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Jitendriya Swain

Publications and source records attributed to Jitendriya Swain.

11 recordsLinked to original sources
arXiv2025

Restriction Theorem and Strichartz estimate for orthonormal functions associated with the Special Hermite Operator

Let $\mathcal{L}$ be the special Hermite operator on $\mathbb{C}^n$. As a continuation of the recent results in \cite{SG}, we establish new Strichartz estimates for systems of orthonormal functions associated with general flows of the form $e^{-itϕ(\mathcal{L})}$, where $ ϕ: \mathbb{R}^{+} \to \mathbb{R} $ is a smooth function. Our approach relies on restriction estimates for the Fourier-special Hermite transform on the class of surfaces $\{(λ, μ, ν)\in \mathbb{R}\times\mathbb{N}_0^n\times\mathbb{N}_0^n : λ=ϕ(2|ν|+n)\}$. We also discuss the endpoint case of the orthonormal Strichartz estimate for the Schrödinger propagator $e^{-it\mathcal{L}}$. Furthermore, we generalize restriction estimates for the special Hermite spectral projections in the context of trace ideals (Schatten spaces).

math.FA↗
arXiv2025

Restriction theorem for the Fourier-Hermite transform associated with the normalized Hermite polynomials and the Ornstein-Uhlenbeck-Schrödinger equation

In this article, we prove the analogue theorems of Stein-Tomas and Srtichartz on the discrete surface restrictions of Fourier-Hermite transforms associated with the normalized Hermite polynomials and obtain the Strichartz estimate for the system of orthonormal functions for the Ornstein-Uhlenbeck operator $L=-\frac{1}{2}Δ+\langle x, \nabla\rangle$ on $\mathbb{R}^n$. Further, we show an optimal behavior of the constant in the Strichartz estimate as limit of a large number of functions.

math.CA↗
arXiv2025

Heisenberg-Pauli-Weyl uncertainty principles for the fractional Dunkl transform on the real line

The aim of the paper is two-fold. First, we provide an explicit form of the functions for which equality holds for the uncertainty inequalities studied in \cite{Fei}. Second, we establish an $L^p$-type Heisenberg-Pauli-Weyl uncertainty principle for the fractional Dunkl transform, with $1 \leq p \leq 2$. For the case $p = 2$, we further derive a sharper uncertainty principle for the fractional Dunkl transform. Furthermore, we derive conditions leading to equality in both the uncertainty principles obtained.

math.FA↗
arXiv2024

On the Schatten exponent in orthonormal Strichartz estimate for the Dunkl operators

In \cite{PRA} and \cite{SSM} the orthonormal Strichartz estimates for the Schrödinger equation associated with the Dunkl Laplacian and the Dunkl-Hermite operator are obtained. In this article, we prove a necessary condition on the Schatten exponent for the above orthonormal Strichartz estimates, which turns out to be optimal for the Schrödinger equations associated with Laplacian and Hermite operator as a special case.

math.AP↗
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↗
arXiv2022

Szegö type limit theorems on the Heisenberg group

Let $\mathcal{H}=-Δ_{\mathbb{H}}+V$ be the Schrödinger operator on the Heisenberg group $\mathbb{H}^n$, where $Δ_{\mathbb{H}}$ is the full laplacian on $\mathbb{H}^n$ and $V$ is a positive smooth potential, bounded below and grows like $|g|^κ, κ>0$ for large $|g|$. Let $\mathcal{P}_{r}$ be the orthogonal projection of $L^2(\mathbb{H}^n)$ onto the space of eigenfunctions of $\mathcal{H}$ with eigenvalue $\leq r$; Let $A$ be a 0-th order self-adjoint pseudo-differential operator on $L^2(\mathbb{H}^n)$ relative to the operator $1+|λ|H+V(g), g\in \mathbb{H}^n, λ\in \mathbb{R}^*$ with symbol $a(g, λ),$ where $H$ is the Hermite operator on $L^2(\mathbb{R}^n)$ then \begin{align*} \lim_{r\to\infty} \frac{tr~{f(\mathcal{P}_rA\mathcal{P}_r)}}{tr~(\mathcal{P}_r)} &= \lim_{r\to\infty} \frac{\int_{G^{r}}f(a_{g, λ}(ξ, x)) \,dξ\,dx \,dg\,dμ(λ) }{\int_{G^{r}} \,dξ\,dx \,dg\,dμ(λ)}, \end{align*} (Assuming one limit exists) where $G^{r}=\{(g, λ, ξ, x)\in \mathbb{H}^n \times \mathbb{R}^*\times \mathbb{R}^n\times \mathbb{R}^n : |λ|(1+|ξ| ^2+|x|^2)+V(g)\leq r \}$, $a(g, λ)=Op^W(a_{g, λ})$, and $μ(λ)$ is the Plancherel measure on the Heisenberg group. Also we show that the above limit on the right hand side remains unaltered under a compact perturbation of the pseudo-differential operator $A$ or a perturbation of the Schrödinger operator $\mathcal{H}$ by bounded self-adjoint operators on $L^2(\mathbb{H}^n)$.

math.FA↗
arXiv2020

Hilbert space valued Gabor frames in weighted amalgam spaces

Let $\mathbb{H}$ be a separable Hilbert space. In this paper we establish a generalization of Walnut's representation and Janssen's representation of the $\mathbb{H}-$valued Gabor frame operator on $\mathbb{H}-$valued weighted amalgam spaces $W_{\mathbb{H}}(L^p,L^q_v)$, $1 \leq p, q \leq \infty$. Also we show that the frame operator is invertible on $W_{\mathbb{H}}(L^p,L^q_v)$, $1 \leq p, q \leq \infty$, if the window function is in the Wiener amalgam space $W_{\mathbb{H}}(L^{\infty},L^1_w)$. Further, we obtain the Walnut representation and invertibility of the frame operator corresponding to Gabor superframes and multi-window Gabor frames on $W_{\mathbb{H}}(L^p,L^q_v)$, $1 \leq p, q \leq \infty,$ as a special case by choosing the appropriate Hilbert space $\mathbb{H}$.

math.FA↗
arXiv2012

Szego limit theorem on the lattice

In this paper, we prove a Szegö type limit theorem on $\ell^2(\ZZ^d)$. We consider operators of the form $H=Δ+V$, $V$ multiplication by a positive sequence $\{V(n), n \in \ZZ^d\}$ with $V(n) \rightarrow \infty, |n| \rightarrow \infty $ on $\ell^2(\ZZ^d)$ and $π_λ$ the orthogonal projection of $\ell^2(\mathbb{Z}^d)$ on to the space of eigenfunctions of $H$ with eigenvalues $\leq λ$. We take $B$ to be a pseudo difference operator of order zero with symbol $b(x,n), (x,n) \in \TT^d\times \ZZ^d$ and show that for nice functions $f$ $$ \lim_{λ\rightarrow \infty} Tr(f(π_λBπ_λ))/Tr(π_λ) = \lim_{λ\rightarrow \infty} \frac{1}{(2π)^d} \frac{\sum_{V(n) \leq λ} \int_{\TT^d} f(b(x,n)) ~ dx}{\sum_{V(n)\leqλ} 1}. $$

math-ph↗