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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