Asymptotic of the density of states for the Schrödinger operator with periodic electric potential
type: Item
instance of: scholarly article
title: Asymptotic of the density of states for the Schrödinger operator with periodic electric potential
zbMATH Open document ID: 0951.35104
DOI: 10.1215/S0012-7094-98-09201-8
author: Bernard Helffer
author: Abderemane Morame
published in: Duke Mathematical Journal
publication date: 2000-03-27T00:00:00Z
review text: Let \(\Gamma\) be a lattice on \(\mathbb{R}^n\), \(V\) be a \(C^\infty\) \(\Gamma\)-periodic potential and \(P_V\) be the Schrödinger operator on \(L^2(\mathbb{R}^n)\), \(n\geq 2: P_V= \sum_{1\leq j\leq n} D^2_j+ V\). The authors prove that for the integrated density of states \(N(\mu, P_V)\), we have for any fixed \(\varepsilon> 0\), the following asymptotic expansion as \(\mu\to\infty\): \[ N(\mu, P_V)= a_n \mu^{n/2}+ a_{n-2} \mu^{(n- 2)/2}+ O(1)+ O(\mu^{(n- 3+\varepsilon)/2}) \] with \[ a_n= (2\pi)^{-n}|S^{n-1}|/n> a_{n- 2}= -{na_n\over 2} |K|^{-1} \int_K V(x) dx \] (\(K\) a periodic cell). They also recover a proof of the Béthe-Sommerfeld conjecture for \(n\leq 4\). The method is related to the one used by Hörmander for the study of the counting function of the eigenvalues of an elliptic operator and needs a periodic pseudodifferential calculus.
Mathematics Subject Classification ID: 35P20
Mathematics Subject Classification ID: 47F05
Mathematics Subject Classification ID: 47N50
Mathematics Subject Classification ID: 81Q10
Mathematics Subject Classification ID: 35J10
zbMATH DE Number: 1425090
reviewed by: Viorel Iftimie
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1425090