04-204 O.A.Veliev

On the Polyharmonic Operator with a Periodic Potential (85K, LATeX 2e) Jun 29, 04

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Abstract. In this paper we obtain asymptotic formulas for eigenvalues and Bloch functions of the polyharmonic operator $L(r,q(x))=-\Delta^{r}+q(x),$ of arbitrary dimension $d$ with periodic, with respect to \ arbitrary lattice, potential $q(x),$ where $r\geq1.$ Then we prove that the number of gaps in the spectrum of the operator $L(r,q(x))$ is finite which is the generalisation of the Bethe -Sommerfeld conjecture for this operator. In particular, taking $r=1$ we get the proof of the Bethe -Sommerfeld conjecture for arbitrary dimension and arbitrary lattice.

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