Resonance near a doubly degenerate embedded eigenvalue
Fuente:
arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2026
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| _version_ | 1866915846924795904 |
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| author | Bansal, Hemant Maharana, Alok Sahu, Lingaraj |
| author_facet | Bansal, Hemant Maharana, Alok Sahu, Lingaraj |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_08554 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Resonance near a doubly degenerate embedded eigenvalue Bansal, Hemant Maharana, Alok Sahu, Lingaraj Spectral Theory Mathematical Physics 47A10, 47A55, 81Q15 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. |
| title | Resonance near a doubly degenerate embedded eigenvalue |
| topic | Spectral Theory Mathematical Physics 47A10, 47A55, 81Q15 |
| url | https://arxiv.org/abs/2603.08554 |