Asymptotics for Christoffel functions associated to continuum Schr\"odinger operators

Abstract

We prove asymptotics of the Christoffel function, $\lambda_L(\xi)$, of a continuum Schr\"odinger 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 Schr\"odinger operators at scale $L$ and obtain clock spacing of the eigenvalues of the finite range truncations.