View source for Inverse scattering, multi-dimensional case

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.

==Inverse potential scattering.==
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>

<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>

<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>

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,

<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>

<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>

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

<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>

<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>

where the coefficient <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130070/i13007016.png" /> 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 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.

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>

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>

<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>

so that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130070/i13007027.png" /> 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>

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

<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>

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.

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" />.

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]]].

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 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.

==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

<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>

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" />.

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 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).

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]]].

==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" />.

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" />.

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" />,

<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>

(see [[#References|[a7]]] for a proof).

====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>