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Pointwise decay for radial solutions of the Schrödinger equation with a repulsive Coulomb potential

Adam Black, Ebru Toprak, Bruno Vergara, Jiahua Zou

Abstract

We study the long-time behavior of solutions to the Schrödinger equation with a repulsive Coulomb potential on $\mathbb{R}^3$ for spherically symmetric initial data. Our approach involves computing the distorted Fourier transform of the action of the associated Hamiltonian $H=-Δ+\frac{q}{|x|}$ on radial data $f$, which allows us to explicitly write the evolution $e^{itH}f$. A comprehensive analysis of the kernel is then used to establish that, for large times, $\|e^{i t H}f\|_{L^{\infty}} \leq C t^{-\frac{3}{2}}\|f\|_{L^1}$. Our analysis of the distorted Fourier transform is expected to have applications to other long-range repulsive problems.

Pointwise decay for radial solutions of the Schrödinger equation with a repulsive Coulomb potential

Abstract

We study the long-time behavior of solutions to the Schrödinger equation with a repulsive Coulomb potential on for spherically symmetric initial data. Our approach involves computing the distorted Fourier transform of the action of the associated Hamiltonian on radial data , which allows us to explicitly write the evolution . A comprehensive analysis of the kernel is then used to establish that, for large times, . Our analysis of the distorted Fourier transform is expected to have applications to other long-range repulsive problems.
Paper Structure (17 sections, 30 theorems, 352 equations)

This paper contains 17 sections, 30 theorems, 352 equations.

Table of Contents

  1. Introduction
  2. Motivation and main result
  3. Prior work
  4. Overview of the proof
  5. Notation and conventions
  6. Organization of the paper
  7. Eigenfunction approximation: Small Energies
  8. Bessel function approximation: $x\ll 1$
  9. Airy function approximation: $x\sim 1$
  10. Oscillatory Airy approximation: $x\gg 1$
  11. Eigenfunction approximation: Large Energies
  12. Proof of Theorem \ref{['thm.main']}
  13. Estimation of $K_t^l(r,s)$
  14. Estimate of $K_t^h(r,s)$
  15. Kernel of the Coulomb evolution
  16. ...and 2 more sections

Key Result

Theorem 1.1

Let $H=-\Delta+\frac{q}{|x|}$ be the Coulomb Hamiltonian in $\mathbb R^{3}$, with $q>0$. Then, for any spherically symmetric function $f$ one has the dispersive estimate with constant $C>0$ independent of $f$ and $t$.

Theorems & Definitions (60)

  • Theorem 1.1
  • Lemma 2.1
  • proof
  • Proposition 2.2
  • Remark 2.3
  • proof
  • Lemma 2.4
  • proof
  • Lemma 2.5
  • proof
  • ...and 50 more