04-198 O.A.Veliev
On the Polyharmonic Operator with a Periodic Potential (87K, LATeX 2e) Jun 23, 04
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Abstract. In this paper we obtain asymptotic formulas for eigenvalues and Bloch functions of the polyharmonic operator $L(l,q(x))=3D-\Delta^{l}+q(x),$ = of arbitrary dimension $d$ with periodic, with respect to \ arbitrary = lattice, potential $q(x),$ where $l\geq1.$ Then we prove that the number of gaps = in the spectrum of the operator $L(l,q(x))$ is finite which is the = generalisation of the Bethe -Sommerfeld conjecture for this operator. In particular, = taking $l=3D1$ we get the proof of the Bethe -Sommerfeld conjecture for = arbitrary dimension and arbitrary lattice.

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