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Shape-Resonance in Spectral density, Scattering Cross-section, Time delay and Bound on Sojourn time

Hemant Bansal, Alok Maharana, Lingaraj Sahu, Kalyan B. Sinha

Abstract

The Friedrichs model~\cite{Friedrichs} is revisited to obtain precise results about the asymptotic behaviour (the so-called Breit-Wigner formula~\cite{Breit}) of a resonance near an embedded eigenvalue and the ``spectral concentration" results as a corollary. Some of the abstract results involved can also be used to address similar questions about a rank-one perturbation of the Laplacian. Exact asymptotic properties are also obtained for the sojourn time, the scattering amplitude and time delay.

Shape-Resonance in Spectral density, Scattering Cross-section, Time delay and Bound on Sojourn time

Abstract

The Friedrichs model~\cite{Friedrichs} is revisited to obtain precise results about the asymptotic behaviour (the so-called Breit-Wigner formula~\cite{Breit}) of a resonance near an embedded eigenvalue and the ``spectral concentration" results as a corollary. Some of the abstract results involved can also be used to address similar questions about a rank-one perturbation of the Laplacian. Exact asymptotic properties are also obtained for the sojourn time, the scattering amplitude and time delay.
Paper Structure (10 sections, 22 theorems, 121 equations)

This paper contains 10 sections, 22 theorems, 121 equations.

Table of Contents

  1. Introduction
  2. Notations and preliminaries
  3. The model
  4. Spectral density and its asymptotic properties
  5. Sojourn time and its properties
  6. Scattering theory and Krein's spectral shift function
  7. An application to perturbation of Laplacian in $\mathbb R^3$
  8. Density asymptotics and other spectral properties
  9. Behaviour of scattering cross-section and time delay
  10. Appendix

Key Result

Proposition 2.2

Let $f \in \mathcal{S}(\mathbb{R})$. Then the following properties hold:

Theorems & Definitions (48)

  • Definition 2.1
  • Proposition 2.2
  • proof
  • Proposition 2.3
  • proof
  • Proposition 2.4: Plemelj-Privalov Theorem
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof
  • ...and 38 more