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A Paley-Wiener type uniqueness result for the electromagnetic Schrödinger equation

Yilin Song, Ying Wang, Jiqiang Zheng, Ruihan Zhou

Abstract

In this paper, we establish a Paley-Wiener type uncertainty principle for Schrödinger equations with bounded electric and magnetic potentials, \begin{align*} i\partial_tu+Δ_Au+V(t,x)u=0,\,\,u(0,x)=u_0(x), \end{align*} where $Δ_A=(\nabla-iA)^2$ denotes the magnetic Schrödinger operator. Specifically, under suitable assumptions on $A$ and $V$, we show that if a solution $u$ exhibits linear exponential decay and support property in one spatial direction at times $t=0$ and $t=1$ respectively, then $u$ must vanish identically. This result extends the theorem of Kenig-Ponce-Vega [Ann. Sci. Éc. Norm. Supér. (4) 47 (2014), 539-557] to the case $A\neq0$. We overcome the difficulty brought by the magnetic potential which breaks the translation invariance in the leading term of Hamiltonian $H=Δ_A+V$. As a direct consequence, we also obtain a uniqueness result for a class of semi-linear Schrödinger equation with electromagnetic potentials.

A Paley-Wiener type uniqueness result for the electromagnetic Schrödinger equation

Abstract

In this paper, we establish a Paley-Wiener type uncertainty principle for Schrödinger equations with bounded electric and magnetic potentials, \begin{align*} i\partial_tu+Δ_Au+V(t,x)u=0,\,\,u(0,x)=u_0(x), \end{align*} where denotes the magnetic Schrödinger operator. Specifically, under suitable assumptions on and , we show that if a solution exhibits linear exponential decay and support property in one spatial direction at times and respectively, then must vanish identically. This result extends the theorem of Kenig-Ponce-Vega [Ann. Sci. Éc. Norm. Supér. (4) 47 (2014), 539-557] to the case . We overcome the difficulty brought by the magnetic potential which breaks the translation invariance in the leading term of Hamiltonian . As a direct consequence, we also obtain a uniqueness result for a class of semi-linear Schrödinger equation with electromagnetic potentials.

Paper Structure

This paper contains 5 sections, 15 theorems, 120 equations.

Table of Contents

  1. Introduction
  2. Main results
  3. Preliminaries
  4. Carleman estimates for covariant Schrödinger flow
  5. Proof of Theorem \ref{['thm-P']}

Key Result

Theorem 1.1

If $f(x)=\mathcal{O}(e^{-\alpha|x|^2})$ and its Fourier transform has Gaussian decay $\widehat{f}(\xi)=\mathcal{O}(e^{-\beta|\xi|^2})$ with $\alpha\beta>\frac{1}{16}$, then $f=0$.

Theorems & Definitions (26)

  • Theorem 1.1: Hardy's uncertainty principle
  • Theorem 1.2: Unique continuation property,EKPV-CPDEEKPV-2008MRLEKPV-JEMSEKPV-Duke
  • Theorem 1.3: KPV-14
  • Remark 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Definition 2.1
  • Lemma 2.2
  • Remark 2.3
  • Remark 2.4
  • ...and 16 more