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Exponential Decay outside of the Light Cone for the Pseudo-Relativistic Non-Autonomous Schrödinger Equation

Sébastien Breteaux, Jérémy Faupin, Viviana Grasselli

TL;DR

This work addresses propagation speed in the semi-relativistic Schrödinger equation $i\partial_t\psi_t=(\langle\nabla\rangle+V_t)\psi_t$ by developing a maximal velocity bound. The authors adapt a unitary-conjugation/analytic framework to the pseudo-relativistic kinetic energy, deriving a precise bound $\|\mathbf{1}_Y U_t \mathbf{1}_X\|_{\mathcal{B}(L^2)}\le e^{t-\mathrm{dist}(X,Y)}$ for convex regions and a sharper decay $\le e^{-2\delta}$ outside the light cone, with $\delta=\mathrm{dist}(Y, X+B(0,t))$. They also provide a practical condition ensuring the existence of a unitary propagator for time-dependent potentials via a decomposition $V_t=V_{\infty,t}+V_{\mathcal B,t}$ and the KLMN theorem, and discuss sharpness and extensions to nonconvex sets. The results have implications for relativistic quantum propagation in the continuum and for nonlinear applications, as highlighted by a companion work on the pseudo-relativistic Hartree equation.

Abstract

We establish a maximal velocity bound for a pseudo-relativistic quantum particle in an external time-dependent potential. Our estimate shows that the probability for the particle, starting in a convex set $X\subset\mathbb{R}^d$ at $t=0$, to reach a convex set $Y\subset\mathbb{R}^d$ at a time $t>0$, is bounded by $e^{-2δ}$ where $δ$ is the distance from $Y$ to the section at time $t$ of the light cone generated by $X$.

Exponential Decay outside of the Light Cone for the Pseudo-Relativistic Non-Autonomous Schrödinger Equation

TL;DR

This work addresses propagation speed in the semi-relativistic Schrödinger equation by developing a maximal velocity bound. The authors adapt a unitary-conjugation/analytic framework to the pseudo-relativistic kinetic energy, deriving a precise bound for convex regions and a sharper decay outside the light cone, with . They also provide a practical condition ensuring the existence of a unitary propagator for time-dependent potentials via a decomposition and the KLMN theorem, and discuss sharpness and extensions to nonconvex sets. The results have implications for relativistic quantum propagation in the continuum and for nonlinear applications, as highlighted by a companion work on the pseudo-relativistic Hartree equation.

Abstract

We establish a maximal velocity bound for a pseudo-relativistic quantum particle in an external time-dependent potential. Our estimate shows that the probability for the particle, starting in a convex set at , to reach a convex set at a time , is bounded by where is the distance from to the section at time of the light cone generated by .
Paper Structure (6 sections, 14 theorems, 116 equations, 1 figure)

This paper contains 6 sections, 14 theorems, 116 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Organisation of the Paper.
  3. Notations and Definitions
  4. Maximal Velocity Estimates
  5. Unitary propagators
  6. Sharpness of the maximal velocity estimate for convex subsets

Key Result

Theorem 1.1

Let $T>0$ and $(V_t)_{t\in[0,T]}$ be a family of real-valued potentials such that the family $(U_t)_{t\in[0,T]}$ is a propagator generated by $(\langle\nabla\rangle+V_t)_{t\in[0,T]}$. If $X$ and $Y$ are convex subsets of $\mathbb{R}^d$, then

Figures (1)

  • Figure 1: Consider $X$ and $Y$ convex subsets of $\mathbb{R}^d$ ($d=2$ in the illustration). Let $B(0,t)$ be the closed ball of radius $t>0$ centered at the origin. If the section $X+B(0,t)$ of the light cone generated by $X$ at a time $t$ is at a distance $\delta$ from $Y$, then a particle in $X$ at time $0$ will reach $Y$ at time $t>0$ with a probability lower than $e^{-2\delta}$.

Theorems & Definitions (34)

  • Theorem 1.1
  • Remark 1.2
  • Remark 1.3
  • Corollary 1.4
  • Remark 1.5
  • Proposition 1.6
  • Remark 1.7
  • Remark 1.8
  • Remark 1.9
  • Definition 2.1
  • ...and 24 more