Perturbation Theory for the Periodic Multidimensional Schrodinger Operator and the Bethe-Sommerfeld Conjecture

TL;DR

This paper develops a comprehensive perturbation theory for the multidimensional periodic Schrödinger operator, providing asymptotic formulas for eigenvalues and eigenfunctions, and confirms the Bethe-Sommerfeld conjecture across all dimensions and lattice types.

Contribution

It introduces arbitrary-order asymptotic formulas for Bloch eigenvalues and functions near diffraction hyperplanes and proves the Bethe-Sommerfeld conjecture for any dimension and lattice.

Findings

Asymptotic formulas for eigenvalues and eigenfunctions near diffraction hyperplanes.

Complete perturbation theory for multidimensional periodic Schrödinger operators.

Validation of the Bethe-Sommerfeld conjecture in arbitrary dimensions.

Abstract

In this paper we obtain asymptotic formulas of arbitrary order for the Bloch eigenvalue and the Bloch function of the periodic Schrodinger operator 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 Schrodinger 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.