Exact mobility edges for almost-periodic CMV matrices via gauge symmetries

TL;DR

This paper introduces a new class of almost-periodic CMV matrices, the mosaic unitary almost-Mathieu operator, and precisely identifies energies that separate different spectral types, advancing understanding of spectral transitions in these operators.

Contribution

It develops a novel approach to symmetries in generalized extended CMV matrices and explicitly constructs a family exhibiting exact mobility edges.

Findings

Existence of energies separating spectral regions with different types.

Explicit calculation of mobility edges for the constructed operator.

Introduction of the mosaic unitary almost-Mathieu operator.

Abstract

We investigate the symmetries of so-called generalized extended CMV matrices. It is well-documented that problems involving reflection symmetries of standard extended CMV matrices can be subtle. We show how to deal with this in an elegant fashion by passing to the class of generalized extended CMV matrices via explicit diagonal unitaries in the spirit of Cantero-Gr\"unbaum-Moral-Vel\'azquez. As an application of these ideas, we construct an explicit family of almost-periodic CMV matrices, which we call the mosaic unitary almost-Mathieu operator, and prove the occurrence of exact mobility edges. That is, we show the existence of energies that separate spectral regions with absolutely continuous and pure point spectrum and exactly calculate them.