Bethe-Sommerfeld conjecture for periodic operators with strong perturbations

We consider a periodic self-adjoint pseudo-differential operator H=(−Δ)m+B, m>0, in ℝd which satisfies the following conditions: (i) the symbol of B is smooth in x, 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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