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

2025/05/22 by Martin Tautenhahn, Tautenhahn, Martin, Ivan Veselić +1
Mathematics · #Advanced Harmonic Analysis Research #Analysis of PDEs (math.AP) #FOS: Mathematics #FOS: Physical sciences #Mathematical Analysis and Transform Methods #Mathematical Physics (math-ph) #Optimization and Control (math.OC) #Spectral Theory in Mathematical Physics

paper · pdf · doi:10.48550/arxiv.2505.16655

openalex publication_date 2025/05/22 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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.

Citations

• Spectral Inequalities for the Schrödinger operator
• Null-controllability and control cost estimates for the heat equation on\n unbounded and large bounded domains
• Lower bounds for Dirichlet Laplacians and uncertainty principles
• Observable set, observability, interpolation inequality and spectral inequality for the heat equation in ℝⁿ
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• On localization in the continuous Anderson-Bernoulli model in higher dimension

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• Analysis of quasi-Monte Carlo methods for elliptic eigenvalue problems\n with stochastic coefficients