Stahl--Totik regularity for continuum Schrödinger operators

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 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 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 $\sqrt{-z}+\frac{a}{2\sqrt{-z}}+o(\frac{1}{\sqrt{-z}})$ as $z\to -\infty$. The constant $a$ in that expansion plays the role of a renormalized Robin constant suited for Schrödinger operators and enters a universal inequality $a\le {\mathrm{liminf}}_{x\to \infty }\frac{1}{x}{\int }_0^{x}V(t)dt$. 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.

33 pages

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Cited by in corpus (1)

• Sturm-Liouville operators with periodically modulated parameters. Part I: Regular case