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Resonance near a doubly degenerate embedded eigenvalue

Hemant Bansal, Alok Maharana, Lingaraj Sahu

Abstract

This paper extends the study of resonance phenomenon initiated by the authors in~\cite{LS} to the case of doubly degenerate embedded eigenvalues (i.e. eigenvalue of multiplicity two). A fundamentally new concept is introduced to resolve the difficulties that arise in this study, beyond the methods of \cite{LS}. We apply a differential topological technique, namely the Morse Lemma, to study the present case. This allows us to understand rank-two self-adjoint perturbations of the Laplacian on $L^{2}(\mathbb{R}^{3})$, and along with methods of \cite{LS}, we obtain asymptotic results for the spectral density near a doubly degenerate embedded eigenvalue. Importantly, we are able to easily handle the threshold eigenvalue case. \par We also analyze important properties which explain such resonance phenomenon, viz., asymptotic behaviour of the sojourn time, scattering cross-section and time delay.

Resonance near a doubly degenerate embedded eigenvalue

Abstract

This paper extends the study of resonance phenomenon initiated by the authors in~\cite{LS} to the case of doubly degenerate embedded eigenvalues (i.e. eigenvalue of multiplicity two). A fundamentally new concept is introduced to resolve the difficulties that arise in this study, beyond the methods of \cite{LS}. We apply a differential topological technique, namely the Morse Lemma, to study the present case. This allows us to understand rank-two self-adjoint perturbations of the Laplacian on , and along with methods of \cite{LS}, we obtain asymptotic results for the spectral density near a doubly degenerate embedded eigenvalue. Importantly, we are able to easily handle the threshold eigenvalue case. \par We also analyze important properties which explain such resonance phenomenon, viz., asymptotic behaviour of the sojourn time, scattering cross-section and time delay.
Paper Structure (16 sections, 24 theorems, 189 equations)

This paper contains 16 sections, 24 theorems, 189 equations.

Table of Contents

  1. Introduction
  2. Notations and some basic results
  3. Spectral representation of the Laplacian on $L^2(\mathbb R^3$)
  4. Rank-two perturbation of the Laplacian on $L^2(\mathbb R^3)$
  5. Calculation of Hessian and its determinant:
  6. The model
  7. Construction of the canonical eigenbasis at $\lambda_0$:
  8. Spectral density and its asymptotic behaviour
  9. Spectral concentration, time decay and sojourn time
  10. Spectral Concentration
  11. Time decay
  12. Sojourn time and its properties
  13. Behaviour of Scattering amplitude and Time delay
  14. Scattering amplitude and total scattering cross section
  15. Krein's spectral shift function and time delay
  16. ...and 1 more sections

Key Result

Proposition 2.2

Let $f \in \mathcal{S}(\mathbb{R})$. Then $\gamma(f,\cdot)\in C^\infty(\mathbb{R})$ and its $k$-th derivative satisfies

Theorems & Definitions (52)

  • Definition 2.1
  • Proposition 2.2: LS
  • Proposition 2.3: Plemelj--Privalov Theorem, DorinaMitreaBook
  • Lemma 3.1
  • proof
  • Lemma 4.1
  • proof
  • Theorem 4.2
  • proof
  • Corollary 4.3
  • ...and 42 more