Unbounded solutions of the stationary Schrödinger equation on Riemannian manifolds
type: Item
instance of: scholarly article
title: Unbounded solutions of the stationary Schrödinger equation on Riemannian manifolds
zbMATH Open document ID: 1138.35306
DOI: 10.1007/BF03321048
author: A. G. Losev
author: E. A. Mazepa
author: Victoriya Y. Chebanenko
published in: Computational Methods and Function Theory
publication date: 2004-11-01T00:00:00Z
review text: This paper deals with the investigation of the behaviour of solutions of the stationary Schrödinger equation in relation to the geometry of the manifold in question. More precisely the authors study question of existence of unbounded solutions of \[ Lu:= \Delta u- cu= 0,\tag{1} \] where \(c\) is a smooth nonnegative function on a non-compact Riemannian manifiold \(M\) without boundary and are interested in the interrelation between problems of existence of solutions of (1) and off some compact \(\mathbb B\subset M\) with the same growth ``at infinity''. To this end, the authors use a new approach based on the consideration of equivalence classes of functions on \(M\) (for bounded solutions this approach was known).
Mathematics Subject Classification ID: 35J10
Mathematics Subject Classification ID: 30F20
Mathematics Subject Classification ID: 31A15
Mathematics Subject Classification ID: 31B15
zbMATH DE Number: 2112949
zbMATH Keywords: non-compact Riemannian manifold
zbMATH Keywords: unbounded solution
zbMATH Keywords: Schrödinger equation
Wikidata QID: Q115390821
reviewed by: Messoud Efendiev
MaRDI profile type: Publication
cites work: Positive harmonic functions on complete manifolds of negative curvature
cites work: On the Dirichlet Problem at Infinity for Manifolds of Nonpositive Curvature
cites work: Q4328567
cites work: Elliptic Partial Differential Equations of Second Order
cites work: Analytic and geometric background of recurrence and non-explosion of the Brownian motion on Riemannian manifolds
cites work: Q3780159
cites work: Elliptic partial differential equations on the warped products of riemannian manifolds
cites work: Bounded solutions of the Schrödinger equation on Riemannian products
cites work: Q4783250
cites work: Some criteria for parabolicity and hyperbolicity of the boundary sets of surfaces
cites work: Uniformly elliptic operators on Riemannian manifolds
cites work: Harmonic functions on complete riemannian manifolds
cites work: Nonlinear analysis in geometry
description: scientific article; zbMATH DE number 2112949