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

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.