The spectrum band structure of the three-dimensional Schrödinger operator with periodic potential
type: Item
instance of: scholarly article
title: The spectrum band structure of the three-dimensional Schrödinger operator with periodic potential
zbMATH Open document ID: 0578.47003
DOI: 10.1007/BF01388550
published in: Inventiones Mathematicae
publication date: 1985-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/143221
review text: It is well known that the spectrum of a one-dimensional Schrödinger operator with a periodic potential is a purely absolutely continuous band spectrum, and generically there are infinitely many gaps [\textit{M. Reed} and \textit{B. Simon}, Methods of modern mathematical physics, Vol. IV, Ch. VIII.16 (1978; Zbl 0401.47001)]. In higher dimensions a particle may have enough leeway to evade a periodic obstacle, and there is the famous Bethe-Sommerfeld conjecture (pertaining, naturally, to dimensions 2 and 3) that such operators have only finitely many gaps. This long-standing conjecture was proved for bounded potentials with rational period lattice in the plane by the author [Sov. Math., Dokl. 20, 956-959 (1979; Zbl 0435.35028)] and for arbitrary planar lattices by \textit{V. N. Popov} and the author in [Zap. Nauchn. Semin. Leningrad Otd. Mat. Inst. Steklova 109, 131-133 (1983; Zbl 0492.47024)] and also by \textit{E. J. Dahlberg} and \textit{E. Trubowitz} [Commentat. Math. Helv. 57, 130-134 (1982; Zbl 0539.35059)]. See also the author's paper [Proc. Steklov Inst. Math. 158, 177-189 (1983; Zbl 0525.47034)] for further informations. In the present paper the conjecture is proved for a Schrödinger operator H with a continuous periodic potential V with arbitrary lattice \(T\subset {\mathbb{R}}^ 3\). It turns out that the number of gaps is primarily determined by the geometric and arithmetic properties of T rather than by the analytic behaviour of V. To describe the author's main estimate we need the following notation and results. For \(k\in {\mathbb{R}}^ 3\) \[ (*)\quad H(k):=(1/i)\nabla -k)^ 2+V \] is a self-adjoint operator with a purely discrete spectrum. Let \(\{E_ j(k)\}\) (j\(\in {\mathbb{N}})\) be a numbering of its eigenvalues in non-decreasing order, counting multiplicities. For \(j\in {\mathbb{N}}\), \(E_ j(.)\) is continuous and periodic with period lattice \(\Gamma\) dual to T. Moreover, \[ \sigma (H)=\cup_{j\in {\mathbb{N}}}\Delta_ j\quad where\quad \Delta_ j:=…
Mathematics Subject Classification ID: 47A10
Mathematics Subject Classification ID: 47F05
Mathematics Subject Classification ID: 35J10
zbMATH DE Number: 3925612
zbMATH Keywords: spectrum band structure
zbMATH Keywords: three-dimensional Schrödinger operator
zbMATH Keywords: with periodic potential
zbMATH Keywords: spectrum of a one-dimensional Schrödinger operator with a
zbMATH Keywords: periodic potential
zbMATH Keywords: purely absolutely continuous band spectrum
zbMATH Keywords: Bethe-Sommerfeld conjecture
zbMATH Keywords: number of gaps
zbMATH Keywords: period lattice
zbMATH Keywords: lattice points
zbMATH Keywords: spectrum of a one-dimensional Schrödinger operator with a periodic potential
author: M. M. Skriganov
reviewed by: Hubert Kalf
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 3925612