Bethe-Sommerfeld Conjecture and Absolutely Continuous Spectrum of Multi-Dimensional Quasi-Periodic Schrödinger Operators

Abstract

We consider Schrödinger operators $H=-\Delta +V(\mathbf{x})$ in ${\mathbb{R}}^d$, $d\ge 2$, with quasi-periodic potentials $V(\mathbf{x})$. We prove that the absolutely continuous spectrum of a generic $H$ contains a semi-axis $[{\lambda }_{\ast },+\infty )$. We also construct a family of eigenfunctions of the absolutely continuous spectrum; these eigenfunctions are small perturbations of the exponentials. The proof is based on a version of the multi-scale analysis in the momentum space with several new ideas introduced along the way.

113 pages; two pictures and several comments and remarks are added following the reviewers' suggestions; matches journal version

References in corpus (3)

• Bethe-Sommerfeld conjecture for periodic operators with strong perturbations
• Quantitative inductive estimates for Green's functions of non-self-adjoint matrices
• Absence of eigenvalues of analytic quasi-periodic Schrodinger operators on ${\mathbb{R}}^d$

Cited by in corpus (5)

• A new spectral analysis of stationary random Schrödinger operators
• Absence of eigenvalues of analytic quasi-periodic Schrodinger operators on ${\mathbb{R}}^d$
• On the spectra of separable 2D almost Mathieu operators
• The rotation number for almost periodic potentials with jump discontinuities and $\delta$-interactions
• Ballistic Transport for Schrödinger Operators with Quasi-periodic Potentials