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Shyam Swarup Mondal

Publications and source records attributed to Shyam Swarup Mondal.

At least 19 recordsLinked to original sources
arXiv2026

Orthonormal Strichartz estimates for the Schrödinger equations with Hamiltonian on Wiener amalgam spaces

The main objective of this paper is to investigate orthonormal Strichartz estimates for Schrödinger equation on the Wiener amalgam space $\mathcal{W}(\mathcal{F} L^p, L^q)$. More precisely, we first examine improvements in the time-integrability exponent of the existing Strichartz estimates established by Cordero and Nicola in \cite{NFC} for Schrödinger equations associated with $\mathcal{H}^{+}=-\frac{1}{4π}Δ+|x|^2$. We then extend these improved Strichartz estimates from a single initial datum to systems of orthonormal families, providing, to the best of our knowledge, the first results in this direction in the setting of Wiener amalgam spaces. We further extend the classical Strichartz estimates in Wiener amalgam spaces from a single initial datum to systems of orthonormal families of initial data for the Schrödinger equation associated with the operator $\mathcal{H}^{-}=-\frac{1}{4π}Δ-|x|^2$ and Hamiltonian operator of the form $\mathcal{H}_{\mathcal{A}} = -\frac{1}{4π} B \nabla \cdot \nabla$, where $\mathcal{A} = \begin{pmatrix} 0 & B \\ 0 & 0 \end{pmatrix} \in \mathrm{Sp}(d,\mathbb{R})$ with $B = B^*$ and $\det B \neq0$. A key ingredient of our approach is Stein's complex interpolation theory for Wiener amalgam spaces, combined with a duality argument inspired by the work of Frank and Sabin. As an application of these orthonormal estimates associated with $\mathcal{H}^{+}, \mathcal{H}^{-}$, and $H_{\mathcal{A}},$ we establish local and small-data global well-posedness for the Hartree equation with infinitely many particles, for non-trace-class initial data.

math.AP↗
arXiv2026

Sharp Time-Decay Estimates for Fractional Heat Semigroups Associated with Polynomial Anharmonic Oscillators

We investigate fractional heat semigroups generated by a class of anharmonic oscillators on $\mathbb R^n$ of the form $\mathcal H_{P,Q}=Q(D)+P(x),$ where $P\in\mathcal P_{2k}$ and $Q\in\mathcal P_{2\ell}$ are real-valued polynomials with anisotropic growth. Using the Weyl--Hörmander calculus associated with the natural metric determined by $(P,Q)$, we show that the fractional powers $\mathcal H_{P,Q}^s$, $s>0$, are pseudo-differential operators with symbols in adapted classes $Σ_{P,Q}^{2s}$. We prove fixed-time decay estimates for the fractional anharmonic heat semigroup $e^{-t\mathcal H_{P,Q}^s}$ on both Lebesgue and modulation spaces. In the Lebesgue setting, we establish sharp $L^p$--$L^q$ estimates for the full range $1\le p,q\le\infty$. For large time, the decay is exponential and governed by the smallest eigenvalue $λ_0$ of $\mathcal H_{P,Q}$, namely through the factor $e^{-tλ_0^s}$, while for small time the estimates reveal two distinct phase-space scales associated with the coercive growth of $P$ and $Q$, leading to anisotropic $L^p$--$L^q$ smoothing. As applications, we study nonlinear fractional heat equations associated with $\mathcal H_{P,Q}^s$. We prove local well-posedness in the supercritical Lebesgue range $ p>\frac{n(β-1)}{2\ell s},$ derive a lower blow-up rate for finite-time blow-up solutions, and obtain critical small-data global existence. We further prove global well-posedness and exponential decay for small initial data in modulation spaces. These results extend the heat semigroup theory for harmonic and model anharmonic oscillators to a broad class of anisotropic polynomial Hamiltonians.

math.FA↗
arXiv2025

Strichartz estimates for higher order Schrödinger equations with Partial regular initial data

In this paper, we establish refined Strichartz estimates for higher-order Schrödinger equations with initial data exhibiting partial regularity. By partial regularity, we mean that the initial data are not required to have full Sobolev regularity but only regularity with respect to a subset of the spatial variables. As an application of these estimates, we investigate the well-posedness of nonlinear Schrödinger equations with power-type nonlinearities. In addition, we extend our analysis to the Dunkl Schrödinger equations under partial regularity, defined with respect to two distinct root systems. This extension poses significant challenges, mainly due to the lack of a suitable stationary phase method in the Dunkl setting. To overcome this difficulty, we develop a new result that provides an adaptation of the stationary phase method to the framework of Dunkl analysis.

math.AP↗
arXiv2025

Orthonormal Strichartz estimates for Dunkl-Schrödinger equation of initial data with Sobolev regularity

Let $Δ_κ$ be the Dunkl-Laplacian on $\mathbb{R}^n$. The main aim of this paper is to investigate the orthonormal Strichartz estimates for the Schrödinger equation with initial data from the homogeneous Dunkl-Sobolev space $\dot{H}_κ^s (\mathbb{R}^n)$. Our approach is based on restricted weak-type orthonormal estimates, frequency-localized estimates for the Dunkl-Schrödinger propagator $e^{itΔ_κ}$, and a series of successive real and complex interpolation techniques.

math.FA↗
arXiv2025

Fractional semilinear damped wave equation on the Heisenberg group

This paper aims to investigate the Cauchy problem for the semilinear damped wave equation for the fractional sub-Laplacian $(-\mathcal{L}_{\mathbb{H}})^α$, $α>0$ on the Heisenberg group $\mathbb{H}^{n}$ with power type non-linearity. With the presence of a positive damping term and nonnegative mass term, we derive $L^2-L^2$ decay estimates for the solution of the homogeneous linear fractional damped wave equation on $\mathbb{H}^{n}$, for its time derivative, and for its space derivatives. We also discuss how these estimates can be improved when we consider additional $L^1$-regularity for the Cauchy data in the absence of the mass term. Also, in the absence of mass term, we prove the global well-posedness for $2\leq p\leq 1+\frac{2α}{(\mathcal{Q}-2α)_{+}}$ $(\text{or }1+\frac{4α}{\mathcal{Q}}<p\leq 1+\frac{2α}{(\mathcal{Q}-2α)_{+}})$ in the case of $L^1\cap L^2$ $(\text{or } L^2)$ Cauchy data, respectively. However, in the presence of the mass term, the global (in time) well-posedness for small data holds for $1<p \leq 1+ \frac{2α}{(\mathcal{Q}-2α)_{+}}$. Finally, as an application of the linear decay estimates, we investigate well-posedness for the Cauchy problem for a weakly coupled system with two semilinear fractional damped wave equations with positive mass term on $\mathbb{H}^{n}$.

math.AP↗
arXiv2024

Orthonormal Strichartz inequalities and their applications on abstract measure spaces

The main objective of this paper is to extend certain fundamental inequalities from a single function to a family of orthonormal systems. In the first part of the paper, we consider a non-negative, self-adjoint operator $L$ on $L^2(X,μ)$, where $(X,μ)$ is a measure space. Under the assumption that the kernel $K_{it}(x,y)$ of the Schrödinger propagator $e^{itL}$ satisfies a uniform $L^\infty$-decay estimate of the form \begin{equation*} \sup_{x,y\in X}|K_{it}(x,y)|\lesssim |t|^{-\frac{n}{2}},\,|t|<T_0, \text{ for some }n\geq1, \end{equation*} where $T_0\in(0,+\infty]$, we establish Strichartz estimates for the Schrödinger propagator $e^{itL}$ and using a duality principle argument by Frank-Sabin \cite{FS}, we extend it for a system of infinitely many fermions on $L^2(X)$. We also obtain orthonormal Strichartz estimates for a class of dispersive semigroup $U(t)=e^{itϕ(L)}ψ(\sqrt{L}),$ where $ϕ: \mathbb{R}^+\rightarrow \mathbb{R}$ is a smooth function and $ψ\in C_c^\infty([\frac{1}{2},2])$. As an application of these orthonormal versions of Strichartz estimates, we prove the well-posedness for the Hartree equation in the Schatten spaces. In the next part of the paper, we obtain some new orthonormal Strichartz estimates, which extend prior work of Kenig-Ponce-Vega \cite{Kenig-Ponce-Vega} for single functions. Using those orthonormal versions of Kenig-Ponce-Vega result, we prove the orthonormal restriction theorem for the Fourier transform on some particular noncompact hypersurface of the form $S=\{(ξ, ϕ(ξ): ξ\in \mathbb{R})\}$, where $ϕ$ satisfies certain growth condition.

math.FA↗
arXiv2024

Higher order hypoelliptic damped wave equations on graded Lie groups with data from negative order Sobolev spaces: the critical case

Let $\mathbb G$ be a graded Lie group with homogeneous dimension $Q$. In this paper, we study the Cauchy problem for a semilinear hypoelliptic damped wave equation involving a positive Rockland operator $\mathcal{R}$ of homogeneous degree $ν\geq 2$ on $\mathbb G$ with power type nonlinearity $|u|^p$ and initial data taken from negative order homogeneous Sobolev space $\dot H^{-γ}(\mathbb G), γ>0,$ for the critical exponent case $p=1+\frac{2ν}{Q+2γ}.$ We also explore the diffusion phenomenon of the higher-order hypoelliptic damped wave equations on graded Lie groups with initial data belonging to Sobolev spaces of negative order. We emphasize that our results are also new, even in the setting of higher-order differential operators on $\mathbb{R}^n$, and more generally, on stratified Lie groups.

math.AP↗
arXiv2024

Non-harmonic analysis of the wave equation for Schrödinger operators with complex potential

This article investigates the wave equation for the Schrödinger operator on $\mathbb{R}^{n}$, denoted as $\mathcal{H}_0:=-Δ+V$, where $Δ$ 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 Hölder 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↗
arXiv2024

Time-fractional discrete diffusion equation for Schrödinger operator

This article aims to investigate the semi-classical analog of the general Caputo-type diffusion equation with time-dependent diffusion coefficient associated with the discrete Schrödinger operator, $\mathcal{H}_{\hbar,V}:=-\hbar^{-2}\mathcal{L}_{\hbar}+V$ on the lattice $\hbar\mathbb{Z}^{n},$ where $V$ is a non-negative multiplication operator and $\mathcal{L}_{\hbar}$ is the discrete Laplacian. We establish the well-posedness of the Cauchy problem for the general Caputo-type diffusion equation with a regular coefficient in the associated Sobolev-type spaces. However, it is very weakly well-posed when the diffusion coefficient has a distributional singularity. Finally, we recapture the classical solution (resp. very weak) for the general Caputo-type diffusion equation in the semi-classical limit $\hbar\to 0$.

math.AP↗
arXiv2024

Decay estimates for a class of Dunkl wave equations

Let $Δ_κ$ be the Dunkl Laplacian on $\mathbb{R}^n$ and $ϕ: \mathbb{R}^+ \to \mathbb{R}$ is a smooth function. The aim of this manuscript is twofold. First, we study the decay estimate for a class of dispersive semigroup of the form $e^{itϕ(\sqrt{-Δ_κ})}$.W e overcome the difficulty arising from the non-homogeneousity of $ϕ$ by frequency localization. As applications, in the next part of the paper, we establish Strichartz estimates for some concrete wave equations associated with the Dunkl Laplacian $Δ_k,$ which corresponds to $ϕ(r)=r, r^2, r^2+r^4, \sqrt{1+r^2}, \sqrt{1+r^4}$, and $r^μ,0<μ\leq 2, μ\neq 1$. More precisely, we unify and simplify all the known dispersive estimates and extend to more general cases. Finally, using the decay estimates, we prove the global-in-time existence of small data Sobolev solutions for the nonlinear Klein-Gordon equation and beam equation with the power type nonlinearities.

math.FA↗
arXiv2024

Higher order hypoelliptic damped wave equations on graded Lie groups with data from negative order Sobolev spaces

Let $\mathbb G$ be a graded Lie group with homogeneous dimension $Q$. In this paper, we study the Cauchy problem for a semilinear hypoelliptic damped wave equation involving a positive Rockland operator $\mathcal{R}$ of homogeneous degree $ν\geq 2$ on $\mathbb G$ with power type nonlinearity $|u|^p$ and initial data taken from negative order homogeneous Sobolev space $\dot H^{-γ}(\mathbb G), γ>0$. In the framework of Sobolev spaces of negative order, we prove that $p_{\text{Crit}}(Q, γ, ν) :=1+\frac{2ν}{Q+2γ}$ is the new critical exponent for $γ\in (0, \frac{Q}{2})$. More precisely, we show the global-in-time existence of small data Sobolev solutions of lower regularity for $p>p_{\text{Crit}}(Q, γ, ν) $ in the energy evolution space $ \mathcal{C}\left([0, T], H^{s}(\mathbb{G})\right), s\in (0, 1]$. Under certain conditions on the initial data, we also prove a finite-time blow-up of weak solutions for $1<p<p_{\text{Crit}}(Q, γ, ν)$. Furthermore, to precisely characterize the blow-up time, we derive sharp upper bound and lower bound estimates for the lifespan in the subcritical cases. We emphasize that our results are also new, even in the setting of higher-order differential operators on $\mathbb{R}^n$, and more generally, on stratified Lie groups.

math.AP↗
arXiv2024

Semilinear damped wave equations on the Heisenberg group with initial data from Sobolev spaces of negative order

In this paper, we focus on studying the Cauchy problem for semilinear damped wave equations involving the sub-Laplacian $\mathcal{L}$ on the Heisenberg group $\mathbb{H}^n$ with power type nonlinearity $|u|^p$ and initial data taken from Sobolev spaces of negative order homogeneous Sobolev space $\dot H^{-γ}_{\mathcal{L}}(\mathbb{H}^n), γ>0$, on $\mathbb{H}^n$. In particular, in the framework of Sobolev spaces of negative order, we prove that the critical exponent is the exponent $p_{\text{crit}}(Q, γ)=1+\frac{4}{Q+2γ},$ for some $γ\in (0, \frac{Q}{2})$, where $Q:=2n+2$ is the homogeneous dimension of $\mathbb{H}^n$. More precisely, we establish a global-in-time existence of small data Sobolev solutions of lower regularity for $p>p_{\text{crit}}(Q, γ)$ in the energy evolution space; a finite time blow-up of weak solutions for $1<p<p_{\text{crit}}(Q, γ)$ under certain conditions on the initial data by using the test function method. Furthermore, to precisely characterize the blow-up time, we derive sharp upper bound and lower bound estimates for the lifespan in the subcritical case.

math.AP↗
arXiv2023

Uncertainty principles for the windowed Opdam--Cherednik transform

In this paper, we study a few versions of the uncertainty principle for the windowed Opdam--Cherednik transform. In particular, we establish the uncertainty principle for orthonormal sequences, Donoho--Stark's uncertainty principle, Benedicks-type uncertainty principle, Heisenberg-type uncertainty principle and local uncertainty inequality for this transform. We also obtain the Heisenberg-type uncertainty inequality using the $k$-entropy of the windowed Opdam--Cherednik transform.

math.FA↗
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 $Ω$ with boundary $\partial Ω$, 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 $M_{ρ, 0, Λ}^{m}, m\in \mathbb{R},$ associated to a suitable weight function $Λ$ on a countable set $\mathcal{I} $ and study elements of the symbolic calculus for pseudo-differential operators associated with $\mathfrak{L}$-symbol class $M_{ρ, 0, Λ}^{m},$ by deriving formulae for the composition, adjoint, and transpose. Using the notion of $M$-ellipticity for symbols belonging to $\mathfrak{L}$-symbol class $M_{ρ, 0, Λ}^{m}$, 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 $σ\in M_{ρ, 0, Λ}^{m}, $ is $M$-elliptic. We provide a necessary and sufficient condition to ensure that the pseudo-differential operators $T_σ$ with symbol in the $\mathfrak{L}$-symbol class $M_{ρ, 0,Λ}^{0} $ is a compact operator in $L^{2}(Ω)$ or a Riesz operator in $L^{p}(Ω).$ Finally, we prove Gärding's inequality for pseudo-differential operators associated with symbol from $M_{ρ, 0,Λ}^{0} $ in the setting of non-harmonic analysis.

math.FA↗
arXiv2023

On local dispersive and Strichartz estimates for the Grushin operator

Let $G=-Δ-|x|^2\partial_{t}^2$ denote the Grushin operator on $\mathbb{R}^{n+1}$. The aim of this paper is two fold. In the first part, due to the non-dispersive phenomena of the Grushin-Schrödinger equation on $\mathbb{R}^{n+1}$, we establish a local dispersive estimate by defining the Grushin-Schrödinger kernel on a suitable domain. As a corollary we obtain a local Strichartz estimate for the Grushin-Schrödinger equation. In the next part, we prove a restriction theorem with respect to the scaled Hermite-Fourier transform on $\mathbb{R}^{n+2}$ for certain surfaces in $\mathbb{N}_0^n\times\mathbb{R^*}\times \mathbb{R}$ and derive anisotropic Strichartz estimates for the Grushin-Schrödinger equation and for the Grushin wave equation as well.

math.AP↗
arXiv2023

Discrete time-dependent wave equation for the Schrödinger operator with unbounded potential

In this article, we investigate the semiclassical version of the wave equation for the discrete Schrödinger operator, $\mathcal{H}_{\hbar,V}:=-\hbar^{-2}\mathcal{L}_{\hbar}+V$ on the lattice $\hbar\mathbb{Z}^{n},$ where $\mathcal{L}_{\hbar}$ is the discrete Laplacian, and $V$ is a non-negative multiplication operator. We prove that $\mathcal{H}_{\hbar,V}$ has a purely discrete spectrum when the potential $V$ satisfies the condition $|V(k)|\to \infty$ as $|k|\to\infty$. We also show that the Cauchy problem with regular coefficients is well-posed in the associated Sobolev type spaces and very weakly well-posed for distributional coefficients. Finally, we recover the classical solution as well as the very weak solution in certain Sobolev type spaces as the limit of the semiclassical parameter $\hbar\to 0$.

math.AP↗
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

Restriction theorem for Fourier-Dunkl transform II: Paraboloid, sphere, and hyperboloid surfaces

This is a continuation of the paper "Restriction theorem for Fourier-Dunkl transform I: Cone surface, J. Pseudo-Differ. Oper. Appl. 14(1), Paper No. 5 (2023)", where the authors introduced and studied the Fourier-Dunkl transform on $\mathbb{R}^{n}\times\mathbb{R}^{d}$. The main novelty of this paper is that we here prove Strichartz's restriction theorem for the Fourier-Dunkl transform for certain surfaces, namely, paraboloid, sphere, and hyperboloid and its generalisation to the family of orthonormal functions. Finally, as an application of these restriction theorems, we establish versions of Strichartz estimates for orthonormal families of initial data associated with Schrödinger's propagator in the case of the Dunkl Laplacian and Klein-Gordon operator.

math.CA↗