Gaussian decay of the magnetic eigenfunctions
type: Item
instance of: scholarly article
title: Gaussian decay of the magnetic eigenfunctions
zbMATH Open document ID: 0863.35089
DOI: 10.1007/BF02247886
author: László Erdős
published in: Geometric and Functional Analysis. GAFA
publication date: 1997-06-03T00:00:00Z
full work available at URL: https://eudml.org/doc/58225
review text: Without magnetic fields it is well-known that the eigenfunctions of Schrödinger operators decay exponentially in the classically forbidden region. In two dimensions with a constant magnetic field and without a potential the decay of the eigenfunctions corresponding to the Landau levels is much stronger, they decay like a Gaussian at infinity. But not much was known in this case with a potential added, except for rotationally invariant potentials. The paper under review fills in this gap. The author investigates the decay of the eigenfunctions of a magnetic Schrödinger operator in two dimensions with a rotationally invariant magnetic field and a potential analytic in the angular variable. For the proof a probabilistic representation of the Schrödinger operator, i.e. a Feynman-Kac formula, in the angular momentum sectors is established. With the help of this representation, it is shown that the eigenfunctions with energy below the essential spectrum exhibit a Gaussian decay at infinity. Due to the probabilistic approach, however, this decay is shown for eigenfunctions with energy below the essential spectrum only. The result has recently been extended to all eigenfunctions with energies in gaps of the essential spectrum, see \textit{S. Nakamura} [Commun. Partial Differ. Equations 21, No. 5-6, 993-1006 (1996)].
Mathematics Subject Classification ID: 35Q40
Mathematics Subject Classification ID: 35J10
Mathematics Subject Classification ID: 81Q10
zbMATH DE Number: 912452
zbMATH Keywords: decay of the eigenfunctions of a magnetic Schrödinger operator
zbMATH Keywords: Feynman-Kac formula
zbMATH Keywords: Gaussian decay
MaRDI profile type: Publication
cites work: CONTINUITY PROPERTIES OF SCHRÖDINGER SEMIGROUPS WITH MAGNETIC FIELDS
cites work: Q3756011
cites work: Probabilistic solution of Pauli type equations
cites work: Q4191141
OpenAlex ID: W2098661948
description: scientific article; zbMATH DE number 912452