# Benjamin Eichinger: Stahl-Totik regularity for continuum Schrödinger operators

Time: Wed 2020-05-06 13.15 - 14.15

Location:

Lecturer: Benjamin Eichinger, Johannes Kepler Universität Linz

### Abstract

We develop a theory of Stahl-Totik regularity for half-line Schrödinger operators $$-\partial_x^2+V$$ with bounded potentials (in a local $$L^1$$ sense).

We prove a universal thickness result for the essential spectrum, $$E$$, in the language of potential theory. Namely, $$E$$ is an Akhiezer-Levin set and the Martin function of the complementary domain at $$\infty$$ obeys an asymptotic expansion $$\sqrt{-z}+\frac{a_ E}{\sqrt{-z}}+o(\frac{1}{\sqrt{-z}})$$ as $$z\to-\infty$$. The constant $$a_E$$ plays the role of a Robin constant suited for Schödinger operators. Stahl-Totik regularity is characterized in terms of the behavior of the averages $$\frac{1}{x}\int_0^xV(t)d t$$ and root asymptotics of the Dirichlet solutions as $$x\to\infty$$. Moreover, it is connected to the zero counting measure for finite truncations. Applications to decaying and ergodic potentials will be discussed.

This talk is based on a joint work with M. Lukić.

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