Stark hamiltonians with periodic potentials

Arne Jensen

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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.

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The lecture is devoted to the problem of absolute continuity of the spectrum of periodic operators. A general approach to this problem was suggested by L. Thomas in 1973 for the case of the Schrödinger operator with periodic electric potential. Further application of his method to concrete operators of mathematical physics met analytic difficulties. In recent years several new problems in this area have been solved. We propose a survey of known results in this area, including very recent,...

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