Variational models for infinite quantum systems: example of the crystal with defects

In this talk I will present a new model for a specific infinite quantum system: a crystal with a defect.

The main idea is to describe at the same time the electrons bound by the defect and the (nonlinear) behavior of the infinite crystal. This leads to a (rather peculiar) bounded-below nonlinear functional whose variable is however an operator of infinite-rank.

I will provide the correct functional setting for this functional and discuss the properties of bound states. I will in particular relate them to the dielectric properties of the crystal. Finally, I will discuss discretization issues and show preliminary numerical results in 1D.

This is a review of joint works with Eric Cancès and Amélie Deleurence (Ecole des Ponts, Paris).