Partial differential equations with periodic coefficients and Bloch waves in crystals

Abstract

Bloch waves are special solutions of Schrödinger's equation with a periodic real potential. They are plane waves multiplied by periodic functions. In this paper we prove the existence and completeness of Bloch waves and of the related Kohn-Luttinger waves in unbounded domains for a class of partial differential equations which includes the Schrödinger equation. In addition, we discuss the dependence of these waves and the corresponding eigenvalues on the wave vector of the associated plane wave. The results may be interpreted as the analogs for certain partial differential equations of Floquet's theory for ordinary differential equations or as the determination of the spectral representation of certain periodic Hamiltonian operators.