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arXiv · 0907.0887

Bethe-Sommerfeld conjecture for periodic operators with strong perturbations

L. Parnovski·A. V. Sobolev
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Abstract

We consider a periodic self-adjoint pseudo-differential operator $H=(-Δ)^m+B$, $m>0$, in $\R^d$ which satisfies the following conditions: (i) the symbol of $B$ is smooth in $\bx$, and (ii) the perturbation $B$ has order less than $2m$. Under these assumptions, we prove that the spectrum of $H$ contains a half-line. This, in particular implies the Bethe-Sommerfeld Conjecture for the Schrödinger operator with a periodic magnetic potential in all dimensions.

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BibTeXRIS

L. Parnovski, A. V. Sobolev. 2009-07-05. Bethe-Sommerfeld conjecture for periodic operators with strong perturbations. https://doi.org/10.1007/s00222-010-0251-1

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