arXiv subjects

Hemant Bansal

Publications and source records attributed to Hemant Bansal.

3 recordsLinked to original sources

arXiv2026

Multi-parameter Perturbations of the Laplacian and Resonance Near a Simple Embedded Eigenvalue

This paper continues the study of resonance phenomena initiated in [3] for rank-one perturbations. We consider finite-rank multi-parameter perturbations $H_\alpha$ of the Laplacian on $L^2(\mathbb{R}^3)$ and establish Breit--Wigner-type asymptotics for the spectral density of $H_\alpha$ along the resonance $\lambda(\alpha)$ near a simple embedded eigenvalue $\lambda_0$ of $H_a$ as $\alpha\to a$. We also obtain similar asymptotic behaviour for the scattering cross-section and the average time delay.

math.SP↗

arXiv2026

Resonance near a doubly degenerate embedded eigenvalue

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.

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arXiv2026

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

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.

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