O.A.Veliev
On the Polyharmonic Operator with a Periodic Potential
(85K, LATeX 2e)
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.