Perturbation theory for the periodic multidimensional Schrödinger Operator and the Bethe-Sommerfeld Conjecture

Abstract

In this paper we obtain asymptotic formulas of arbitrary order for the Bloch eigenvalue and the Bloch function of the periodic Schr¨odinger operatör −Δ + q(x), of arbitrary dimension, when corresponding quasimomentum lies near a diffraction hyperplane. Besides, writing the asymptotic formulas for the Bloch eigenvalue and the Bloch function, when corresponding quasimomentum lies far from the diffraction hyperplanes, obtained in my previous papers in improved and enlarged form, we obtain the complete perturbation theory for the multidimensional Schr¨odinger operator with a periodic potential. Moreover, we estimate the measure of the isoenergetic surfaces in the high energy region which implies the validity of the Bethe-Sommerfeld conjecture for arbitrary dimension and arbitrary lattice.