Bethe-Sommerfeld conjecture for periodic operators with strong perturbations

Details Bethe-Sommerfeld conjecture for periodic operators with strong perturbations Bethe-Sommerfeld conjecture for periodic operators with strong perturbations

2009-07-05

arxiv.org/abs/0907.0887v1

Mathematics

Spectral Theory

61 pages

Scientific paper

We consider a periodic self-adjoint pseudo-differential operator $H=(-\Delta)^m+B$, $m>0$, 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 Schr\"odinger operator with a periodic magnetic potential in all dimensions.

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