Oscillation theory and the density of states for the Schrödinger operator in odd dimension
type: Item
instance of: scholarly article
title: Oscillation theory and the density of states for the Schrödinger operator in odd dimension
zbMATH Open document ID: 0734.34027
DOI: 10.1016/0022-0396(91)90068-K
author: Russell Johnson
published in: Journal of Differential Equations
publication date: 1991-01-01T00:00:00Z
review text: The purpose of the paper is to introduce a rotation number for the Schrödinger operator \[ h=-\Delta +q(x)=- \sum^{n}_{i=1}\frac{\partial^ 2}{\partial x_ i^ 2}+q(x_ 1,x_ 2,...,x_ n) \] in odd dimension \(n=1,3,5,... \). q(x) is a bounded \(C^{\infty}\) potential defined on \({\mathbb{R}}^ n\). This rotation number generalizes the one defined by Johnson-Moser when \(n=1\). The author introduces a measure of the number of ``changes of sign'' of the Dirichlet Green's function g of the operator h restricted to bounded smooth domains \(D\subset {\mathbb{R}}^ n\). Then the rotation number is defined by an averaging procedure, letting D tend to \({\mathbb{R}}^ n\). The main result of the paper is that the rotation number, thus defined, equals a constant times the integrated density of states of h, similar to that in one dimension. It starts with collecting some elementary facts about the Green's function for h on a bounded domain \(\bar D\subset {\mathbb{R}}^ n\) with \(C^{\infty}\) boundary \(\partial D\). This is followed by a geometric method of counting the eigenvalues of h. Finally the main result is obtained.
Mathematics Subject Classification ID: 34C10
Mathematics Subject Classification ID: 35J10
Mathematics Subject Classification ID: 34L40
Mathematics Subject Classification ID: 34B27
zbMATH DE Number: 4214408
zbMATH Keywords: rotation number
zbMATH Keywords: Schrödinger operator
zbMATH Keywords: Dirichlet Green's function
zbMATH Keywords: averaging procedure
MaRDI profile type: Publication
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full work available at URL: 0022-0396(91)90068-k
OpenAlex ID: W1975203344
description: scientific article; zbMATH DE number 4214408