Stahl–Totik regularity for continuum Schrödinger operators

Eichinger, Benjamin; Lukić, Milivoje

doi:10.2140/apde.2025.18.591

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Stahl–Totik regularity for continuum Schrödinger operators

Benjamin Eichinger and Milivoje Lukić

Vol. 18 (2025), No. 3, 591–628

DOI: 10.2140/apde.2025.18.591

Abstract

We develop a theory of regularity for continuum Schrödinger 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’s function with a pole at $\infty$ , logarithmic capacity, and the equilibrium measure, notions which do not extend to unbounded spectra. For any half-line Schrödinger 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 $\sqrt{- z} + a ∕ (2 \sqrt{- z}) + o (1 ∕ \sqrt{- z})$ as $z \to - \infty$ . The constant $a$ in that expansion has not appeared in the literature before; we show that it can be used to measure the size of the spectrum in a potential-theoretic sense and that it should be thought of as a renormalized Robin constant suited for semibounded sets. We prove that it enters a universal inequality $a \leq {liminf}_{x \to \infty} (1 ∕ x) \int_{0}^{x} V (t) d t$ , which 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.

Keywords

Schrödinger operators, Stahl–Totik regularity

Mathematical Subject Classification

Primary: 34L40

Secondary: 31C35, 35J10

Milestones

Received: 13 June 2022

Revised: 17 September 2023

Accepted: 30 November 2023

Published: 3 March 2025

Authors

Benjamin Eichinger
Institute for Analysis and Scientific Computing TU Wien Vienna Austria	Department of Mathematics and Statistics Flyde College Lancaster University Lancaster United Kingdom

Milivoje Lukić
Department of Mathematics Rice University Houston, TX United States

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Vol. 18, No. 3, 2025