Skip to content
DraconiticResearch
Search Sources API
186,125 studies31 sourcesUpdated 29 min ago
PreprintCC BY-NC-ND 4.0HAL

INVISIBLE SOLUTIONS FOR SCHRÖDINGER AND WAVE EQUATIONS

  • Spyridon Filippas
  • Camille Laurent
  • Matthieu Léautaud
Open at sourcePDFCite

Abstract

On the Euclidean disk, we construct a nontrivial solution of the Schrödinger (resp. wave) equation with Dirichlet boundary conditions which vanishes identically on a smaller concentric disk for all times. The equation has an additional potential term which is a smooth function having Gevrey 2 + ε (resp. 1 + ε) regularity in time. This shows optimality of the local and global results of the authors [FLL25] (resp. of Tataru-Robbiano-Zuily-Hörmander [Tat95, RZ98, Hör97, Tat99]), stating that Gevrey 2 (resp. Gevrey 1) regularity of the potential yields unique continuation. In turn, this proves that Gevrey 2 (resp. Gevrey 1) regularity in time is critical for the approximate controllability of the Schrödinger (resp. wave) equation. Our construction builds on those of Kumano-go [Kg63] and Takase [Tak21].

Related

  1. Reconstruction of solutions to abstract Schrodinger, diffusion, and wave-type equations from snapshots
    Isaac Z. Pesenson
    arXivPreprintCC BY 4.0Full textAnalysis & PDEsarXiv
  2. Normalized solutions to mass supercritical Schrödinger equations with radial potentials
    Pablo Carrillo, Louis Jeanjean
    Journal of Differential EquationsPeer-reviewedCC BY 4.0Full textGeneral mathematicsHAL
  3. Global Energy Solutions for Normalized Linear Stochastic Schrödinger Equations
    Théo Hérouard
    HALPreprintCC BY 4.0Full textAnalysis & PDEsMathematical physicsHALarXiv
  4. Orbital stability of solitary wave solutions of the derivative nonlinear Schrödinger equation in the endpoint case
    Jia Shen, Yifei Wu, Changping Yang
    arXivPreprintarXiv licenceFull textAnalysis & PDEsarXiv
  5. Pullback dynamics and statistical solutions for semi-dissipative nonlinear Schrödinger equations
    Zhao Caidi, Dou Jing, Jiang Huite
    Advances in Nonlinear AnalysisPeer-reviewedCC BYFull textGeneral mathematicsDOAJ

Cite

BibTeXRIS
BibTeX
@misc{filippas2026invisible,
  title = {{INVISIBLE SOLUTIONS FOR SCHRÖDINGER AND WAVE EQUATIONS}},
  author = {Spyridon Filippas and Camille Laurent and Matthieu Léautaud},
  howpublished = {HAL},
  year = {2026},
  month = oct,
  url = {https://hal.science/hal-05830608v1},
  note = {CC BY-NC-ND 4.0},
}
RIS
TY  - UNPB
TI  - INVISIBLE SOLUTIONS FOR SCHRÖDINGER AND WAVE EQUATIONS
AU  - Spyridon Filippas
AU  - Camille Laurent
AU  - Matthieu Léautaud
PY  - 2026
DA  - 2026/10/21
PB  - HAL
UR  - https://hal.science/hal-05830608v1
AB  - On the Euclidean disk, we construct a nontrivial solution of the Schrödinger (resp. wave) equation with Dirichlet boundary conditions which vanishes identically on a smaller concentric disk for all times. The equation has an additional potential term which is a smooth function having Gevrey 2 + ε (resp. 1 + ε) regularity in time. This shows optimality of the local and global results of the authors [FLL25] (resp. of Tataru-Robbiano-Zuily-Hörmander [Tat95, RZ98, Hör97, Tat99]), stating that Gevrey 2 (resp. Gevrey 1) regularity of the potential yields unique continuation. In turn, this proves that Gevrey 2 (resp. Gevrey 1) regularity in time is critical for the approximate controllability of the Schrödinger (resp. wave) equation. Our construction builds on those of Kumano-go [Kg63] and Takase [Tak21].
ER  -