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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