Reconstruction from boundary measurements"  ''Ann. Math.'' , '''128'''  (1988)  pp. 531–578</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  R. Newton,  "Inverse Schrödinger scattering in three dimensions" , Springer  (1989)</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  D. Pearson,  "Quantum scattering and spectral theory" , Acad. Press  (1988)</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top">  A.G. Ramm,  "Random fields estimation theory" , Longman/Wiley  (1990)</TD></TR><TR><TD valign="top">[a7]</TD> <TD valign="top">  A.G. Ramm,  "Multidimensional inverse scattering problems" , Longman/Wiley  (1992)</TD></TR><TR><TD valign="top">[a8]</TD> <TD valign="top">  A.G. Ramm,  "Recovery of the potential from fixed energy scattering data"  ''Inverse Probl.'' , '''4'''  (1988)  pp. 877–886  (See also: Ibid. 3 (1987), L77-82)</TD></TR><TR><TD valign="top">[a9]</TD> <TD valign="top">  A.G. Ramm,  "Stability estimates in inverse scattering"  ''Acta Applic. Math.'' , '''28''' :  1  (1992)  pp. 1–42</TD></TR><TR><TD valign="top">[a10]</TD> <TD valign="top">  A.G. Ramm,  "Stability of solutions to inverse scattering problems with fixed-energy data"  ''Rend. Sem. Mat. e Fisico''  (2001)  pp. 135–211</TD></TR></table>
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<table><tr><td valign="top">[a1]</td> <td valign="top">  H. Cycon,  R. Froese,  W. Kirsch,  B. Simon,  "Schrödinger operators" , Springer  (1986)</td></tr><tr><td valign="top">[a2]</td> <td valign="top">  L. Hörmander,  "Analysis of linear partial differential operators" , '''IV''' , Springer  (1985)</td></tr><tr><td valign="top">[a3]</td> <td valign="top">  A. Nachman,  "Reconstruction from boundary measurements"  ''Ann. Math.'' , '''128'''  (1988)  pp. 531–578</td></tr><tr><td valign="top">[a4]</td> <td valign="top">  R. Newton,  "Inverse Schrödinger scattering in three dimensions" , Springer  (1989)</td></tr><tr><td valign="top">[a5]</td> <td valign="top">  D. Pearson,  "Quantum scattering and spectral theory" , Acad. Press  (1988)</td></tr><tr><td valign="top">[a6]</td> <td valign="top">  A.G. Ramm,  "Random fields estimation theory" , Longman/Wiley  (1990)</td></tr><tr><td valign="top">[a7]</td> <td valign="top">  A.G. Ramm,  "Multidimensional inverse scattering problems" , Longman/Wiley  (1992)</td></tr><tr><td valign="top">[a8]</td> <td valign="top">  A.G. Ramm,  "Recovery of the potential from fixed energy scattering data"  ''Inverse Probl.'' , '''4'''  (1988)  pp. 877–886  (See also: Ibid. 3 (1987), L77-82)</td></tr><tr><td valign="top">[a9]</td> <td valign="top">  A.G. Ramm,  "Stability estimates in inverse scattering"  ''Acta Applic. Math.'' , '''28''' :  1  (1992)  pp. 1–42</td></tr><tr><td valign="top">[a10]</td> <td valign="top">  A.G. Ramm,  "Stability of solutions to inverse scattering problems with fixed-energy data"  ''Rend. Sem. Mat. e Fisico''  (2001)  pp. 135–211</td></tr></table>

Revision as of 16:45, 1 July 2020

There are many multi-dimensional inverse scattering problems. Below, inverse potential scattering and inverse geophysical scattering are briefly discussed; see Obstacle scattering for inverse obstacle scattering problems.

Inverse potential scattering.

To formulate the inverse potential scattering problem, consider first the direct scattering problem (see [a1], [a2], [a4], [a5], [a6], Appendix):

\begin{equation} \tag{a1} [ - \nabla ^ { 2 } + q ( x ) - k ^ { 2 } ] u = 0\, \operatorname { in } \mathbf{R} ^ { 3 } ,\, k = \text{const} > 0, \end{equation}

\begin{equation} \tag{a2} u = e ^ { i k \alpha x } + v , \alpha \in S ^ { 2 }, \end{equation}

\begin{equation} \tag{a3} \operatorname { lim } _ { r \rightarrow \infty } \int _ { |x| = r } \left| \frac { \partial v } { \partial r } - i k v \right| ^ { 2 } d s = 0, \end{equation}

where $\alpha$ is given, $S ^ { 2 }$ is the unit sphere, $v$ is the scattered field, $u$ is the scattering solution, condition (a3) is called the (outgoing) radiation condition, $e ^ { i k \alpha x}$ is the incident plane wave, and $q ( x )$ is a real-valued function, called a potential,

\begin{equation*} q ( x ) \in L ^ { 2 }_\text { loc } ( \mathbf{R} ^ { 3 } ), \end{equation*}

\begin{equation*} | q ( x ) | \leq c ( 1 + | x | ) ^ { - b } , b > 2,\text{ for large }|x|. \end{equation*}

The existence and uniqueness of the solution to (a1)–(a3) has been proved under less restrictive assumptions on $q ( x )$ [a2]. The function $v$ has the form

\begin{equation*} v ( x , \alpha , k ) = A ( \alpha ^ { \prime } , \alpha , k ) \frac { e ^ { i k r} } { r } + o \left( \frac { 1 } { r } \right), \end{equation*}

\begin{equation*} r \rightarrow \infty , \frac { x } { r } = \alpha ^ { \prime }, \end{equation*}

where the coefficient $A ( \alpha ^ { \prime } , \alpha , k )$ is called the scattering amplitude.

The inverse potential scattering problem consists of finding $q ( x )$ given $A ( \alpha ^ { \prime } , \alpha , k )$ on some subsets of $S ^ { 2 } \times S ^ { 2 } \times \mathbf{R} _ { + }$.

The first result is simple: If $A ( \alpha ^ { \prime } , \alpha , k )$ is known for all $\alpha ^ { \prime } , \alpha \in S ^ { 2 }$ and all $k > 0$, then $q ( x )$ is uniquely determined.

If

\begin{equation*} q \in Q _ { m } : = \left\{ \begin{array} { c } { q = \overline { q } }, \\ { q : | q ( x ) | + | \nabla ^ { m } q | \leq c ( 1 + | x | ) ^ { - b } }, \\ { b > 3 } \end{array} \right\}, \end{equation*}

then it is known (e.g. [a6], p. 233, see also [a4]) that

\begin{equation*} A ( \alpha ^ { \prime } , \alpha , k ) = - \frac { 1 } { 4 \pi } \int _ { \mathbf{R} ^ { 3 } } e ^ { i k (\alpha - \alpha ^ { \prime } ) x } q ( x ) d x + O \left( \frac { 1 } { k } \right), \end{equation*}

\begin{equation*} k \rightarrow \infty, \end{equation*}

so that $\tilde{q} ( \xi ) : = \int _ { \mathbf{R} ^ { 3 } } e ^ { - i \xi x } q ( x ) d x$ can be found:

The second result is much more difficult.

For decades it was not known if the data $A ( \alpha ^ { \prime } , \alpha ) : = A ( \alpha ^ { \prime } , k _ { 0 } )$, $\forall \alpha ^ { \prime }$, $\alpha \in S ^ { 2 }$ and $k _ { 0 } > 0$ fixed, determine $q ( x )$ uniquely. In 1987 the uniqueness result has been established by A.G. Ramm (see [a7], [a8]) under the assumptions $q ( x ) \in L ^ { 2 } ( \mathbf{R} ^ { 3 } )$, $q ( x ) = 0$ for $| x | > a$, where $a > 0$ is an arbitrary large fixed number, and in 1988 inversion procedures were published; see [a7]. One of them, proposed by Ramm, is based on the formula

where $M : = \left\{ \theta : \theta \in \mathbf{C} ^ { 3 } , \theta . \theta = k ^ { 2_0 } \right\}$, $\theta . w : = \sum _ { j = 1 } ^ { 3 } \theta _ { j } .w _ { j }$, $v ( \alpha , \theta ) \in L ^ { 2 } ( S ^ { 2 } )$, and $\xi \in \mathbf{R} ^ { 3 }$ is an arbitrary point.

Another inversion procedure ([a3], [a7]) is based on the reconstruction of the Dirichlet-to-Neumann mapping and then finding $q ( x )$.

Error estimates for Ramm's inversion procedure in the case of noisy data and an algorithm for calculating the function $v ( \alpha , \theta )$ in the inversion formula are obtained in [a9].

The uniqueness problem for inverse potential scattering with the data $A ( \alpha ^ { \prime } , \alpha_0 , k )$, $\forall \alpha ^ { \prime } \in S ^ { 2 }$, $\forall k > 0$, $\alpha _ { 0 } \in S ^ { 2 }$, fixed, is still open (as of 2000).

The same is true for the uniqueness problem for inverse potential scattering with the (backscattering) data $A ( - \alpha , \alpha , k )$, $\forall \alpha \in S ^ { 2 }$, $\forall k > 0$, although for this problem a uniqueness theorem for small $q ( x )$ holds.

Inverse geophysical scattering.

The inverse geophysical scattering problem consists of finding the unknown coefficient $v ( x )$ in the equation

\begin{equation} \tag{a4} ( \nabla ^ { 2 } + k ^ { 2_0 } + k ^ { 2_0 }v ( x ) ) u ( x , y , k _ { 0 } ) = - \delta ( x - y ) \text { in } \mathbf{R} ^ { 3 }, \end{equation}

where $u : = u ( x , y ) : = u ( x , y , k _ { 0 } )$ satisfies the outgoing radiation condition (a3), $k _ { 0 } = \text { const } > 0$ is fixed, and $v ( x )$ is a real-valued $L _ { \text{loc} } ^ { 2 }$ function with compact support in $\mathbf{R} _ { - } ^ { 3 } : = \{ x : x _ { 3 } < 0 \}$.

The scattering data are the values $u ( x , y )$, $\forall x , y \in P : = \{ x : x_ {3} = 0 \}$, that is, the values of $u$ on the surface of the Earth. The function $v ( x )$ describes an inhomogeneity in the velocity profile (in the refraction coefficient), $u$ can be an acoustic pressure. Uniqueness of the solution to inverse geophysical scattering problem was proved in 1987 [a8], [a7].

The uniqueness problem for inverse geophysical scattering with data $u ( x , y_{0} , k )$, $\forall x \in P$, $\forall k > 0$, and $y _ { 0 } \in P$ fixed, is open (as of 2000).

A reduction of the inverse geophysical scattering problem with the data $u ( x , y , k _ { 0 } )$, $\forall x , y \in P$, to the inverse potential scattering problem with the data $A ( \alpha ^ { \prime } , \alpha , k _ { 0 } )$, $\forall \alpha , \alpha ^ { \prime } \in S _ { + } ^ { 2 }$, $k _ { 0 } > 0$ fixed, $S _ { + } ^ { 2 } : = \left\{ \alpha : \alpha \in S ^ { 2 } , \alpha \cdot e _ { 3 } > 0 \right\}$, with $e_3$ the unit vector along $x _ { 3 }$-axis, is done in [a7].

Inverse potential scattering: Open problem.

An interesting open problem (as of 2000) in inverse potential scattering is the problem of finding discontinuities of $q ( x )$ and the number of bound states of the Schrödinger operator generated by the expression $- \nabla ^ { 2 } + q ( x )$ in $L ^ { 2 } ( \mathbf{R} _ { 3 } )$ from the knowledge of fixed energy scattering data $A ( \alpha ^ { \prime } , \alpha , k _ { 0 } )$, $\forall \alpha ^ { \prime } , \alpha \in S ^ { 2 }$.

If $q \in L ^ { 2_0 } (\mathbf{ R} ^ { 3 } )$, then $A ( \alpha ^ { \prime } , \alpha )$ is an analytic function of $\alpha ^ { \prime } , \alpha \in M$. Therefore, knowledge of $A ( \alpha ^ { \prime } , \alpha )$ on an open set in $S ^ { 2 } \times S ^ { 2 }$, however small, allows one to recover $A ( \alpha ^ { \prime } , \alpha )$ on $M \times M$.

The assumption concerning compactness of the support of $q ( x )$ is natural in inverse potential scattering because the scattering data are always noisy and it is not possible in principle to recover the tail of a $q ( x ) \in Q$ (that is, $q ( x )$ for $| x | > R$, where $R > 0$ is sufficiently large) from knowledge of noisy data $A _ { \delta } ( \alpha ^ { \prime } , \alpha )$,

\begin{equation*} \operatorname { sup } _ { \alpha , \alpha ^ { \prime } \in S ^ { 2 } } | A _ { \delta } ( \alpha ^ { \prime } , \alpha ) - A ( \alpha ^ { \prime } , \alpha ) | < \delta \end{equation*}

(see [a7] for a proof).

References

[a1] H. Cycon, R. Froese, W. Kirsch, B. Simon, "Schrödinger operators" , Springer (1986)
[a2] L. Hörmander, "Analysis of linear partial differential operators" , IV , Springer (1985)
[a3] A. Nachman, "Reconstruction from boundary measurements" Ann. Math. , 128 (1988) pp. 531–578
[a4] R. Newton, "Inverse Schrödinger scattering in three dimensions" , Springer (1989)
[a5] D. Pearson, "Quantum scattering and spectral theory" , Acad. Press (1988)
[a6] A.G. Ramm, "Random fields estimation theory" , Longman/Wiley (1990)
[a7] A.G. Ramm, "Multidimensional inverse scattering problems" , Longman/Wiley (1992)
[a8] A.G. Ramm, "Recovery of the potential from fixed energy scattering data" Inverse Probl. , 4 (1988) pp. 877–886 (See also: Ibid. 3 (1987), L77-82)
[a9] A.G. Ramm, "Stability estimates in inverse scattering" Acta Applic. Math. , 28 : 1 (1992) pp. 1–42
[a10] A.G. Ramm, "Stability of solutions to inverse scattering problems with fixed-energy data" Rend. Sem. Mat. e Fisico (2001) pp. 135–211
How to Cite This Entry:
Inverse scattering, multi-dimensional case. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Inverse_scattering,_multi-dimensional_case&oldid=16465
This article was adapted from an original article by A.G. Ramm (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article