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From resolvent expansions at zero to long time wave expansions

T. J. Christiansen, K. Datchev, M. Yang

TL;DR

The paper develops an abstract resolvent-based framework to convert zero-energy resolvent expansions into precise long-time wave asymptotics under a high-energy polynomial bound. It proves a main theorem that decomposes the wave evolution into discrete, zero-energy, and exponentially small remainder parts via contour integrals of the resolvent, and then applies it to obstacle scattering, Aharonov--Bohm systems, cones, and Schrödinger operators with compactly supported potentials. The results yield explicit leading terms, including logarithmic and power-law decays with computable constants, and rigorous remainder estimates, thereby unifying diverse 2D scattering problems under a single methodology. This framework provides sharp, model-specific asymptotics with potential for broad extension to other black-box perturbations and inhomogeneous problems, enhancing understanding of how low-energy spectral data governs long-time wave propagation.

Abstract

We prove a general abstract theorem deducing wave expansions as time goes to infinity from resolvent expansions as energy goes to zero, under an assumption of polynomial boundedness of the resolvent at high energy. We give applications to obstacle scattering, to Aharonov--Bohm Hamiltonians, to scattering in a sector, and to scattering by a compactly supported potential.

From resolvent expansions at zero to long time wave expansions

TL;DR

The paper develops an abstract resolvent-based framework to convert zero-energy resolvent expansions into precise long-time wave asymptotics under a high-energy polynomial bound. It proves a main theorem that decomposes the wave evolution into discrete, zero-energy, and exponentially small remainder parts via contour integrals of the resolvent, and then applies it to obstacle scattering, Aharonov--Bohm systems, cones, and Schrödinger operators with compactly supported potentials. The results yield explicit leading terms, including logarithmic and power-law decays with computable constants, and rigorous remainder estimates, thereby unifying diverse 2D scattering problems under a single methodology. This framework provides sharp, model-specific asymptotics with potential for broad extension to other black-box perturbations and inhomogeneous problems, enhancing understanding of how low-energy spectral data governs long-time wave propagation.

Abstract

We prove a general abstract theorem deducing wave expansions as time goes to infinity from resolvent expansions as energy goes to zero, under an assumption of polynomial boundedness of the resolvent at high energy. We give applications to obstacle scattering, to Aharonov--Bohm Hamiltonians, to scattering in a sector, and to scattering by a compactly supported potential.
Paper Structure (17 sections, 10 theorems, 50 equations, 1 figure)

This paper contains 17 sections, 10 theorems, 50 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Scattering by several convex obstacles
  3. Aharonov--Bohm Hamiltonians
  4. Laplacians on Cones
  5. Schrödinger operators
  6. Further background and context
  7. Further results
  8. Plan of the paper
  9. The Main Expansion
  10. The abstract theorem
  11. Asymptotics in terms of elementary functions
  12. Tail estimate
  13. Applications of the main expansion
  14. Scattering by convex obstacles
  15. Aharonov--Bohm operators
  16. ...and 2 more sections

Key Result

Theorem 1.1

Let $\chi$, $f \in C_c^\infty(\mathbb{R}^2\setminus \overline{\mathcal{O}})$, and $u$ be as in eq:waveIntro. Let $G\in \mathcal{D}_{loc}$ satisfy $PG=0$ and $G(x,y)= \log|(x,y)| +O(1)$ as $(x,y)\rightarrow \infty$. Then, for any $s \in \mathbb N$, we have

Figures (1)

  • Figure 1: The first contour is $\gamma(\delta,c)$, the second is $\Gamma(\delta,c)$, and the third is $\gamma(\delta,\infty)$. They are all oriented from left to right.

Theorems & Definitions (14)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 2.1
  • Corollary 2.2
  • proof : Proof of Theorem \ref{['t:main']}
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • ...and 4 more