Eigenvalue repulsion estimates and some applications for the one-dimensional Anderson model

TL;DR

This paper establishes a lower bound on eigenvalue spacing for 1D Hamiltonians with arbitrary bounded potentials, ensuring no degeneracy, and applies this to demonstrate exponential localization in a nonlinear perturbed Anderson model.

Contribution

It provides explicit eigenvalue spacing bounds for 1D Hamiltonians and applies these results to prove localization and eigenfunction stability in nonlinear Anderson models.

Findings

Eigenvalue spacing is bounded below by Ce^{-bN}.

Eigenvalues of the 1D Hamiltonian are non-degenerate.

Solutions remain exponentially localized under nonlinear perturbations.

Abstract

We show that the spacing between eigenvalues of the discrete 1D Hamiltonian with arbitrary potentials which are bounded, and with Dirichlet or Neumann Boundary Conditions is bounded away from zero. We prove an explicit lower bound, given by $Ce^{-bN}$, where $N$ is the lattice size, and $C$ and $b$ are some finite constants. In particular, the spectra of such Hamiltonians have no degenerate eigenvalues. As applications we show that to leading order in the coupling, the solution of a nonlinearly perturbed Anderson model in one-dimension (on the lattice) remains exponentially localized, in probability and average sense for initial conditions given by a unique eigenfunction of the linear problem. We also bound the derivative of the eigenfunctions of the linear Anderson model with respect to a potential change.