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

Martin Tautenhahn, Ivan Veselic

TL;DR

This work advances scale-free unique continuation for elliptic second-order operators with Lipschitz leading coefficients, proving sampling and equidistribution theorems on $\mathbb{R}^d$ and cubes that are robust to coefficient variation. It develops a three-annuli Carleman-based framework and a chaining argument to overcome large Lipschitz constants, enabling quantitative spectral results such as eigenvalue lifting and lower bounds for the essential spectrum, plus a short-energy projector uncertainty principle. The paper further derives a Hölder Wegner estimate for random potentials and outlines control-theoretic and homogenization applications, while proposing extensions to random divergence-type operators. Overall, the results provide coefficient-robust tools for spectral and probabilistic analysis of elliptic operators, with potential impact on random Schrödinger operators and multi-scale control problems.

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.

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

TL;DR

This work advances scale-free unique continuation for elliptic second-order operators with Lipschitz leading coefficients, proving sampling and equidistribution theorems on and cubes that are robust to coefficient variation. It develops a three-annuli Carleman-based framework and a chaining argument to overcome large Lipschitz constants, enabling quantitative spectral results such as eigenvalue lifting and lower bounds for the essential spectrum, plus a short-energy projector uncertainty principle. The paper further derives a Hölder Wegner estimate for random potentials and outlines control-theoretic and homogenization applications, while proposing extensions to random divergence-type operators. Overall, the results provide coefficient-robust tools for spectral and probabilistic analysis of elliptic operators, with potential impact on random Schrödinger operators and multi-scale control problems.

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.

Paper Structure

This paper contains 15 sections, 26 theorems, 234 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Notation and main results
  3. Applications and Discussion
  4. Throughout this section we consider the following type of self-adjoint operators
  5. Uncertainty relation for short energy intervals and lower bounds on the lifting of spectra
  6. An abstract uncertainty relation for low energy spectral projectors
  7. Uniform uncertainty relations for spectral projectors of elliptic operators as in § \ref{['sec:Self-adjoint']}
  8. Wegner estimate for elliptic second order operator plus random potential
  9. Outlook and further research goals
  10. Control theory for the heat equation
  11. Wegner estimates for random divergence type operators with small support
  12. Three annuli inequality
  13. Intermezzo: Short proof for the case of small Lipschitz constant coefficients
  14. Chaining argument and the proof of Theorems \ref{['thm:sampling']} and \ref{['thm:equidistribution']}
  15. Extension of the differential inequality

Key Result

Theorem 2.2

Let $G > 0$ and assume Then for all measurable and bounded $V \colon \mathbb{R}^d \to \mathbb{R}$, all $\psi \in H^2 (\mathbb{R}^d)$ and $\zeta \in L^2 (\mathbb{R}^d)$ satisfying $\lvert H \psi \rvert \leq \lvert V\psi \rvert + \lvert \zeta \rvert$ almost everywhere on $\mathbb{R}^d$, all $\delta \in (0,G/2)$ and all $(G, where and $D_1$, $D_2$, and $D_3$ are positive constants depending only on

Figures (2)

  • Figure 1: Within two steps we can reach any point in $\overline{B (b-a , z^k)}$
  • Figure 2: Illustration of a sequence $\tau$ with $\mu = 8$

Theorems & Definitions (55)

  • Definition 2.1
  • Theorem 2.2: BorisovTV-17
  • Theorem 2.3: Sampling Theorem
  • Remark 2.4
  • Corollary 2.5: Scaled Sampling Theorem
  • Theorem 2.6: Equidistribution Theorem
  • Remark 2.7
  • Corollary 2.8: Scaled Equidistribution Theorem
  • Theorem 3.1: Uncertainty relation for arbitrary positioned short intervals
  • proof
  • ...and 45 more