Homogenization of the Schrödinger Equation and Effective Mass Theorems

TLDR: In this paper, the authors derived effective mass theorems in solid state physics for a Schrodinger equation with a large periodic potential, denoting by ∈ the period, the potential is scaled as ∈−2.

Abstract: We study the homogenization of a Schrodinger equation with a large periodic potential: denoting by ∈ the period, the potential is scaled as ∈−2. We obtain a rigorous derivation of so-called effective mass theorems in solid state physics. More precisely, for well-prepared initial data concentrating on a Bloch eigenfunction we prove that the solution is approximately the product of a fast oscillating Bloch eigenfunction and of a slowly varying solution of an homogenized Schrodinger equation. The homogenized coefficients depend on the chosen Bloch eigenvalue, and the homogenized solution may experience a large drift. The homogenized limit may be a system of equations having dimension equal to the multiplicity of the Bloch eigenvalue. Our method is based on a combination of classical homogenization techniques (two-scale convergence and suitable oscillating test functions) and of Bloch waves decomposition.