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Asymptotics of the integrated density of states for periodic elliptic pseudo-differential operators in dimension one.

Alexander V. Sobolev — 2006

Revista Matemática Iberoamericana

We consider a periodic pseudo-differential operator on the real line, which is a lower-order perturbation of an elliptic operator with a homogeneous symbol and constant coefficients. It is proved that the density of states of such an operator admits a complete asymptotic expansion at large energies. A few first terms of this expansion are found in a closed form.

On the Bethe-Sommerfeld conjecture

Leonid Parnovski;  Alexander V. Sobolev — 2000

Journées équations aux dérivées partielles

We consider the operator in ℝ d , d ≥ 2 , of the form H = ( - Δ ) l + V , l > 0 with a function V periodic with respect to a lattice in ℝ d . We prove that the number of gaps in the spectrum of H is finite if 8 l > d + 3 . Previously the finiteness of the number of gaps was known for 4 l > d + 1 . Various approaches to this problem are discussed.

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