Stahl--Totik regularity for continuum Schr\"odinger operators

TL;DR

This paper develops a new regularity theory for continuum Schrödinger operators using Martin compactification, extending concepts from Stahl--Totik regularity to unbounded spectra, and establishes key asymptotic and spectral properties.

Contribution

It introduces a novel regularity framework for continuum Schrödinger operators based on Martin compactification, including asymptotic expansions and inequalities involving the potential.

Findings

Essential spectrum obeys the Akhiezer--Levin condition.

Martin function at infinity has a specific asymptotic expansion.

Universal inequality relating the constant a to the potential's average.

Abstract

We develop a theory of regularity for continuum Schr\"odinger operators based on the Martin compactification of the complement of the essential spectrum. This theory is inspired by Stahl--Totik regularity for orthogonal polynomials, but requires a different approach, since Stahl--Totik regularity is formulated in terms of the potential theoretic Green function with a pole at $\infty$, logarithmic capacity, and the equilibrium measure for the support of the measure, notions which do not extend to the case of unbounded spectra. For any half-line Schr\"odinger operator with a bounded potential (in a locally $L^1$ sense), we prove that its essential spectrum obeys the Akhiezer--Levin condition, and moreover, that the Martin function at $\infty$ obeys the two-term asymptotic expansion $\sqrt{-z} + \frac{a}{2\sqrt{-z}} + o(\frac 1{\sqrt{-z}})$ as $z \to -\infty$. The constant $a$ in that expansion plays the role of a renormalized Robin constant suited for Schr\"odinger operators and enters a universal inequality $a \le \liminf_{x\to\infty} \frac 1x \int_0^x V(t)dt$. This leads to a notion of regularity, with connections to the root asymptotics of Dirichlet solutions and zero counting measures. We also present applications to decaying and ergodic potentials.