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Benjamin Eichinger: Stahl-Totik regularity for continuum Schrödinger operators

Time: Wed 2020-05-06 13.15 - 14.15

Location: Zoom, Meeting ID: 694 6938 6866

Participating: Benjamin Eichinger, Johannes Kepler Universität Linz

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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.