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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 $ℝ^{d}, d \geq 2$ , 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.

Variation of the number of lattice points in large balls

Maxim M. Skriganov; Alexander V. Sobolev — 2005

Acta Arithmetica

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

On the Bethe-Sommerfeld conjecture

Variation of the number of lattice points in large balls