Stahl-Totik Regularity for Dirac Operators

TL;DR

This paper develops a regularity theory for Dirac operators with locally square-integrable data, extending Stahl--Totik regularity concepts and revealing new spectral phenomena and inequalities.

Contribution

It introduces a new regularity framework for Dirac operators, including asymptotic spectral analysis and universal inequalities, expanding prior theories from orthogonal polynomials and Schrödinger operators.

Findings

Asymptotic expansion of the symmetric Martin function at infinity.

A universal inequality involving the lower average L^2-norm of the operator data.

Characterization of Dirac operator regularity through a family of equalities.

Abstract

We develop a theory of regularity for Dirac operators with uniformly locally square-integrable operator data. This is motivated by Stahl--Totik regularity for orthogonal polynomials and by recent developments for continuum Schr\"odinger operators, but contains significant new phenomena. We prove that the symmetric Martin function at $\infty$ for the complement of the essential spectrum has the two-term asymptotic expansion $\Im \left( z - \frac{b}{2 z}\right) + o(\frac 1z)$ as $z \to i \infty$, which is seen as a thickness statement for the essential spectrum. The constant $b$ plays the role of a renormalized Robin constant and enters a universal inequality involving the lower average $L^2$-norm of the operator data. However, we show that regularity of Dirac operators is not precisely characterized by a single scalar equality involving $b$ and is instead characterized by a family of equalities. This work also contains a sharp Combes--Thomas estimate (root asymptotics of eigensolutions), a study of zero counting measures, and applications to ergodic and decaying operator data.