Non-random perturbations of the Anderson Hamiltonian

TL;DR

This paper investigates how adding a non-random, decaying potential to the Anderson Hamiltonian affects the number of negative eigenvalues, identifying a decay rate threshold for finiteness.

Contribution

It establishes the critical decay rate of the perturbing potential that determines whether the number of negative eigenvalues is finite or infinite.

Findings

Finiteness of negative eigenvalues depends on decay rate of perturbation.

Critical decay rate identified as O(ln^{-2/d} |x|).

Results apply to Anderson Hamiltonian with Bernoulli potential.

Abstract

The Anderson Hamiltonian $H_0=-\Delta+V(x,\omega)$ is considered, where $V$ is a random potential of Bernoulli type. The operator $H_0$ is perturbed by a non-random, continuous potential $-w(x) \leq 0$, decaying at infinity. It will be shown that the borderline between finitely, and infinitely many negative eigenvalues of the perturbed operator, is achieved with a decay of the potential $-w(x)$ as $O(\ln^{-2/d} |x|)$.