Sampling and equidistribution theorems for elliptic second order operators, lifting of eigenvalues, and applications

TL;DR

This paper establishes scale-free sampling and equidistribution theorems for eigenfunctions of elliptic second order operators with Lipschitz coefficients, with applications to eigenvalue lifting, spectrum analysis, and spectral inequalities.

Contribution

It introduces quantitative, scale-free estimates for eigenfunctions of variable coefficient elliptic operators and applies these to spectral theory and random operators.

Findings

Uniform eigenfunction estimates on growing cubes

Lifting of eigenvalues and essential spectrum bounds

Spectral inequalities for energy intervals

Abstract

We consider elliptic second order partial differential operators with Lipschitz continuous leading order coefficients on finite cubes and the whole Euclidean space. We prove quantitative sampling and equidistribution theorems for eigenfunctions. The estimates are scale-free, in the sense that for a sequence of growing cubes we obtain uniform estimates. These results are applied to prove lifting of eigenvalues as well as the infimum of the essential spectrum, and an uncertainty relation (aka spectral inequality) for short energy interval spectral projectors. Several application including random operators are discussed. In the proof we have to overcome several challenges posed by the variable coefficients of the leading term.