Stahl-Totik regularity for continuum Schr"odinger operators

Abstract: This talk describes joint work with Benjamin Eichinger: a theory of regularity for one-dimensional continuum Schr"odinger
operators based on the Martin compactification of the complement of the essential spectrum. For a half-line Schr"odinger operator
$-\partial_x^2+V$ with a bounded potential $V$ (in a local $L^1$ sense), it was previously known that the spectrum can have zero
Lebesgue measure and even zero Hausdorff dimension; however, we prove universal thickness statements in the language of potential theory. Namely, the essential spectrum is not polar, it obeys the Akhiezer–Levin condition, and moreover, 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 its asymptotic expansion plays the role of a renormalized Robin constant suited for Schr"odinger operators and
enters a universal inequality $a \le \liminf_{x\to\infty} \frac 1x \int_0^x V(t) dt$ where $V$ denotes the potential. This leads to a
notion of regularity, with connections to the exponential growth rate of Dirichlet solutions and the zero counting measures for finite
truncations of the operator. We also present applications to decaying and ergodic potentials.