Vernny Ccajma | Searcharxiv

arXiv2022

On a Characterization of the Rellich-Kondrachov Theorem on Groups

Motivated by an eigenvalue-eigenfunction problem posed in IR^n x Ω, where Ω is a probability space, we are concerned in this paper with the Sobolev space on groups. Hence it is established an equivalence between locally compact Abelian groups and the space of solutions to the associated variational problem. Then, we study some conditions which characterize in a precisely manner the Rellich-Kondrachov Theorem, the principal ingredient to solve the variational problem.

math.AP↗

arXiv2022

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

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.

math.AP↗