Work

Perturbative Diagonalization and Spectral Gaps of Quasiperiodic Operators on $$\ell ^2(\mathbb Z^d)$$ with Monotone Potentials

Year: 2025

Type: article

Abstract: We obtain a perturbative proof of localization for quasiperiodic operators on $$\ell ^2(\mathbb Z^d)$$ ℓ 2 ( Z d ) with one-dimensional phase space and monotone sampling functions, in the regime of small hopping. The proof is based on an iterative scheme which can be considered as a local (in the energy and the phase) and convergent version of KAM-type diagonalization, whose result is a covariant family of uniformly localized eigenvalues and eigenvectors. We also prove that the spectra of such operators contain infinitely many gaps.

Source: Communications in Mathematical Physics

DOI: 10.1007/s00220-025-05280-y

Authors: Ilya Kachkovskiy, Leonid Parnovski, Roman Shterenberg

Institutions: Michigan State University, University College London, University of Alabama at Birmingham

Language: English

Topic: Spectral Theory in Mathematical Physics

Subfield: Mathematical Physics

Field: Mathematics

Domain: Physical Sciences

Open Access status: hybrid

Estimated APC paid: $2,990

Funders: National Science Foundation, Alfred P. Sloan Foundation, Leverhulme Trust, Engineering and Physical Sciences Research Council