Growth order of eigenfunctions of Schrödinger operators with potentials admitting some integral conditions. I: General theory
type: Item
instance of: scholarly article
title: Growth order of eigenfunctions of Schrödinger operators with potentials admitting some integral conditions. I: General theory
zbMATH Open document ID: 0879.35116
author: Masaharu Arai
author: Jun Uchiyama
published in: Publications of the Research Institute for Mathematical Sciences, Kyoto University
publication date: 1998-01-29T00:00:00Z
review text: The authors consider the lower bounds of the growth order of a solution \(u(x)\in H^2_{\text{loc}}(\Omega)\) of a second-order elliptic equation: \[ -\sum^n_{j=1}(\partial_j+ \sqrt{-1}b_j(x))^2u(x)+ (q_1(x)+ q_2(x))u(x)= 0\tag{\(*\)} \] for \(x\in\Omega:= \{x|x\in\mathbb{R}^n, |x|>R_0\}\), where \(b_j(x)\) and \(q_1(x)\) are real functions, \(q_2(x)\) is a complex-valued function. To obtain the lower bounds of the growth order, is one of the main tools to show the nonexistence of eigenvalues embedded in the continuous spectrum. Among many other assumptions, \(q_j(x)\), \(j=1,2\), are supposed to satisfy: for any \(w(x)\in H^1_{\text{loc}}(\Omega)\), \(q_j(x)|w(x)|\), \(\partial_rq_1(x)\in L^1_{\text{loc}}(\Omega)\). Under some additional assumptions on \(q_j(x)\) the author proves lower bounds for \(u(x)\).
Mathematics Subject Classification ID: 35P05
Mathematics Subject Classification ID: 35J10
Mathematics Subject Classification ID: 35B05
zbMATH DE Number: 1002130
zbMATH Keywords: nonexistence of eigenvalues
zbMATH Keywords: lower bounds
reviewed by: Akira Tsutsumi
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.2977/prims/1195162713
OpenAlex ID: W2023726047
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DOI: 10.2977/PRIMS/1195162713
description: scientific article; zbMATH DE number 1002130