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Weakly localized states of one dimensional Schrodinger equations have localized energy

Gavin Stewart, Avy Soffer

TL;DR

This work analyzes the one-dimensional Schrödinger equation with a time-dependent, rapidly decaying potential and proves a precise long-time decomposition of solutions into free radiation and a weakly bound component with localized energy. The authors extract a free channel via a time-dependent projection, employ an incoming/outgoing phase-space decomposition, and establish energy localization for the weakly bound part, including a higher-order symbol analysis that yields localization of derivatives up to order $n$ under symbol-type bounds on $V$. The key contributions are (i) the $\dot{H}^1$-localized refinement $u_{wb}=u_{loc}+o_{\dot{H}^1}(1)$, (ii) the construction and analysis of the free channel $\Omega_{free}$ and the projections $P^{in/out/low}$, and (iii) an inductive scheme producing $u_{\textup{loc},n}$ with sharp derivative localization and a remainder $u_{\textup{rem},n}$ that remains remainder-type. These results extend prior higher-dimensional/ radial theories to the 1D setting, clarifying the role of resonances and localization in low dimensions and contributing to the understanding of scattering and long-time dynamics under time-dependent perturbations.

Abstract

We study the asymptotics of the Schrödinger equation with time-dependent potential in dimension one. Assuming that the potential decays sufficiently rapidly as $|x| \to \infty$, we prove that the solution can be written as the sum of a free wave $e^{-itΔ} u_+$ and a weakly bound component $u_{\text{wb}}(t)$. Moreover, we show that the weakly bound part decomposes as $u_{\text{wb}}(t) = u_{\text{loc}}(t) + o_{\dot{H}^1}(1)$, where $\partial_x u_\text{loc}(t)$ is localized near the origin uniformly in time. Since decay conditions on the potential do not preclude resonances unless $d \geq 5$, our results can be seen as a natural extension of [Terence Tao. Dynamics of Partial Differential Equations, 5(2), 2008] and [Avy Soffer, Xiaoxu Wu. arXiv:2304.04245] to the lower-dimensional case.

Weakly localized states of one dimensional Schrodinger equations have localized energy

TL;DR

This work analyzes the one-dimensional Schrödinger equation with a time-dependent, rapidly decaying potential and proves a precise long-time decomposition of solutions into free radiation and a weakly bound component with localized energy. The authors extract a free channel via a time-dependent projection, employ an incoming/outgoing phase-space decomposition, and establish energy localization for the weakly bound part, including a higher-order symbol analysis that yields localization of derivatives up to order under symbol-type bounds on . The key contributions are (i) the -localized refinement , (ii) the construction and analysis of the free channel and the projections , and (iii) an inductive scheme producing with sharp derivative localization and a remainder that remains remainder-type. These results extend prior higher-dimensional/ radial theories to the 1D setting, clarifying the role of resonances and localization in low dimensions and contributing to the understanding of scattering and long-time dynamics under time-dependent perturbations.

Abstract

We study the asymptotics of the Schrödinger equation with time-dependent potential in dimension one. Assuming that the potential decays sufficiently rapidly as , we prove that the solution can be written as the sum of a free wave and a weakly bound component . Moreover, we show that the weakly bound part decomposes as , where is localized near the origin uniformly in time. Since decay conditions on the potential do not preclude resonances unless , our results can be seen as a natural extension of [Terence Tao. Dynamics of Partial Differential Equations, 5(2), 2008] and [Avy Soffer, Xiaoxu Wu. arXiv:2304.04245] to the lower-dimensional case.
Paper Structure (17 sections, 14 theorems, 232 equations)

This paper contains 17 sections, 14 theorems, 232 equations.

Table of Contents

  1. Introduction
  2. Previous work
  3. Main result and sketch of the proof
  4. Preliminaries
  5. Notation and conventions
  6. Basic estimates
  7. Propagation estimates
  8. Construction of the free channel
  9. Energy localization
  10. Incoming/outgoing projection
  11. Decomposition of the weakly bound state
  12. Bounds for u_rem
  13. Bounds for u_loc
  14. Higher-order symbol bounds
  15. Definition of $u_{\textup{loc},n}$
  16. ...and 2 more sections

Key Result

Theorem 1

Suppose $u$ solves eqn:main-eqn with a potential $V$ satisfying and that $\sup_t\lVert u(t) \rVert_{H^1} < \infty$, and that $\theta \in \left(0, 1\right)$ is a constant. Then, we have the decomposition where $u_+ \in H^1$ and $u_\textup{loc}$ has a localized derivative in the sense that Moreover, if $V$ satisfies the symbol-type bounds for $n > 1$ and $\theta \in \left(0, 1\right)$ then we ca

Theorems & Definitions (26)

  • Theorem 1
  • Remark 1
  • Lemma 2
  • proof
  • Corollary 3
  • proof
  • Lemma 4
  • proof
  • Lemma 5
  • proof
  • ...and 16 more