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Unique continuation for Schr{ö}dinger operators with partially Gevrey coefficients

Spyridon Filippas, Camille Laurent, Matthieu Léautaud

Abstract

We prove a local unique continuation result for Schr\''odinger operators with time independent Lipschitz metrics and lower order terms which are Gevrey 2 in time and bounded in space. This implies global unique continuation from any open set in a connected Riemannian manifold. These results relax in the same geometric setting the analyticity assumption in time of the Tataru-Robbiano-Zuily-H\''ormander theorem for these operators. The proof is based on (i) a Tataru-Robbiano-Zuily-H\''ormander type Carleman estimate with a nonlocal weight adapted to the anisotropy of the Schr\''odinger operator and (ii) the description of the conjugation of the Schr\''odinger operator with Gevrey coefficients by this nonlocal weight.

Unique continuation for Schr{ö}dinger operators with partially Gevrey coefficients

Abstract

We prove a local unique continuation result for Schr\''odinger operators with time independent Lipschitz metrics and lower order terms which are Gevrey 2 in time and bounded in space. This implies global unique continuation from any open set in a connected Riemannian manifold. These results relax in the same geometric setting the analyticity assumption in time of the Tataru-Robbiano-Zuily-H\''ormander theorem for these operators. The proof is based on (i) a Tataru-Robbiano-Zuily-H\''ormander type Carleman estimate with a nonlocal weight adapted to the anisotropy of the Schr\''odinger operator and (ii) the description of the conjugation of the Schr\''odinger operator with Gevrey coefficients by this nonlocal weight.
Paper Structure (31 sections, 32 theorems, 290 equations, 1 figure)

This paper contains 31 sections, 32 theorems, 290 equations, 1 figure.

Table of Contents

  1. Introduction and main results
  2. Background and results
  3. Application to controllability and observability
  4. Approximate controllability
  5. Observability, exact controllability
  6. Remarks
  7. Remarks on Gevrey regularity
  8. Remarks on the divergence
  9. More general lower order terms
  10. Idea of the proof, structure of the paper
  11. The Carleman estimate
  12. Toolbox of Riemannian geometry
  13. The Carleman weight
  14. From the subelliptic estimate to the Carleman estimate
  15. Proof of the subelliptic estimate
  16. ...and 16 more sections

Key Result

Theorem 1.2

Assume $\Omega = I \times V$ where $I \subset {\mathbb{R}}$ is an open interval and $V\subset {\mathbb{R}}^d$ an open set, and let $(t_0,x_0) \in \Omega$. Assume $g^{jk}\in W^{1,\infty}(V)$ satisfies e:elliptic, that $\mathsf{b}^j, \mathsf{q} \in \mathcal{G}^2(I; L^\infty(V;{\mathbb{C}}))$. Let $\Ps

Figures (1)

  • Figure 1: The domain $\Omega$ where we apply Stokes' theorem in case $\xi >0$ (the picture in case $\xi <0$ is the symmetric about the real axis). Notice that $\partial \Omega=\Gamma_1\cup \Gamma_2 \cup \Gamma_3 \cup [-r,r]$. Recall as well that in this regime we have $\xi\sim h^{-2/3}$ and therefore $h \xi \sim h^{1/3}$. As $h$ goes to $0$ the domain $\Omega$ collapses to the segment $[-r,r]$.

Theorems & Definitions (61)

  • Definition 1.1
  • Theorem 1.2: Local unique continuation for Schrödinger operators
  • Theorem 1.3: Local unique continuation for $L^2$ solutions
  • Remark 1.4
  • Theorem 1.5
  • Corollary 1.6
  • Theorem 1.7
  • Corollary 1.8
  • Lemma 2.1: Lemma 3.12 in LL:23notes
  • Lemma 2.2: Lemma 3.14 in LL:23notes
  • ...and 51 more