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Lalit Mohan

Publications and source records attributed to Lalit Mohan.

7 recordsLinked to original sources
arXiv2025

Structurally damped semilinear evolution equation for positive operators on Hilbert space

In this study, we analyze a semilinear damped evolution equation under different damping conditions, including the undamped $(\theta=0)$, effectively damped $(0<2\theta<\sigma)$, critically damped $(2\theta=\sigma)$, and non-effectively damped $(\sigma<2\theta\leq 2\sigma)$. The analysis is conducted in two parts; the present article is devoted to examining decay estimates of solutions to the linear evolution equation governed by a self-adjoint, positive operator $\mathcal{L}$ with discrete spectrum subject to initial Cauchy data of minimal regularity. Specifically, we consider the Cauchy problem: \begin{equation*} \left\{\begin{array}{l} u_{tt}(t)+\mathcal{L}^{\theta}u_{t}(t)+\mathcal{L}^{\sigma}u(t) =0, \quad t>0, u(0)=u_{0}\in\mathcal{H},\quad u_{t}(0)=u_{1}\in\mathcal{H}, \end{array}\right. \end{equation*} in different damping conditions. %More precisely, we study decay estimates for a solution, its time derivative, and space derivative in both cases. Furthermore, we demonstrate that the decay rates of the associated solutions improve with the regularity of the initial Cauchy data. As an application of the decay estimates, we also demonstrate the global existence (in time) of the solution in certain cases, taking into account polynomial-type nonlinearity. In the 2nd article, we will address the remaining instances where global existence cannot be assured and instead present findings on local existence and possible blow-up results.

math.AP↗
arXiv2024

Non-harmonic analysis of the wave equation for Schr\"{o}dinger operators with complex potential

This article investigates the wave equation for the Schr\"{o}dinger operator on $\mathbb{R}^{n}$, denoted as $\mathcal{H}_0:=-\Delta+V$, where $\Delta$ is the standard Laplacian and $V$ is a complex-valued multiplication operator. We prove that the operator $\mathcal{H}_0$, with $\operatorname{Re}(V)\geq 0$ and $\operatorname{Re}(V)(x)\to\infty$ as $|x|\to\infty$, has a purely discrete spectrum under certain conditions. In the spirit of Colombini, De Giorgi, and Spagnolo, we also prove that the Cauchy problem with regular coefficients is well-posed in the associated Sobolev spaces, and when the propagation speed is H\"{o}lder continuous (or more regular), it is well-posed in Gevrey spaces. Furthermore, we prove that it is very weakly well-posed when the coefficients possess a distributional singularity.

math.AP↗
arXiv2023

Non-harmonic $M$-elliptic pseudo differential operators on manifolds

In this article, we introduce and study $M$-elliptic pseudo-differential operators in the framework of non-harmonic analysis of boundary value problems on a manifold $\Omega$ with boundary $\partial \Omega$, introduced by Ruzhansky and Tokmagambetov ( Int. Math. Res. Not. IMRN, (12), 3548-3615, 2016) in terms of a model operator $\mathfrak{L}$. More precisely, we consider a weighted $\mathfrak{L}$-symbol class $M_{\rho, 0, \Lambda}^{m}, m\in \mathbb{R},$ associated to a suitable weight function $\Lambda$ on a countable set $\mathcal{I} $ and study elements of the symbolic calculus for pseudo-differential operators associated with $\mathfrak{L}$-symbol class $M_{\rho, 0, \Lambda}^{m},$ by deriving formulae for the composition, adjoint, and transpose. Using the notion of $M$-ellipticity for symbols belonging to $\mathfrak{L}$-symbol class $M_{\rho, 0, \Lambda}^{m}$, we construct the parametrix of $M$-elliptic pseudo-differential operators. Further, we investigate the minimal and maximal extensions for $M$-elliptic pseudo-differential operators and show that they coincide when the symbol $\sigma\in M_{\rho, 0, \Lambda}^{m}, $ is $M$-elliptic. We provide a necessary and sufficient condition to ensure that the pseudo-differential operators $T_{\sigma}$ with symbol in the $\mathfrak{L}$-symbol class $M_{\rho, 0,\Lambda}^{0} $ is a compact operator in $L^{2}(\Omega)$ or a Riesz operator in $L^{p}(\Omega).$ Finally, we prove G\"arding's inequality for pseudo-differential operators associated with symbol from $M_{\rho, 0,\Lambda}^{0} $ in the setting of non-harmonic analysis.

math.FA↗
arXiv2023

Multilinear Fourier Integral operators on modulation spaces

In this article, we study properties of multilinear Fourier integral operators on weighted modulation spaces. In particular, using the theory of Gabor frames, we study boundedness of multilinear Fourier integral operators on products of weighted modulation spaces. Further, we investigate the periodic multilinear Fourier integral operator. Finally, we study continuity of bilinear pseudo-differential operators on modulation spaces for certain symbol classes, namely $\textbf{SG}$-class.

math.FA↗
arXiv2022

Weighted periodic and discrete Pseudo-Differential Operators

In this paper, we study elements of symbolic calculus for pseudo-differential operators associated with the weighted symbol class $M_{\rho, \Lambda}^m(\mathbb{ T}\times \mathbb{Z})$ (associated to a suitable weight function $\Lambda$ on $\mathbb{Z}$) by deriving formulae for the asymptotic sums, composition, adjoint, transpose. We also construct the parametrix of $M$-elliptic pseudo-differential operators on $\mathbb{ T}$. Further, we prove a version of Gohberg's lemma for pseudo-differetial operators with weighted symbol class $M_{\rho, \Lambda}^0(\mathbb{ T}\times \mathbb{Z})$ and as an application, we provide a sufficient and necessary condition to ensure that the corresponding pseudo-differential operator is compact on $L^2(\mathbb{T})$. Finally, we provide G\r{a}rding's and Sharp G\r{a}rding's inequality for $M$-elliptic operators on $\mathbb{Z}$ and $\mathbb{T}$, respectively, and present an application in the context of strong solution of the pseudo-differential equation $T_{\sigma} u=f$ in $L^{2}\left(\mathbb{T}\right)$.

math.FA↗
arXiv2003

Contents of Physics Related E-Print Archives

The frontiers of physics related e-print archives (1994-2002) at http://www.arxiv.org/archives/physics web service are explored from 7770 submissions. No. of e-prints in the six research disciplines besides physics (5390) were: Condensed matter(754), Quantum physics(279), Astrophysics(222), Chemical physics(129), High energy physics Phenomenology(118), and High energy physics-Theory(100)). By keyword contents following major sub-fields have high frequency: Atomic physics(1258), General physics(1121), Chemical physics(892), Accelerator physics(769), Optics(686), Biological physics(674), and Computational physics(607). Interdomainary co-word cluster analysis revealed higher e-print contents for: Classical physics-General physics(108), Quantum physics-Optics(53), and High energy physics (Phenomenology)Atomic physics(49). Prominent contributors were B. G. Sidharth (India), V. V. Flambaum (Australia), Antonina N. Fedorova (Russia), and Michael G. Zeitlin (Russia).

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