Some quantum operators with discrete spectrum but classically continuous spectrum
type: Item
instance of: scholarly article
title: Some quantum operators with discrete spectrum but classically continuous spectrum
zbMATH Open document ID: 0547.35039
DOI: 10.1016/0003-4916(83)90057-X
author: B. Simon
published in: Annals of Physics
publication date: 1983-01-01T00:00:00Z
review text: This paper gives some examples for the breakdown of the following rule of thumb in quantum mechanics: If the volume (i.e. the Lebesgue measure) of the set \(\{(p,q)\in {\mathbb{R}}^{2\nu}| p^ 2+V(q)\leq E\}\) for some \(E<\infty\) is infinite then the classical intuition says that the spectrum of the Hamiltonian \(H=-\Delta +V(x)\) in \(L^ 2({\mathbb{R}}^{\nu})\) is not purely discrete. The author considers the two dimensional Hamiltonian \(H_ 1=-\partial^ 2/\partial x^ 2- \partial^ 2/\partial y^ 2+x^ 2y^ 2\) which is closely related to the Dirichlet operator \(H_ 2=-\Delta_ D^{\Omega}\) with zero boundary conditions on \(\Omega =\{(x,y)\in {\mathbb{R}}^ 2| | xy|\leq 1\}.\) He gives five proofs that \(H_ 2\) has discrete spectrum, where three of them work directly for \(H_ 1\). The different proofs have different virtues and use different reasoning. The arguments run from estimates for the zero point harmonic oscillator, Dirichlet-Neumann bracketing, path integral representations and a ''sliced bread'' inequality, to a theorem of Feffermann and Phong which estimates the number of eigenvalues by the number of cubes \(\Delta_ j^{\lambda}\) centered at \(\lambda^{- {1\over2}}j\) and of side \(\lambda^{-{1\over2}}\) with the property \(\max\{V(x)|\quad x\in\Delta_ j^{\lambda}\}\leq b\lambda\) (for suitable b).
Mathematics Subject Classification ID: 35J10
Mathematics Subject Classification ID: 35P15
Mathematics Subject Classification ID: 47B25
zbMATH DE Number: 3871879
zbMATH Keywords: Schrödinger operators
zbMATH Keywords: quantum mechanics
zbMATH Keywords: discrete spectrum
zbMATH Keywords: number of eigenvalues
MaRDI profile type: Publication
cites work: The uncertainty principle and sharp gårding inequalities
cites work: Q4185363
cites work: Classical boundary conditions as a technical tool in modern mathematical physics
cites work: Q3872705
cites work: Scattering theory for Schrödinger equation with potential periodic in time
full work available at URL: 0003-4916(83)90057-x
OpenAlex ID: W2026341320
description: scientific article; zbMATH DE number 3871879