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Quantitative propagation of smallness and spectral estimates for the Schrödinger operator

Kévin Le Balc'h, Jérémy Martin

Abstract

In this paper, we investigate quantitative propagation of smallness properties for the Schrödinger operator on a bounded domain in $\mathbb R^d$. We extend Logunov, Malinnikova's results concerning propagation of smallness for $A$-harmonic functions to solutions of divergence elliptic equations perturbed by a bounded zero order term. We also prove similar results for gradient of solutions to some particular equations. This latter result enables us to follow the recent strategy of Burq, Moyano for the obtaining of spectral estimates on rough sets for the Schrödinger operator. Applications to observability estimates and to the null-controllability of associated parabolic equations posed on compact manifolds or the whole euclidean space are then considered.

Quantitative propagation of smallness and spectral estimates for the Schrödinger operator

Abstract

In this paper, we investigate quantitative propagation of smallness properties for the Schrödinger operator on a bounded domain in . We extend Logunov, Malinnikova's results concerning propagation of smallness for -harmonic functions to solutions of divergence elliptic equations perturbed by a bounded zero order term. We also prove similar results for gradient of solutions to some particular equations. This latter result enables us to follow the recent strategy of Burq, Moyano for the obtaining of spectral estimates on rough sets for the Schrödinger operator. Applications to observability estimates and to the null-controllability of associated parabolic equations posed on compact manifolds or the whole euclidean space are then considered.
Paper Structure (17 sections, 14 theorems, 147 equations)

This paper contains 17 sections, 14 theorems, 147 equations.

Table of Contents

  1. Introduction
  2. Main results
  3. Propagation of smallness for the Schrödinger operator
  4. Spectral estimates on compact manifolds and applications
  5. Spectral estimates on the Euclidean space and applications
  6. Proof of propagation of smallness for Schrödinger operators
  7. Existence of a positive multiplier and reduction to divergence form
  8. Propagation of smallness
  9. Proof of the spectral estimates
  10. Spectral estimates on compact manifolds
  11. Reduction of spectral estimates to sets of positive Hausdorff measures
  12. Local spectral estimates
  13. Propagation to the whole manifold
  14. The double manifold
  15. Spectral estimates on the Euclidean space
  16. ...and 2 more sections

Key Result

Theorem 2.1

Let $\rho, m, \delta >0$ and $\mathcal{K}, E \subset \Omega$, be measurable subsets such that There exist $C=C(\Omega, \Lambda_1, \Lambda_2, \|V\|_{\infty}, \rho, m, \delta)>0$ and $\alpha=\alpha(\Omega, \Lambda_1, \Lambda_2, \|V\|_{\infty}, \rho, m, \delta) \in (0,1)$ such that for every weak solution $u \in W^{1,2}(\Omega) \cap L^{\infty}(\Omega)$ of the elliptic equation we have

Theorems & Definitions (21)

  • Theorem 2.1
  • Theorem 2.2
  • Theorem 2.3
  • Theorem 2.4
  • Theorem 2.5
  • Definition 2.6
  • Theorem 2.7
  • Theorem 2.8
  • Theorem 2.9
  • Lemma 3.1
  • ...and 11 more