Stahl--Totik regularity for continuum Schrödinger operators
Benjamin Eichinger, Milivoje Lukić
TL;DR
This work extends Stahl–Totik regularity to continuum Schrödinger operators on the half-line by exploiting the Martin function M_E for the essential spectrum E = σ_ess(L_V). It establishes a universal Akhiezer–Levin-type structure M_E(z) = Re(√{-z} + a_E/(2√{-z})) + o(1/√{|z|}) with a_E ≤ liminf_{x→∞} (1/x)∫_0^x V(t) dt, and connects regularity to Dirichlet-solution root asymptotics, zero-counting measures, and density of states. The theory yields Widom-type regularity criteria, links to ergodic and decaying potentials, and a conformal-maps framework (comb mappings) that characterizes the Martin function and absolute continuity through sector conditions, including explicit formulas in finite-gap settings. Collectively, the results provide a robust, general regularity paradigm for continuum Schrödinger operators beyond compact-spectrum constraints, with broad implications for spectral types and ergodic models.
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 $a \le \liminf_{x\to\infty} \frac 1x \int_0^x V(t)dt$. 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.