Partial identification of the potential from phase shifts
type: Item
instance of: scholarly article
title: Partial identification of the potential from phase shifts
zbMATH Open document ID: 1221.34229
author: Barnabás Apagyi
published in: Journal of Mathematical Analysis and Applications
publication date: 2011-05-16T00:00:00Z
review text: Consider the inverse scattering problem for the operator \[ \phi ^{^{\prime \prime }}(r,\lambda)-\frac{\lambda ^{2}-\frac{1 }{4}}{r^{2}}\phi (r,\lambda)+(1-q(r))\phi (r,\lambda)=0,\quad r\geq 0, \] \[ \phi (r,\lambda)=\gamma (\lambda) r^{\lambda + \frac{1}{2}}(1+o(1))\text{ as }r\rightarrow 0+, \] \[ \phi(r,\lambda)=\sin \left( r-\tfrac{\pi }{2}\left( \lambda - \tfrac{1}{2}\right) +\delta (\lambda)\right) +o(1) \text{ as }r\rightarrow +\infty \] with fixed energy \(k^{2}=1\). The quantities \(\delta \left( n+\frac{1}{2} \right) =\delta _{n}\) are called phase shifts and \(\gamma \left( n+\frac{1}{2 }\right) =\gamma _{n}\) for \(n\geq 0.\) The inverse scattering problem here consists of the recovery of the potential \(q\)\ from the phase shifts \(\delta _{n}.\) The scattering amplitude \(F(t)\) can be expressed by the phase shifts: \[ F(t)=\sum_{n=0}^{\infty }(2n+1) F_{n}P_{n}(t),\quad F_{n}=e^{i\delta _{n}}\sin \delta _{n}, \] where the functions \(P_{n}(t)\) are the Legendre polynomials. This paper shows that the tail of the potential can be determined by the asymptotical behavior of the scattering amplitude and of the phase shifts. The following results are obtained: Assume that the potential \(q\) satisfies \(rq(r),rq^{\ast }(r)\in L_{1}(0,\infty)\) and let \(0<a<\infty \). (a) If \(q=q^{\ast }\) a.e. on \((a,\infty)\) then \(F(t) -F^{\ast }(t)\) is an entire function and satisfies \[ |F(t)-F^{\ast }(t)| \leq c(1+|t|)\exp (a\sqrt{2|t|}). \] Conversely, if \(q\) and \(q^{\ast }\) have compact support, \(F(t) -F^{\ast }(t)\) is an entire function and \[ F(t) -F^{\ast }(t) =O(\exp (a_{1}\sqrt{ 2|t|})) \] for all \(a_{1}>a\), then \(q=q^{\ast }\) a.e. on \((a,\infty)\). (b) If \(q=q^{\ast }\) a.e. on \((a,\infty),\) then, for all sufficiently large \(n\), it holds \[ |\delta _{n}-\delta _{n}^{\ast }| \leq \frac{c}{n^{2}} \left( \frac{ae}{2n}\right) ^{2n}. \] Conversely, if \(q\) and \(q^{\ast }\) have compact support and \[ \delta_{n}-\delta _{n}^{\ast }=O\left( \lef…
reviewed by: Kh. R. Mamedov
Mathematics Subject Classification ID: 34L25
Mathematics Subject Classification ID: 34D05
Mathematics Subject Classification ID: 34D20
Mathematics Subject Classification ID: 34A55
zbMATH DE Number: 5888558
zbMATH Keywords: inverse scattering
zbMATH Keywords: phase shifts
zbMATH Keywords: scattering amplitude
MaRDI profile type: Publication
full work available at URL: j.jmaa.2010.10.071
OpenAlex ID: W1981734568
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DOI: 10.1016/J.JMAA.2010.10.071
description: scientific article; zbMATH DE number 5888558