On the density of states for the periodic Schrödinger operator
type: Item
instance of: scholarly article
title: On the density of states for the periodic Schrödinger operator
zbMATH Open document ID: 1021.35027
DOI: 10.1007/BF02384494
author: Yulia Karpeshina
published in: Arkiv för Matematik
publication date: 2003-06-16T00:00:00Z
review text: The operators considered in the paper under review are of the form \[ H_V=(-\Delta)^l+V \qquad \text{in} L^2(\mathbb{R}^n), n\geq 2, \] with \(l\geq 1/2\). Here \(V\) is the operators of multiplication by a real potential that is periodic with respect to the lattice \(\Gamma=\{(a_1k_1,\dots, a_nk_n)\mid (k_1,\dots k_n)\in \mathbb{Z}^n\}\), with \(a_1, \dots, a_n>0\); \(V\) is supposed to be of the form \[ V(x)=\sum\limits_{{m\in \mathbb{Z}^n}\atop{m\neq 0}}v_m\exp{(ip_m(0)\cdot x)}, \] with \(p_m(0)=2\pi (m_1a_1^{-1}, \dots, m_na_n^{-1})\). Moreover, \(V\) satisfies the following condition of smoothness: \[ \sum\limits_{{m\in \mathbb{Z}^n}\atop{m\neq 0}}|v_m||m|^{\nu}<\infty, \quad \text{for some }\nu>0. \tag{1} \] Let \(Q\), \(K\) be the elementary cells of the lattice \(\Gamma\) and of the dual lattice, respectively. For \(t\in K\), \(H(t)\) is, as usual, the reduced operator acting in \(L^2(Q)\) with quasiperiodic boundary conditions depending on \(t\). Let \(N_V(\lambda, t)\) be the counting function of the eigenvalues associated to \(H(t)\). The paper deals with the density of states of \(H_V\), that is, \[ D_V(k^{2l})=\int\limits N_V(k^{2l}, t) \text{d}t. \] The main results of the paper are the following. (i) The estimate \[ D_V(k^{2l})=\omega_nk^n+{\mathcal O}(k^{-\xi}\log k), \quad \xi=4l-n-1 \] holds provided \(l>1/2\) and \(V\) satisfies (1) with \(\nu>\frac{1}{2}(n-1)\), if \(n\geq 3\), or \(\nu>\frac{2}{3}(l+1)\), if \(n=2\). (Here \(\omega_n\) is a constant that depends on \(n\) and the volume of \(K\).) This estimate is proved using a perturbation technique and is a generalization of a previous result of \textit{B. Helffer} and \textit{A. Mohamed} [Duke Math. J. 92, 1-60 (1998; Zbl 0951.35104)]. The estimate also holds for \(V\) belonging to a certain class of periodic differential operator; yet, in the author remarks that, in the particular case of magnetic perturbations, the estimate is not optimal, and that, for this particular case, a better estim…
Mathematics Subject Classification ID: 35J10
Mathematics Subject Classification ID: 35P20
zbMATH DE Number: 1928891
zbMATH Keywords: perturbation technique
zbMATH Keywords: periodic differential operator
reviewed by: Ingrid Beltiţă
MaRDI profile type: Publication
cites work: Almost periodic Schrödinger operators. II: The integrated density of states
cites work: The periodic Dirac operator is absolutely continuous
cites work: Distribution of the error term in the Weyl asymptotics for the Laplace operator on a two-dimensional torus and related lattice problems
cites work: Q5524507
cites work: Nombre de points entiers dans une famille homothétique de domaines de ${R}$
cites work: A remark on two dimensional periodic potentials
cites work: On isospectral periodic potentials in Rn
cites work: On isospectral periodic potentials in Rn. II
cites work: Asymptotic of the density of states for the Schrödinger operator with periodic electric potential
cites work: Q3736112
cites work: Perturbation theory for the Schrödinger operator with a periodic potential
cites work: Q5526485
cites work: Floquet theory for partial differential equations
cites work: Asymptotic of the density of states for the Schrödinger operator with periodic electromagnetic potential
cites work: Q3741933
cites work: Q4092320
cites work: Q3868351
cites work: The spectrum band structure of the three-dimensional Schrödinger operator with periodic potential
cites work: Improved two–term asymptotics for the eigenvalue distribution function of an elliptic operator on a compact manifold
cites work: Q5733385
full work available at URL: https://doi.org/10.1007/bf02384494
OpenAlex ID: W1984184089
description: scientific article; zbMATH DE number 1928891