Bethe-Sommerfeld conjecture for periodic operators with strong perturbations

By L. Parnovski, A. Sobolev. 2009

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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.

Subfields: Mathematical Physics, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/560c76ae31396a5e2507ba0a15add4c3b0eedc49