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Unique continuation and Hardy's uncertainty principle for hyperbolic Schrödinger equations

Torunn Jensen

TL;DR

The paper extends the Hardy uncertainty principle and related unique continuation results to hyperbolic Schrödinger models by combining Carleman estimates with a careful hyperbolic-geometry analysis. It proves that Gaussian decay at two times forces a solution to vanish under suitable conditions on either the potential or the nonlinearity, mirroring EKPV-type results in the elliptic setting but with new hyperbolic-specific technical adaptations. The approach hinges on a two-pronged strategy: (i) a conformal/Appell transformation to equalize Gaussian weights, and (ii) rigorous Carleman and energy estimates for a parabolic regularization, ensuring persistence of Gaussian-weighted norms and enabling a log-convexity argument. The results yield sharp two-time decay rigidity for the hyperbolic Schrödinger equation with potential and extend to the hyperbolic NLS, offering a definitive uncertainty-principle-type rigidity in the hyperbolic context and broadening the scope of unique continuation phenomena in dispersive PDEs.

Abstract

We prove unique continuation properties related to the Hardy uncertainty principle for solutions of the hyperbolic nonlinear Schrödinger equation and the hyperbolic Schrödinger equation with potential. Under suitable conditions on the nonlinearity, or the potential, we show that if $u$ is a solution with Gaussian decay at two different times, then $u\equiv 0$. These results extend to the hyperbolic setting the work of Escauriaza, Kenig, Ponce, and Vega (JEMS, 10, 2008) for the classical Schrödinger equation. The proofs rely on Carleman estimates based on calculus and convexity arguments, with the main challenge being to provide a rigorous justification of these estimates. Although our approach follows the general strategy of Escauriaza, Kenig, Ponce, and Vega, several technical modifications are required to handle the hyperbolic character of the equation.

Unique continuation and Hardy's uncertainty principle for hyperbolic Schrödinger equations

TL;DR

The paper extends the Hardy uncertainty principle and related unique continuation results to hyperbolic Schrödinger models by combining Carleman estimates with a careful hyperbolic-geometry analysis. It proves that Gaussian decay at two times forces a solution to vanish under suitable conditions on either the potential or the nonlinearity, mirroring EKPV-type results in the elliptic setting but with new hyperbolic-specific technical adaptations. The approach hinges on a two-pronged strategy: (i) a conformal/Appell transformation to equalize Gaussian weights, and (ii) rigorous Carleman and energy estimates for a parabolic regularization, ensuring persistence of Gaussian-weighted norms and enabling a log-convexity argument. The results yield sharp two-time decay rigidity for the hyperbolic Schrödinger equation with potential and extend to the hyperbolic NLS, offering a definitive uncertainty-principle-type rigidity in the hyperbolic context and broadening the scope of unique continuation phenomena in dispersive PDEs.

Abstract

We prove unique continuation properties related to the Hardy uncertainty principle for solutions of the hyperbolic nonlinear Schrödinger equation and the hyperbolic Schrödinger equation with potential. Under suitable conditions on the nonlinearity, or the potential, we show that if is a solution with Gaussian decay at two different times, then . These results extend to the hyperbolic setting the work of Escauriaza, Kenig, Ponce, and Vega (JEMS, 10, 2008) for the classical Schrödinger equation. The proofs rely on Carleman estimates based on calculus and convexity arguments, with the main challenge being to provide a rigorous justification of these estimates. Although our approach follows the general strategy of Escauriaza, Kenig, Ponce, and Vega, several technical modifications are required to handle the hyperbolic character of the equation.

Paper Structure

This paper contains 16 sections, 15 theorems, 286 equations.

Table of Contents

  1. Introduction
  2. The hyperbolic Schrödinger equation
  3. Uncertainty principle for the Schrödinger equation
  4. Statement of the result
  5. Outline of the proof of Theorem \ref{['hyperbolic result']}.
  6. Notation and Preliminaries
  7. General notation
  8. The ' ' hyperbolic'' Laplacian
  9. The orthogonal and the indefinite orthogonal group
  10. Some basic lemmas
  11. Proof of Theorem \ref{['theorem 3']}
  12. Energy estimate
  13. Carleman estimates
  14. L2-weighted bound under a smallness assumption
  15. Conclusion of the proof of Theorem \ref{['theorem 3']}
  16. ...and 1 more sections

Key Result

Theorem 1

Let $u\in C([0,1],L^2(\mathbb{R}^n))$ be a solution to where $V$ is bounded, and either $V(x,t)= V_1(x)+V_2(x,t)$ with $V_1$ real valued and or If there exist constants $\alpha,\beta >0$ such that $\alpha \beta < 2$ and $\|e^{\frac{|x|^2}{\beta^2}}u(0)\|_{L^2(\mathbb{R}^n)}$ and $\|e^{\frac{|x|^2}{\alpha^2}}u(1)\|_{L^2(\mathbb{R}^n)}$ are finite, then $u\equiv 0$.

Theorems & Definitions (37)

  • Theorem 1: escauriaza_hardys_2008
  • Remark 1.1
  • Theorem 2: escauriaza_hardys_2008
  • Theorem 3
  • Remark 1.2
  • Theorem 4
  • Theorem 5
  • Remark 1.3
  • Remark 1.4
  • Remark 2.1
  • ...and 27 more