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The band structure of the general periodic Schrödinger operator with point interactions. (English) Zbl 0966.47023
The Bethe-Sommerfeld conjecture predicts that the periodic Schrödinger operator in \(L^2(\mathbb{R}^d)\), \(d= 2\) or \(3\), has a finite number of spectral gaps. For point potentials of a special kind, this conjecture has been confirmed by A. Grossmann, R. Hoegh-Krohn, M. Mebkhout and Yu. E. Karpeshina. These authors assumed that the support of the point potential has exactly one point in the Wigner-Seitz cell of the period lattice. Their proof is rather straightforward due to the following observation: For every lattice \(\Lambda\subset \mathbb{R}^d\) there exists \(E_0> 0\) such that every \(E> E_0\) is at least twice degenerate eigenvalue of \(-\Delta\) for a fixed value of the quasimomentum from the Brillouin zone for \(\Lambda\). This fact does not help in the case of an arbitrary number of point sources in the Wigner-Seitz cell, and the problem (explicitly stated by W. Kirsch) to confirm the Bethe-Sommerfeld conjecture in this case has remained open.
The main result of the paper under review is the affirmative solution to this problem. Moreover, it is shown that in the case of non-local periodic point perturbations of \(-\Delta\) the spectrum may contain a fractal component similar to the Cantor set. This component may be the support of the singular continuous spectrum or it may contain a dense set of eigenvalues. For a generic local point perturbation the spectrum remains absolutely continuous. For details we refer the reader to this interesting paper.

MSC:
47B65 Positive linear operators and order-bounded operators
46A55 Convex sets in topological linear spaces; Choquet theory
47F05 General theory of partial differential operators (should also be assigned at least one other classification number in Section 47-XX)
47N50 Applications of operator theory in the physical sciences
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