Asymptotic expansion for the density of states of the magnetic Schrödinger operator with a random potential
type: Item
instance of: scholarly article
title: Asymptotic expansion for the density of states of the magnetic Schrödinger operator with a random potential
zbMATH Open document ID: 0851.35145
DOI: 10.1007/BF02099434
author: W.-M. Wang
published in: Communications in Mathematical Physics
publication date: 1996-11-25T00:00:00Z
review text: The author presents a study of the asymptotics for the density of states of the magnetic Schrödinger operator with a random potential defined on \(L^2(\mathbb{R}^2)\) \[ P^\omega_{B, V}= (D_x+ By)^2+ D^2_y+ V^\omega(x, y), \] where \(D_x= {1\over i} \partial_x\), \(D_y= {1\over i} \partial_y\) and \(B> 0\) is a constant. If \(v\) is a function in \(C^\infty_0(\mathbb{R}^2)\), then the potential \(V^\omega\) is defined as \[ V^\omega(x, y)= \sum_{(i, j)\in \mathbb{Z}^2} \alpha_{i, j} v(x- i, y- j), \] where \(\alpha= \{\alpha_{i, j}\}_{(i, j)\in \mathbb{Z}^2}\) is a family of random variables on a probability space \((\Omega, P)\). By using the method of effective Hamiltonian [see for example the reviewer and \textit{J. Sjöstrand}, Lect. Notes Phys. 345, 118-197 (1989; Zbl 0699.35189)], complex dilation and complex translation, the author obtains in the large magnetic field limit, the asymptotic expansion for the density of states measure considered as distribution. This justifies the Wegner approximation in some weak sense [cf. \textit{F. Wegner}, Z. Phys. B. Condensed Matter 51, No. 4, 279-285 (1983)].
reviewed by: Bernard Helffer
Mathematics Subject Classification ID: 35S05
Mathematics Subject Classification ID: 82B44
Mathematics Subject Classification ID: 35R60
zbMATH DE Number: 850571
zbMATH Keywords: asymptotics
zbMATH Keywords: density of states
zbMATH Keywords: magnetic Schrödinger operator
zbMATH Keywords: random potential
zbMATH Keywords: complex dilation
zbMATH Keywords: complex translation
zbMATH Keywords: Wegner approximation
MaRDI profile type: Publication
cites work: Q3477353
cites work: Spectral properties of disordered systems in the one-body approximation
cites work: Q4016480
full work available at URL: https://doi.org/10.1007/bf02099434
OpenAlex ID: W1983163957
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description: scientific article; zbMATH DE number 850571