# Events for 05/01/2020 from all calendars

## Mathematical Physics and Harmonic Analysis Seminar

**Time: ** 1:50PM - 2:50PM

**Location: ** Zoom seminar

**Speaker: **Milivoje Lukic, Rice University

**Title: ***Zoom Seminar: 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 $V$, it was previously
known that the spectrum can have zero Lebesgue measure and even zero
Hausdorff dimension; however, we obtain universal thickness statements
in the language of potential theory.
Namely, we prove that 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 $a \le \liminf_{x\to\infty} \frac 1x
\int_0^x V(t) dt$. This leads to a notion of regularity, with
connections to the exponential growth rate of Dirichlet solutions and
the zero counting measures for finite restrictions of the operator. We
also present applications to decaying and ergodic potentials.