Homogenization of Schrodinger equations. Extended Effective Mass Theorems for non-crystalline matter

TL;DR

This paper extends homogenization theory for Schrödinger equations to non-crystalline materials, establishing effective mass theorems that generalize Bloch theory beyond periodic structures and include limited disorder.

Contribution

It introduces new effective mass theorems for non-crystalline matter, extending Bloch theory to stochastic deformations and beyond classical ergodic assumptions.

Findings

Established a general effective mass theorem for non-crystalline matter.

Extended Bloch theory to stochastic deformations.

Proved effective mass results for quasi-perfect materials with limited disorder.

Abstract

This paper concerns the homogenization of Schrodinger equations for non-crystalline matter, that is to say the coefficients are given by the composition of stationary functions with stochastic deformations. Two rigorous results of so-called effective mass theorems in solid state physics are obtained: a general abstract result (beyond the classical stationary ergodic setting), and one for quasi-perfect materials (i.e. the disorder in the non-crystalline matter is limited). The former relies on the double-scale limits and the wave function is spanned on the Bloch basis. Therefore, we have extended the Bloch Theory which was restrict until now to crystals (periodic setting). The second result relies on the Perturbation Theory and a special case of stochastic deformations, namely stochastic perturbation of the identity.