  In this paper we obtain the asymptotic formulas of arbitrary order for the
Bloch eigenvalues and Bloch functions of the multidimensional polyharmonic
operator with periodic, with respect to arbitrary lattice, potential. Then we
prove that the number of gaps in the spectrum of the operator is finite. In
particular, we get the proof of the Bethe -Sommerfeld conjecture for arbitrary
dimension and arbitrary lattice.
