On the Darboux transformation of \(N\)th order
type: Item
instance of: scholarly article
title: On the Darboux transformation of \(N\)th order
zbMATH Open document ID: 0996.34069
published in: Russian Mathematics
publication date: 2001-03-07T00:00:00Z
review text: Let \(h_0\Psi(x)=E\Psi\), \(h_0\equiv -D^2+V(x)\), \(D\equiv {d\over dx}\), with \(x\in(a,b)\). \(V(x)\) is a sufficiently smooth real-valued function in \((a,b)\). A linear differential \(N\)th-order operator \(\widehat L^{(N)}\), with coefficient of \(D^N=1\), acting from \(T_0\equiv \{\Psi:(h_0-E) \Psi(x)=0\}\) to \(T_{N1}= \{\varphi:\varphi= \widehat L\Psi,\;\Psi\in T_0\}\) is a Darboux transformation of order \(N\), if \(\widehat L^{(N)}h_0= (h_0+A_N(x)\widehat L^{(N)})\), where \(A_N(x)\) is a sufficiently smooth function. It is established, that ``The action of any nontrivial operator \(\widehat L^{(N)}\) is equivalent to the resulting action of a certain chain of \(k(\leq N)\) Darboux transformations of first order''.
Mathematics Subject Classification ID: 34L40
Mathematics Subject Classification ID: 81Q05
zbMATH DE Number: 1541486
zbMATH Keywords: Schrödinger operator
zbMATH Keywords: Darboux transformation
author: Boris F. Samsonov
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1541486