DOI:10.1007/s00023-013-0246-8
Corpus ID: 424390
Complete Asymptotic Expansion of the Integrated Density of States of Multidimensional Almost-Periodic Pseudo-Differential Operators
@article{Morozov2012CompleteAE,
title={Complete Asymptotic Expansion of the Integrated Density of States of Multidimensional Almost-Periodic Pseudo-Differential Operators},
author={Sergey Morozov and Leonid Parnovski and Roman Shterenberg},
journal={Annales Henri Poincar{\'e}},
S. Morozov, L. Parnovski, R. Shterenberg
Annales Henri Poincaré
We obtain a complete asymptotic expansion of the integrated density of states of operators of the form $${H = (-\Delta)^w+ B}$$ in $${\mathbb{R}^d}$$ . Here w > 0 and B belong to a wide class of almost-periodic self-adjoint pseudo-differential operators of order less than 2w. In particular, we obtain such an expansion for magnetic Schrödinger operators with either smooth periodic or generic almost-periodic coefficients.
View on Springer
Complete asymptotic expansion of the spectral function of multidimensional almost-periodic Schrodinger operators
L. Parnovski, R. Shterenberg
We prove the existence of a complete asymptotic expansion of the spectral function (the integral kernel of the spectral projection) of a Schrodinger operator H=−Δ+bH=−Δ+b acting in RdRd when the…
The almost periodic gauge transform -- An abstract scheme with applications to Dirac Operators
Jean Lagac'e, S. Morozov, L. Parnovski, Bernhard Pfirsch, R. Shterenberg
One of the main tools used to understand both qualitative and quantitative spectral behaviour of periodic and almost periodic Schrodinger operators is the method of gauge transform. In this paper, we…
Complete Semiclassical Spectral Asymptotics for Periodic and Almost Periodic Perturbations of Constant Operators
V. Ivrii
Microlocal Analysis, Sharp Spectral Asymptotics and Applications V
Under certain assumptions we derive a complete semiclassical asymptotics of the spectral function \(e_{h,\varepsilon }(x, x,\lambda )\) for a scalar operator $$\begin{aligned} A_\varepsilon (x,…
Complete asymptotic expansions of the spectral function for symbolic perturbations of almost periodic Schrödinger operators in dimension one
J. Galkowski
Journal of Spectral Theory
In this article we consider asymptotics for the spectral function of Schrodinger operators on the real line. Let $P:L^2(\mathbb{R})\to L^2(\mathbb{R})$ have the form $$ P:=-\tfrac{d^2}{dx^2}+W, $$…
An overview of periodic elliptic operators
P. Kuchment
The article surveys the main techniques and results of the spectral theory of periodic operators arising in mathematical physics and other areas. Close attention is paid to studying analytic…
Extended states for polyharmonic operators with quasi-periodic potentials in dimension two
Yulia Karpeshina, R. Shterenberg
We consider a polyharmonic operator H = ( − Δ)l + V(x) in dimension two with l ⩾ 2, l being an integer, and a quasi-periodic potential V(x). We prove that the absolutely continuous spectrum of H…
Multiscale analysis in momentum space for quasi-periodic potential in dimension two
Mathematics, Physics
We consider a polyharmonic operator H=(−Δ)l+V(x) in dimension two with l ⩾ 2, l being an integer, and a quasi-periodic potential V(x). We prove that the absolutely continuous spectrum of H contains a…
Perturbation Theory for Almost-Periodic Potentials I: One-Dimensional Case
We consider the family of operators $${H^{(\varepsilon)}:=-\frac{d^2}{dx^2}+\varepsilon V}$$H(ε):=-d2dx2+εV in $${\mathbb{R}}$$R with almost-periodic potential V. We study the behaviour of the…
Sharp asymptotics of the quasimomentum
E. Korotyaev
Asymptot. Anal.
The sharp asymptotics of the quasimomentum and the Titchmarsh-Weyl functions, the Bloch functions at high energy are determined.
Bethe-Sommerfeld Conjecture in Semiclassical Settings
Under certain assumptions (including \(d\ge 2)\) it is proved that the spectrum of a scalar operator in \(\mathscr {L}^2({\mathbb {R}}^d)\) is scalar-valued.
Complete asymptotic expansion of the integrated density of states of multidimensional almost-periodic Schrödinger operators
We obtain a complete asymptotic expansion of the integrated density of states of operators of the form \({H = (-\Delta)^w+ B}\) in \({\mathbb{R}^d}\) . Here w > 0 and B belong to a wide class of…
View 12 excerpts, references background
Asymptotics of the integrated density of states for periodic elliptic pseudo-differential operators in dimension one
A. Sobolev
We consider a periodic pseudo-differential operator on the real line, which is a lower-order perturbation of an elliptic operator with a homogeneous symbol and constant coefficients. It is proved…
On the density of states for the periodic Schrödinger operator
Yulia Karpeshina
An asymptotic formula for the density of states of the polyharmonic periodic operator (−δ)l+V inRn,n≥2,l>1/2 is obtained. Special consideration is given to the case of the Schrödinger…
Asymptotic of the density of states for the Schrödinger operator with periodic electromagnetic potential
Abderemane Mohamed
For the Schrodinger operator in L2(Rn), n>1, with C∞ periodic electromagnetic potential, we give an asymptotic formula of the integrate density of states of the form N(μ)=anμn/2+O(μ(n−2+e)/2), ∀e>0.…
Bethe-Sommerfeld conjecture for periodic operators with strong perturbations
L. Parnovski, A. Sobolev
We consider a periodic self-adjoint pseudo-differential operator H=(−Δ)m+B, m>0, in ℝd which satisfies the following conditions: (i) the symbol of B is smooth in x, and (ii) the perturbation B has…
Asymptotic expansion of the integrated density of states of a two-dimensional periodic Schrödinger operator
We prove the complete asymptotic expansion of the integrated density of states of a two-dimensional Schrödinger operator with a smooth periodic potential.
ASYMPTOTIC EXPANSION OF THE STATE DENSITY AND THE SPECTRAL FUNCTION OF A HILL OPERATOR
D. Shenk, M. Shubin
Complete asymptotic expansions are obtained for the integrated state density and the spectral function of a Hill operator with smooth potential. These expansions can be differentiated any number of…
On the High‐Energy Asymptotics of the Integrated Density of States
E. Korotyaev, A. Pushnitski
Assuming that the integrated density of states of a Schrödinger operator admits a high‐energy asymptotic expansion, the authors give explicit formulae for the coefficients of this expansion in terms…
Regularized Traces and Taylor Expansions for the Heat Semigroup
M. Hitrik, I. Polterovich
The coefficients in asymptotics of regularized traces and associated trace (spectral) distributions for Schrödinger operators with short and long range potentials are computed. A kernel expansion for…
Spectrum of multidimensional periodic operators
O. Veliev
Let Ω be a lattice in the n-dimensional Euclidean space Rn and let F be the fundamental domain of the lattice Ω. We denote by H the Schrödinger operator generated in L2(Rn) by the expression −‡u +…