 04198 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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