Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Resonances and resonance expansions for point interactions on the half-space

Diego Noja, Francesco Raso Stoia

TL;DR

This work provides a complete spectral-resonance analysis for a single δ-interaction on the half-space with Dirichlet or Neumann boundary conditions. It explicitly characterizes the infinite resonance set, derives a modified Weyl law for resonance counting, and develops semiclassical resonance asymptotics, all via a transparent Krein-type resolvent description with zeros of Γ_{α,y}^{D/N}(z). The paper then translates these spectral insights into concrete time-domain results by establishing resonance expansions for both the wave and Schrödinger evolutions, including zero-energy threshold phenomena and precise residue-based representations. Collectively, these results illuminate the fine-structure of resonances in domains not equal to ℝ^3 and demonstrate how boundary conditions and singular perturbations shape long-time dynamics.$$

Abstract

In this paper we describe the resonances of the singular perturbation of the Laplacian on the half space $Ω=\mathbb R^3_+$ given by the self-adjoint operator named $δ$-interaction. We will assume Dirichlet or Neumann boundary conditions on $\partial Ω$. At variance with the well known case of $\mathbb R^3$, the resonances constitute an infinite set, here completely characterized. Moreover, we prove that resonances have an asymptotic distribution satisfying a modified Weyl law and we give the semiclassical asymptotics. Finally we give applications of the results to the asymptotic behavior of the abstract wave and Schrödinger dynamics generated by the Laplacian with a point interaction on the half-space

Resonances and resonance expansions for point interactions on the half-space

TL;DR

This work provides a complete spectral-resonance analysis for a single δ-interaction on the half-space with Dirichlet or Neumann boundary conditions. It explicitly characterizes the infinite resonance set, derives a modified Weyl law for resonance counting, and develops semiclassical resonance asymptotics, all via a transparent Krein-type resolvent description with zeros of Γ_{α,y}^{D/N}(z). The paper then translates these spectral insights into concrete time-domain results by establishing resonance expansions for both the wave and Schrödinger evolutions, including zero-energy threshold phenomena and precise residue-based representations. Collectively, these results illuminate the fine-structure of resonances in domains not equal to ℝ^3 and demonstrate how boundary conditions and singular perturbations shape long-time dynamics.$$

Abstract

In this paper we describe the resonances of the singular perturbation of the Laplacian on the half space given by the self-adjoint operator named -interaction. We will assume Dirichlet or Neumann boundary conditions on . At variance with the well known case of , the resonances constitute an infinite set, here completely characterized. Moreover, we prove that resonances have an asymptotic distribution satisfying a modified Weyl law and we give the semiclassical asymptotics. Finally we give applications of the results to the asymptotic behavior of the abstract wave and Schrödinger dynamics generated by the Laplacian with a point interaction on the half-space

Paper Structure

This paper contains 8 sections, 25 theorems, 231 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries: Point Interaction on the half-space
  3. Resonances of point interactions in the half-space
  4. Asymptotics of the Resonance Counting Function
  5. Semiclassical Asymptotics of Resonances for the Half-Space
  6. Dirichlet Boundary Condition
  7. Neumann Boundary Condition
  8. Resonance Expansion for the wave and Schrödinger Equation with Point Interaction

Key Result

Proposition 2.1

Let $\alpha\in \mathbb R^*:=\mathbb R\cup\{\infty\}$ and $-\Delta_{\alpha, y}^{\textup{D}}$ be the one parameter family of self-adjoint extensions of the closed symmetric non self-adjoint operator $-\mathring\Delta_y^{\textup{D}}$ with domain $D(-\mathring\Delta_y^{\textup{D}})=\Set{\psi\in L^2(\mat

Figures (1)

  • Figure 1: Plot of the imaginary part versus real part of the first 100 values $z$ with $\mathop{\mathrm{Re}}\nolimits{z}>\pi$ for which $z^{2}$ is a resonance for $\alpha=0$ and $y_{3}=1$ in the Dirichlet half-space case.

Theorems & Definitions (59)

  • Proposition 2.1
  • Remark 2.1
  • Proposition 2.2
  • Remark 2.2
  • Remark 2.3
  • Proposition 3.1
  • Definition 3.1
  • Remark 3.1
  • Remark 3.2
  • Definition 3.2
  • ...and 49 more