On the determination of differential equations with singularities and turning points
type: Item
instance of: scholarly article
title: On the determination of differential equations with singularities and turning points
zbMATH Open document ID: 1039.34008
DOI: 10.1007/BF03322770
published in: Results in Mathematics
publication date: 2003-10-29T00:00:00Z
review text: The paper deals with an inverse problem for stratified media, namely, one considers the system \[ \begin{cases} \frac{dy_1}{dx}=i\rho R(x)y_2,\;\frac{dy_2}{dx}=i\rho\frac{1}{R(x)}y_1,\quad x\in[0,T],&\\ y_1(0,\rho)=1,\;y_2(0,\rho)=-1,&\\ \end{cases} \tag{1} \] where \(T>0\), \(\rho\in{\mathbb C}\) is a spectral parameter and \(R\) is a real nonnegative function called the wave resistance. The function \(R\) is supposed to have the form \[ R(x)=\prod_{j=1}^N| x-\gamma_j|^{p_j-1}R_0(x), \] where \(0<\gamma_1<\cdots<\gamma_N<T\), \(p_j\in{\mathbb R}\), \(R_0,R_0',R_0^{\prime\prime} \in L^1[0,T]\), \(R_0>0\), \(R(0)=1\), \(R'(0)=0\). The inverse problem the paper deals with is: given the coefficient \[ r(\rho)=\frac{y_1(T,\rho)+R(T)y_2(T,\rho)}{y_2(T,\rho)-R(T)y_1(T,\rho)}, \] construct \(R(x)\). The authors show that \(R\) is uniquely determined by \(r\) and give a method of reconstruction of \(R\) from \(r\). The main idea is to associate to the above inverse problem an inverse problem for a Schrödinger operator on a half-axis, with a singular potential.
Mathematics Subject Classification ID: 34A55
Mathematics Subject Classification ID: 34B05
Mathematics Subject Classification ID: 34L40
Mathematics Subject Classification ID: 47E05
zbMATH DE Number: 1997871
zbMATH Keywords: inverse problem
zbMATH Keywords: stratified media
zbMATH Keywords: reflection coefficient
author: Gerhard Freiling
author: V. A. Yurko
reviewed by: Ingrid Beltiţă
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1997871