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Lingaraj Sahu

Publications and source records attributed to Lingaraj Sahu.

11 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.

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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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arXiv2015

Some Quantum Dynamical Semi-groups with Quantum Stochastic Dilation

We consider the GNS Hilbert space $\mathcal{H}$ of a uniformly hyper-finite $C^*$- algebra and study a class of unbounded Lindbladian arises from commutators. Exploring the local structure of UHF algebra, we have shown that the associated Hudson-Parthasarathy type quantum stochastic differential equation admits a unitary solution. The vacuum expectation of homomorphic co-cycle, implemented by the Hudson-Parthasarathy flow, is conservative and gives the minimal semi-group associated with the formal Lindbladian. We also associate conservative minimal semi-groups to another class of Lindbladian by solving the corresponding Evan-Hudson equation.

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arXiv2010

Unitary Processes with Independent Increments

In this paper, we study unitary Gaussian processes with independent increments with which the unitary equivalence to a Hudson-Parthasarathy evolution systems is proved. This gives a generalization of results in [16] and [17] in the absence of the stationarity condition.

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arXiv2008

Characterization of unitary processes with independent and stationary increments

This is a continuation of the earlier work \cite{SSS} to characterize stationary unitary increment Gaussian processes. The earlier assumption of uniform continuity is replaced by weak continuity and with a technical assumption on the domain of the generator, unitary equivalence of the processes to the solution of Hudson-Parthasarathy equation is proved.

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arXiv2007

Quantum random walks and vanishing of the second Hochschild cohomology

Given a conditionally completely positive map $\mathcal L$ on a unital $\ast$-algebra $\A$, we find an interesting connection between the second Hochschild cohomology of $\A$ with coefficients in the bimodule $E_{\mathcal L}=\B^a(\A \oplus M)$ of adjointable maps, where $M$ is the GNS bimodule of $\mathcal L$, and the possibility of constructing a quantum random walk (in the sense of \cite{AP,LP,L,KBS}) corresponding to $\mathcal L$.

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arXiv2005

Quantum random walks and their convergence

Using coordinate-free basic operators on toy Fock spaces \cite{AP}, quantum random walks are defined following the ideas in \cite{LP,AP}. Strong convergence of quantum random walks associated with bounded structure maps is proved under suitable assumptions, extendings the result obtained in \cite{KBS} in case of one dimensional noise. To handle infinite dimensional noise we have used the coordinate-free language of quantum stochastic calculus developed in \cite{GS1}.

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