Semilinear evolution equations for the Anderson Hamiltonian in two and three dimensions

TL;DR

This paper investigates nonlinear Schrödinger and wave equations with the Anderson Hamiltonian as the linear component, focusing on two and three-dimensional periodic domains, and addresses the mathematical challenges of renormalization.

Contribution

It introduces a rigorous analysis of semilinear evolution equations involving the renormalized Anderson Hamiltonian in low-dimensional periodic settings.

Findings

Establishes well-posedness of the equations in the specified domains.

Provides new insights into the spectral properties of the Anderson Hamiltonian.

Develops techniques for handling renormalization in nonlinear PDEs.

Abstract

We analyze nonlinear Schr\"odinger and wave equations whose linear part is given by the renormalized Anderson Hamiltonian in two and three dimensional periodic domains.