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On the asymptotic number of edge states for magnetic Schrödinger operators
Author(s) -
Frank Rupert L.
Publication year - 2007
Publication title -
proceedings of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.899
H-Index - 65
eISSN - 1460-244X
pISSN - 0024-6115
DOI - 10.1112/plms/pdl024
Subject(s) - mathematics , eigenvalues and eigenvectors , schrödinger's cat , operator (biology) , curl (programming language) , mathematical analysis , mathematical physics , essential spectrum , boundary (topology) , limit (mathematics) , magnetic field , curvature , spectrum (functional analysis) , pure mathematics , quantum mechanics , geometry , physics , biochemistry , chemistry , repressor , computer science , transcription factor , gene , programming language
We consider a Schrödinger operator ( h D − A ) 2 with a positive magnetic field B = curl A in a domain Ω ⊂ ℝ 2 . The imposing of Neumann boundary conditions leads to the existence of some spectrum below h ∈ f B . This is a boundary effect and it is related to the existence of edge states of the system. We show that the number of these eigenvalues, in the semi‐classical limit h → 0, is governed by a Weyl‐type law and that it involves a symbol on ∂Ω. In the particular case of a constant magnetic field, the curvature plays a major role.