Uniform Anderson localization, unimodal eigenstates and simple spectra in a class of "haarsh" deterministic potentials

TL;DR

This paper proves Anderson localization, eigenstate unimodality, and spectrum simplicity for a class of deterministic multi-dimensional lattice Schrödinger operators generated by Haar wavelet-based potentials, extending previous results and setting the stage for multi-particle analysis.

Contribution

It introduces a novel class of deterministic potentials generated by Haar wavelets and proves localization, eigenstate unimodality, and spectrum simplicity in the strong disorder regime.

Findings

Proved Anderson localization for the model.

Established unimodality and exponential decay of eigenstates.

Demonstrated spectrum simplicity using a deterministic Minami estimate.

Abstract

We study a particular class of families of multi-dimensional lattice Schr\"o\-dinger operators with deterministic (including quasi-periodic) potentials generated by the "hull" given by an orthogonal series over the Haar wavelet basis on the torus, of arbitrary dimension, with expansion coefficients considered as independent parameters. In the strong disorder regime, we prove Anderson localization for generic operator families, using a variant of the Multi-Scale Analysis, and show that all localized eigenfunctions are unimodal and feature uniform exponential decay away from their respective localization centers. Using the Klein--Molchanov argument and a variant of the Minami estimate for deterministic potentials, we prove the simplicity of the spectrum in our model. NOTE: This text completes our earlier manuscript (math-ph/0907.1494), originally uploaded in 2009 and revised in 2011, which is kept in \textbf{arXiv} in a reduced form, merely to avoid broken references in earlier works. Compared to [\texttt{math-ph/0907.1494}], we add the results on unimodality of the eigenstates, uniform dynamical localization, and simplicity of p.p. spectra. Compared to earlier versions of this preprint, the presentation has been adapted to the future extension of the main results (uniform localization, unimodality of the eigenfunctions) to the multi-particle Anderson Hamiltonians with a nontrivial interaction between the particles, which we plan to publish in a forthcoming paper.