Isospectral sets for AKNS systems on the unit interval with generalized periodic boundary conditions
type: Item
instance of: scholarly article
title: Isospectral sets for AKNS systems on the unit interval with generalized periodic boundary conditions
zbMATH Open document ID: 0879.34075
DOI: 10.1007/BF02246765
published in: Geometric and Functional Analysis. GAFA
publication date: 1998-01-19T00:00:00Z
full work available at URL: https://eudml.org/doc/58216
review text: The authors consider the differential operator \(H(p,q)\) generated by the differential expression \[ \left(\left( \begin{matrix} 0 & -1 \\ 1 & 0\end{matrix} \right) {d\over dx} +\left( \begin{matrix} -q & p \\ p & q\end{matrix} \right) \right) F(x) \] and the generalized periodic boundary conditions \[ BF(1) =F(0) \] with square integrable potentials. The spectrum of \(H(p,q)\) for the boundary condition generated by \(B\) is the unbounded collection \(\{\lambda (p,q,B)\}_{k\in \mathbb{Z}}\). The objective of the paper is an inverse spectral theory for \(H(p,q)\). The set \[ \biggl\{ (p,q) \in L^2_{\mathbb{R}} \bigl([0,1] \bigr)^2,\;B\in SL(2,\mathbb{R}):\;\lambda_k(p,q,B)= \lambda_k(p_0,q_0,B_0),\;k\in \mathbb{Z} \biggr\} \] is called the isospectral set associated with \((p_0,q_0,B_0)\). The main results of the paper concern a description of the isospectral sets. The authors prove that if \(B_0\) is a rotation matrix then the generalized periodic spectrum is the usual periodic one and the isospectral set is equal to its section for \(B_0\). When \(B\) is not a rotation, the isospectral sets are cylindrical real analytic submanifolds of \(L^2_{\mathbb{R}}([0,1])^2SL(2,\mathbb{R})\) and their sections for fixed boundary conditions are real analytic submanifolds of \(L^2_{\mathbb{R}}([0,1])^2\).
Mathematics Subject Classification ID: 34L05
Mathematics Subject Classification ID: 34A55
zbMATH DE Number: 881243
zbMATH Keywords: differential operators
zbMATH Keywords: generalized periodic boundary conditions
zbMATH Keywords: inverse spectral theory
zbMATH Keywords: isospectral set
author: Laurent Amour
author: Jean-Claude Guillot
reviewed by: Teresa Regińska
MaRDI profile type: Publication
cites work: Inverse spectral theory for the AKNS system with separated boundary conditions
cites work: Q4856125
cites work: The inverse Sturm–Liouville problem III
cites work: Gaps of one dimensional periodic AKNS systems
cites work: The inverse Sturm–Liouville problem. II
cites work: The inverse Sturm‐Liouville problem. I
cites work: Hill's operator and hyperelliptic function theory in the presence of infinitely many branch points
cites work: Q3882770
cites work: Q3760860
cites work: Isospectral sets for boundary value problems on the unit interval
OpenAlex ID: W2032591983
description: scientific article; zbMATH DE number 881243