Integrable Systems and Random Matrix Theory Seminar

In this talk we survey recent results on asymptotics for the eigenvalues of finite range truncations of continuum Schrödinger operators onto $[0,L]$ as $L\to\infty$. We will study them on different scale levels.

First, we will introduce the notion of Stahl-Totik regularity for continuum Schödinger operators. This theory is crucially different from the discrete setting, since for unbounded operators $\infty$ becomes a boundary point of the resolvent domain. Recently, we showed that the potential theoretic Green function with pole at $\infty$ and the equilibrium measure, which appear in the theory of bounded operators, should be substituted by the Martin function and the Martin measure. We show that on a macroscopic scale, Stahl-Totik regularity implies that the normalized eigenvalue counting measure converges to the Martin measure.

We will then turn to the microscopic scale and study the local eigenvalue spacing. We show that bulk universality of the Christoffel-Darboux kernel holds for any point $\xi$ where the imaginary part of the $m$-function has a positive finite nontangential limit. In particular, by the Freud-Levin theorem this implies equal asymptotic eigenvalue spacing around $\xi$, where the rate is given by the Christoffel function $\lambda_L(\xi)$. Finally, we turn to asymptotics of the Christoffel function and show that for Stahl-Totik regular measures, $L\lambda_L(\xi)$ has a limit as $L\to\infty$.
In the course of the talk we will put emphasize on understanding which of those phenomena are local properties and which are global.

The talk is partly based on joint work with Milivoje Lukić and Brian Simanek. Speaker(s): Benjamin Eichinger (Vienna University of Technology)