Bethe–Sommerfeld Conjecture for Pseudodifferential Perturbation

We consider a periodic pseudodifferential operator H = (? Δ) ^l  + A (l > 0) in R ^d which satisfies the following conditions: (i) the symbol of H is smooth in x, and (ii) the perturbation A has order smaller than 2l ? 1. Under these assumptions, we prove that the spectrum of H contains a half-line.     

Asymptotic expansion of the integrated density of states of a two-dimensional periodic Schrödinger operator

We prove the complete asymptotic expansion of the integrated density of states of a two-dimensional Schrödinger operator with a smooth periodic potential.