GAP estimates of Schrödinger operator
type: Item
instance of: scholarly article
title: GAP estimates of Schrödinger operator
zbMATH Open document ID: 0886.35102
author: Roger Chen
published in: Pacific Journal of Mathematics
publication date: 1998-05-10T00:00:00Z
review text: The aim of this paper is to give a lower bound for the gap \(\Gamma_k:= (\eta_k-\eta_1)\) of \(k\)th eigenvalues \(\eta_k\) and \(\eta_1\) of the following Neumann eigenvalue problem \[ \Delta u-qu= -\eta u \quad\text{in }M \qquad\text{and}\quad \frac{\partial u}{\partial\nu} =0\quad\text{on }\partial M. \] Here \(M\) is an \(m\)-dimensional compact Riemannian manifold \((m\geq 3)\) and \(\partial M\neq\emptyset\) the boundary on \(M\) and \(q\in C^2(M)\). More precisely, the author proves, under some geometrical assumptions on \(M\) and convexity conditions for \(\partial M\), that the gap \(\Gamma_k\) satisfies \(\Gamma_k\geq \alpha_1 k^{2/m}\), for all \(k=2,3,\dots\) and some constant \(\alpha_1\) depending on the geometrical quantities of \(M\), \(m\), and the oscillation of \(q\) over \(M\). The techniques used are global estimates for the first eigenfunction and similar arguments as used by \textit{S.-Y. Cheng} and \textit{P. Li} [Comment. Math. Helv. 56, 327-338 (1981; Zbl 0484.53034)].
Mathematics Subject Classification ID: 35P15
Mathematics Subject Classification ID: 35J10
Mathematics Subject Classification ID: 58J50
zbMATH DE Number: 1054053
zbMATH Keywords: Neumann eigenvalue problem
reviewed by: A. Rhandi
MaRDI profile type: Publication
full work available at URL: pjm.1997.178.225
OpenAlex ID: W2106631267
DOI: 10.2140/PJM.1997.178.225
description: scientific article; zbMATH DE number 1054053