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The spectrum band structure of the three-dimensional Schrödinger operator with periodic potential. (English) Zbl 0578.47003

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 [M. Reed and 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 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 E. J. Dahlberg and 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, \[ \sigma (H)=\cup_{j\in {\mathbb{N}}}\Delta_ j\quad where\quad \Delta_ j:=R(E_ j(.)). \] Given \(\lambda\in {\mathbb{R}}\), let M(H,\(\lambda)\) be the multiplicity of the covering of \(\lambda\), i.e. the number of j such that \(\lambda \in \Delta_ j\). The author shows that there is a number \(c_ T>0\) with \[ (*)\quad M(H,\lambda)>c_ T\lambda^{1/2}\quad as\quad \lambda \to \infty. \] As a consequence, H has only finitely many gaps. An inspection of \(c_ T\) reveals that there are no gaps at all when \(\| V\|_{\infty}\) is sufficiently small.
The derivation of (*) is very intricate, a central role being played by the number \(N_{\Gamma}(\lambda,k)\) of lattice points of \(\Gamma\) in a ball with radius \(\lambda^{1/2}\) and centre \(k\in {\mathbb{R}}^ 3\), which was estimated in an elegant way by Dahlberg and Trubowitz [loc. cit.]. The author’s presentation is reasonably self-contained. Occasionally reference is made to a paper of his which has meanwhile appeared in Trudy Mat. Inst. Steklov. 171, (1985; Zbl 0567.47004)]. An estimate analogous to (*) for dimensions \(n\geq 4\) seems to have been established so far for special lattices only [cf. the author’s paper in Math. USSR Izv. 22, 619- 645 (1984; Zbl 0545.35027)].
{Reviewer’s remark. For smooth V in \({\mathbb{R}}^ 3\) another proof of the Bethe-Sommerfeld conjecture has been given by O. A. Veliev in Sov. Math. Dokl. 27, 234-237 (1983; Zbl 0541.47017).}
Reviewer: H.Kalf

MSC:

47A10 Spectrum, resolvent
47F05 General theory of partial differential operators
35J10 Schrödinger operator, Schrödinger equation
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