Absolute continuity of a two-dimensional magnetic periodic Schrödinger operator with potentials of the type of measure derivative
type: Item
instance of: scholarly article
title: Absolute continuity of a two-dimensional magnetic periodic Schrödinger operator with potentials of the type of measure derivative
zbMATH Open document ID: 1040.35049
published in: Journal of Mathematical Sciences (New York)
publication date: 2003-08-31T00:00:00Z
review text: The author discusses the absolute continuity of the spectrum for the periodic (with respect to the lattice \(\Gamma\)) Schrödinger operator \(H\) in \(L_2({\mathbb R}^2)\) with magnetic potential, electric potential and variable metric \[ H = (i\overrightarrow{\nabla} - \overrightarrow{A}(x))^* g(x) (i\overrightarrow{\nabla} - \overrightarrow{A}(x)) + V(x). \] An electric potential \(V\) is assumed to be a distribution formally given by an expression \({d\nu} \over {dx}\), where \(\nu\) is a periodic signed measure with a locally finite variation form bounded with respect to \(\Delta\). The magnetic potential \(\overrightarrow{A}\) is assumed to be a real \(\Gamma\)-periodic function from \(L_r, r > 2\). The metric \(g\) is defined by a \((2\times 2)\)-matrix valued function with \(\Gamma\)-periodic real-valued entries from Sobolev space \(W^2_r\), \(r > 2\), such that \(c_1 I \leq g \leq c_2I\). The proof of absolute continuity is based on the method due to L. Thomas (1973).
Mathematics Subject Classification ID: 35P05
Mathematics Subject Classification ID: 35J10
Mathematics Subject Classification ID: 35Q40
Mathematics Subject Classification ID: 35Q60
Mathematics Subject Classification ID: 81Q10
zbMATH DE Number: 1971380
zbMATH Keywords: absolute continuity
zbMATH Keywords: spectrum
zbMATH Keywords: Schrödinger operator
zbMATH Keywords: magnetic potential
zbMATH Keywords: electric potential
author: Roman Shterenberg
reviewed by: Michael Perelmuter
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1971380