Sampling inequality for $L^2$-norms of eigenfunctions, spectral projectors, and Weyl sequences of Schr\"odinger operators

TL;DR

This paper establishes explicit quantitative equidistribution estimates for $L^2$-eigenfunctions, Weyl sequences, and eigenfunction combinations of Schrödinger operators, relating total and localized $L^2$-norms in multidimensional space.

Contribution

It provides the first explicit, quantitative bounds on the distribution of eigenfunctions and Weyl sequences for Schrödinger operators with bounded potentials.

Findings

Explicit equidistribution estimates for eigenfunctions.

Similar bounds for Weyl sequences and linear combinations.

Dependence of estimates on potential norm and energy.

Abstract

We consider a Schr\"odinger operator with bounded, measurable potential in multidimensional Euclidean space. We prove for every $L^2$-eigenfunction a quantitative equidistribution estimate. It compares the total $L^2$-norm with the $L^2$-norm over an equidistributed collection of balls. Our estimate is explicit with respect to the radius of the balls, norm of the potential and the energy of the eigenfunction. Similar estimates also hold for Weyl sequences and for linear combinations of eigenfunctions, as long as the associated eigenvalues are sufficiently close.