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

TL;DR

This paper establishes the asymptotic behavior of Christoffel functions for continuum Schrödinger operators, linking spectral measures to eigenvalue spacing and universality limits, advancing understanding of spectral theory in quantum mechanics.

Contribution

It provides the first asymptotic characterization of Christoffel functions for continuum Schrödinger operators and connects spectral measure derivatives to eigenvalue spacing universality.

Findings

Lλ_L(ξ) converges to the Radon–Nikodym derivative of the spectral measure.

Universality limits at scale λ_L(ξ) are computed for continuum Schrödinger operators.

Eigenvalues of finite range truncations exhibit clock spacing.

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.