Nonhomogeneity of Picard dimensions of rotation free hyperbolic densities
type: Item
instance of: scholarly article
title: Nonhomogeneity of Picard dimensions of rotation free hyperbolic densities
zbMATH Open document ID: 0846.35037
author: Toshimasa Tada
published in: Hiroshima Mathematical Journal
publication date: 1996-10-06T00:00:00Z
review text: Consider the time-independent Schrödinger equation \((- \Delta+ P(x)) u(x)= 0\) on the punctured disc \(\Omega= \{z\in C; |z|< 1\}\), for some potential \(P\) which is assumed locally Hölder continuous on \(0< |z|\leq 1\). We denote the set \(\{z; |z|= 1\}\) by \(\Gamma\) and consider the convex set \(PP_1(\Omega; \Gamma)\) of nonnegative solutions of the equations which satisfy \((- 1/2\pi) \int^{2\pi}_0 [(\partial/\partial r)u(r\exp [i\theta])]_{r= 1} d\theta= 1\) and call ``Picard dimension'' of \(P\) at \(0\) the cardinal number of the set of extremal points of \(PP_1(\Omega; \Gamma)\). A potential is called hyperbolic if the associated Picard dimension is \(\geq 1\) and if the Schrödinger equation on \(\Omega\) admits a Green function. The author shows that there are radial hyperbolic potentials \(P\) so that the Picard dimension of \(P\) is 1, whereas the Picard dimension of \(P/4\) is the cardinal number of the continuum.
Mathematics Subject Classification ID: 35J10
zbMATH DE Number: 811372
zbMATH Keywords: nonnegative solutions
zbMATH Keywords: Picard dimension
reviewed by: Otto Liess
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.32917/hmj/1206127711
OpenAlex ID: W2118859151
Wikidata QID: Q128899692
DOI: 10.32917/HMJ/1206127711
description: scientific article; zbMATH DE number 811372