Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Resolvent bounds for repulsive potentials

Andrés Larraín-Hubach, Yulong Li, Jacob Shapiro, Joseph Tiller

TL;DR

The paper proves weighted limiting absorption (resolvent) bounds for the semiclassical Schrödinger operator $P(h) = -h^2 \Delta + V(x)$ in dimensions $n \ge 3$ with nonnegative, repulsive potentials that may have a singularity at the origin. Using a spherical energy method and careful weighted estimates, it derives nontrapping and low-frequency resolvent bounds in weighted $L^2$ spaces, valid for $s,s_1,s_2>1/2$ with $s_1+s_2>2$, and extends known one-dimensional results to higher dimensions including singularities like $r^{-1}$. These resolvent bounds feed into norm bounds for weighted Helmholtz resolvents and their $\lambda$-derivatives, which, via Fourier analysis and Duhamel's formula, yield time decay of the weighted energy for the associated wave equation with a short-range repulsive potential and compactly supported initial data. The results provide a rigorous link between low-energy resolvent behavior and long-time decay, with implications for dispersive and scattering phenomena in quantum mechanics and wave propagation.

Abstract

We prove limiting absorption resolvent bounds for the semiclassical Schrödinger operator with a repulsive potential in dimension $n\ge 3$, which may have a singularity at the origin. As an application, we obtain time decay for the weighted energy of the solution to the associated wave equation with a short range repulsive potential and compactly supported initial data.

Resolvent bounds for repulsive potentials

TL;DR

The paper proves weighted limiting absorption (resolvent) bounds for the semiclassical Schrödinger operator in dimensions with nonnegative, repulsive potentials that may have a singularity at the origin. Using a spherical energy method and careful weighted estimates, it derives nontrapping and low-frequency resolvent bounds in weighted spaces, valid for with , and extends known one-dimensional results to higher dimensions including singularities like . These resolvent bounds feed into norm bounds for weighted Helmholtz resolvents and their -derivatives, which, via Fourier analysis and Duhamel's formula, yield time decay of the weighted energy for the associated wave equation with a short-range repulsive potential and compactly supported initial data. The results provide a rigorous link between low-energy resolvent behavior and long-time decay, with implications for dispersive and scattering phenomena in quantum mechanics and wave propagation.

Abstract

We prove limiting absorption resolvent bounds for the semiclassical Schrödinger operator with a repulsive potential in dimension , which may have a singularity at the origin. As an application, we obtain time decay for the weighted energy of the solution to the associated wave equation with a short range repulsive potential and compactly supported initial data.

Paper Structure

This paper contains 9 sections, 13 theorems, 138 equations.

Table of Contents

  1. Introduction and statement of results
  2. Proof of Theorem \ref{['nontrap est thm repulsive V']}
  3. Resolvent bounds for wave decay
  4. Proof of Theorem \ref{['wave decay thm']}
  5. Phragmén-Lindelöf Theorem
  6. Density argument: proof of \ref{['resolv est in sector']} and \ref{['resolv est in sector low freq']}
  7. Justification of Remark \ref{['sharpness remark']}
  8. Proof of Lemma \ref{['lap free resolv square lem']}
  9. Useful lemmas

Key Result

Theorem 1.1

Suppose $n \ge 3$ and $V$, satisfies nonneg through V prime cond. Define $P(h)$ by P, and equip it with the domain $H^2(\mathbb{R}^n)$. For all $s, \, s_1, \, s_2 > 1/2$ with $s_1 + s_2 > 2$, there is $C > 0$ such that for all $z \in \mathbb{C} \setminus [0, \infty)$ and $h > 0$,

Theorems & Definitions (23)

  • Theorem 1.1
  • Remark 1.2
  • Theorem 1.3
  • Remark 1.4
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • Remark 3.3
  • Lemma 3.4
  • proof
  • ...and 13 more