Asymptotics for Christoffel functions associated to continuum Schrödinger operators

Abstract

We prove asymptotics of the Christoffel function, ${\lambda }_L(\xi )$, of a continuum Schrödinger operator for points in the interior of the essential spectrum under some mild conditions on the spectral measure. It is shown that $L{\lambda }_L(\xi )$ has a limit and that this limit is given by the Radon--Nikodym derivative of the spectral measure with respect to the Martin measure. Combining this with a recently developed local criterion for universality limits at scale ${\lambda }_L(\xi )$, we compute universality limits for continuum Schrödinger operators at scale $L$ and obtain clock spacing of the eigenvalues of the finite range truncations.