Asymptotic behavior of solutions of \(-\Delta v+qv=\lambda v\) and the distance of \(\lambda\) to the essential spectrum
type: Item
instance of: scholarly article
title: Asymptotic behavior of solutions of \(-\Delta v+qv=\lambda v\) and the distance of \(\lambda\) to the essential spectrum
zbMATH Open document ID: 0622.35053
DOI: 10.1007/BF01161966
author: Andreas M. Hinz
published in: Mathematische Zeitschrift
publication date: 1987-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/173825
review text: The main results of the paper are upper bounds for the distance of \(\lambda\in {\mathbb{R}}\) to the essential spectrum of an operator \(\overline{-\Delta +q},\) depending on growth or decay properties of distributional solutions v for the equation \(-\Delta v+qv=\lambda v\). From these bounds it is derived that if \(q(x) \geq -o(| x|^ 2)\) outside a ball, then the existence of a bounded solution guarantees \(\lambda \in \sigma (\overline{-\Delta +q}),\) and every eigenfunction to a discrete eigenvalue decays exponentially. Furthermore, an example of an eigenvalue in the essential spectrum and lower bounds for \(non\)-L\({}_ 2\)-solutions are given. \(q(x) \sim -O(| x|^ 2)\) appears as a borderline case, and the question is raised if, in that case, there may be a \(\lambda\) outside the spectrum of \(\overline{-\Delta +q}\) with a bounded solution v of \(-\Delta v+qv=\lambda v.\)
Mathematics Subject Classification ID: 35P20
Mathematics Subject Classification ID: 35J10
Mathematics Subject Classification ID: 35B40
zbMATH DE Number: 4008962
zbMATH Keywords: Schrödinger operators
zbMATH Keywords: essential spectrum
zbMATH Keywords: growth
zbMATH Keywords: decay
zbMATH Keywords: distributional solutions
zbMATH Keywords: existence
MaRDI profile type: Publication
cites work: Q4073917
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OpenAlex ID: W2911664925
description: scientific article; zbMATH DE number 4008962