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Jayanta Sarkar

Publications and source records attributed to Jayanta Sarkar.

At least 19 recordsLinked to original sources
arXiv2025

$L^p$- Heisenberg--Pauli--Weyl uncertainty inequalities on certain two-step nilpotent Lie groups

This article presents the $L^p$-Heisenberg--Pauli--Weyl uncertainty inequality for the group Fourier transform on a class of two-step nilpotent Lie groups, specifically the M\'etivier groups. This inequality quantitatively demonstrates that on M\'etivier groups, a nonzero function and its group Fourier transform cannot both be sharply localized. The proof primarily relies on utilizing the dilation structure inherent to two-step nilpotent Lie groups and estimating the Schatten class norms of the group Fourier transform. The inequality we establish is new, even in the simplest case of Heisenberg groups. Our result significantly sharpens all previously known $L^p$-Heisenberg--Pauli--Weyl uncertainty inequalities for $1 \leq p < 2$ on M\'etivier groups.

math.FA↗
arXiv2024

$L^p$-asymptotic behaviour of solutions of the heat equation on Riemannian symmetric spaces of noncompact type

For Riemannian symmetric spaces $X=G/K$ of noncompact type, we show that for all left $K$-invariant $f\in L^1(X)$, the functions $\|h_t\|_{L^p(X)}^{-1}(f\ast h_t-M_p(f)h_t)$ (with $h_t$ being the heat kernel of $X$) converges to zero in $L^p(X)$, $p\in [1,\infty]$, as $t\to\infty$, with the constant $M_p(f)$ depending only on $p$ and $f$. We also prove an analogous result for the fractional heat kernels $h_t^{\alpha}$, $\alpha\in (0,1)$. The above results have recently been proved for the important special cases $p=1$ and $\alpha= 1,\frac 12$.

math.CA↗
arXiv2023

Unique Continuation Inequalities for the Schrödinger equations associated with the Special Hermite operators

We investigate unique continuation inequalities for solutions of the Schrödinger equations associated with special Hermite operators. Our main result establishes that if the solution remains small at two distinct time points outside sets of finite measure, then the solution also remains small throughout the entire space. We also explore analogous results for the Hermite-Schrödinger equations.

math.CA↗
arXiv2023

Weighted estimates for Hardy-Littlewood maximal functions on Harmonic $NA$ groups

Our aim in this article is to study the weighted boundedness of the centered Hardy-Littlewood maximal operator in Harmonic $NA$ groups. Following Ombrosi et al. \cite{ORR}, we define a suitable notion of $A_p$ weights, and for such weights, we prove the weighted $L^p$-boundedness of the maximal operator. Furthermore, as an endpoint case, we prove a variant of the Fefferman-Stein inequality, from which vector-valued maximal inequality has been established. We also provide various examples of weights to substantiate many aspects of our results. In particular, we have shown certain spherical functions of the Harmonic $NA$ group constitute examples of $A_p$ weights. The purely exponential volume growth property of the Harmonic $NA$ group has played a crucial role in our proofs.

math.CA↗
arXiv2023

Generalized Hausdorff metric on $S_{b}$-metric space and some fixed point results

In this paper, a metric on $S_b$-metric space analogous to the Hausdorff metric has been introduced and some basic properties are obtained on multi-valued $S_b$-metric space. Further, the fundamental multi-valued contraction of Nadler(1962) has been extended to the $S_b$-metric space setting, and two results have been established. The entire study is supported by suitable examples.

math.FA↗
arXiv2022

An analogue of Ingham's theorem on the Heisenberg group

We prove an exact analogue of Ingham's uncertainty principle for the group Fourier transform on the Heisenberg group. This is accomplished by explicitly constructing compactly supported functions on the Heisenberg group whose operator-valued Fourier transforms have suitable Ingham type decay and proving an analogue of Chernoff's theorem for the family of special Hermite operators.

math.CA↗
arXiv2022

Fatou theorem and its converse for positive eigenfunctions of the Laplace-Beltrami operator on Harmonic $NA$ groups

We prove a Fatou-type theorem and its converse for certain positive eigenfunctions of the Laplace-Beltrami operator $\mathcal{L}$ on a Harmonic $NA$ group. We show that a positive eigenfunction $u$ of $\mathcal{L}$ with eigenvalue $β^2-ρ^2$, $β\in (0,\infty)$, has an admissible limit in the sense of Korányi, precisely at those boundary points where the strong derivative of the boundary measure of $u$ exists. Moreover, the admissible limit and the strong derivative are the same. This extends a result of Ramey and Ullrich regarding nontangential convergence of positive harmonic functions on the Euclidean upper half space.

math.CA↗
arXiv2022

Boundary behavior of positive solutions of the heat equation on a stratified Lie group

In this article, we are concerned with a certain type of boundary behavior of positive solutions of the heat equation on a stratified Lie group at a given boundary point. We prove that a necessary and sufficient condition for the existence of the parabolic limit of a positive solution $u$ at a point on the boundary is the existence of the strong derivative of the boundary measure of $u$ at that point. Moreover, the parabolic limit and the strong derivative are equal. We also construct an example of a positive measure on the Heisenberg group to show that the set of all points where strong derivative exists is strictly larger than the set of Lebesgue points of the measure.

math.AP↗
arXiv2021

Suppression of $1/f$ noise in graphene due to non-scalar mobility fluctuations induced by impurity motion

Low frequency resistance variations due to mobility fluctuations is one of the key factors of $1/f$ noise in metallic conductors. According to theory, such noise in a two-dimensional (2D) device can be suppressed to zero at small magnetic fields, implying important technological benefits for low noise 2D devices. In this work, we provide direct evidence of anisotropic mobility fluctuations by demonstrating a strong field-induced suppression of noise in a high-mobility graphene Corbino disk, even though the device displays only a tiny amount of $1/f$ noise inherently. The suppression of the $1/f$ noise depends on charge density, showing less non-uniform mobility fluctuations away from the Dirac point with charge puddles. We model our results using a new approach based on impurity clustering dynamics and find our results consistent with the $1/f$ noise induced by scattering of carriers on mobile impurities forming clusters.

cond-mat.mes-hall↗
arXiv2021

Electrical low-frequency $1/f^γ$ noise due to surface diffusion of scatterers on an ultra low noise graphene platform

Low-frequency $1/f^γ$ noise is ubiquitous, even in high-end electronic devices. For qubits such noise results in decrease of their coherence times. Recently, it was found that adsorbed O$_2$ molecules provide the dominant contribution to flux noise in superconducting quantum interference devices. To clarify the basic principles of such adsorbant noise, we have investigated the formation of low-frequency noise while the mobility of surface adsorbants is varied by temperature. In our experiments, we measured low-frequency current noise in suspended monolayer graphene samples under the influence of adsorbed Ne atoms. Owing to the extremely small intrinsic noise of graphene in suspended Corbino geometry, we could resolve a combination of $1/f^γ$ and Lorentzian noise spectra induced by the presence of Ne. We find that the $1/f^γ$ noise is caused by surface diffusion of Ne atoms and by temporary formation of few-Ne-atom clusters. Our results support the idea that clustering dynamics of defects is relevant for understanding of $1/f$ noise in general metallic systems.

cond-mat.mes-hall↗
arXiv2021

Strong magnetoresistance in a graphene Corbino disk at low magnetic fields

We have measured magnetoresistance of suspended graphene in the Corbino geometry at magnetic fields up to $B=0.15\,$T, i.e., in a regime uninfluenced by Shubnikov-de Haas oscillations. The low-temperature relative magnetotoresistance $[R(B)-R(0)]/R(0)$ amounts to $4000 B^2\% $ at the Dirac point ($B$ in Tesla), with a quite weak temperature dependence below $30\,$K. A decrease in the relative magnetoresistance by a factor of two is found when charge carrier density is increased to $|n| \simeq 3 \times 10^{-10}$ cm$^{-2}$. The gate dependence of the magnetoresistance allows us to characterize the role of scattering on long-range (Coulomb impurities, ripples) and short-range potential, as well as to separate the bulk resistance from the contact one. Furthermore, we find a shift in the position of the charge neutrality point with increasing magnetic field, which suggests that magnetic field changes the screening of Coulomb impurities around the Dirac point. The current noise of our device amounts to $10^{-23}$ A$^2$/$\sqrt{\textrm{Hz}}$ at $1\,$kHz at $4\,$K, which corresponds to a magnetic field sensitivity of $60$ nT/$\sqrt{\textrm{Hz}}$ in a background field of $0.15\,$T.

cond-mat.mes-hall↗
arXiv2021

A note on $σ$-point and nontangential convergence

In this article, we generalize a theorem of Victor L. Shapiro concerning nontangential convergence of the Poisson integral of a $L^p$-function. We introduce the notion of $σ$-points of a locally finite measure and consider a wide class of convolution kernels. We show that convolution integrals of a measure have nontangential limits at $σ$-points of the measure. We also investigate the relationship between $σ$-point and the notion of the strong derivative introduced by Ramey and Ullrich. In one dimension, these two notions are the same.

math.CA↗
arXiv2020

On parabolic convergence of positive solutions of the heat equation

In this article, we study certain type of boundary behaviour of positive solutions of the heat equation on the upper half-space of $\R^{n+1}$. We prove that the existence of the parabolic limit of a positive solution of the heat equation at a point in the boundary is equivalent to the existence of the strong derivative of the boundary measure of the solution at that point. Moreover, the parabolic limit and the strong derivative are equal.

math.CA↗
arXiv2020

On Pointwise converse of Fatou's theorem for Euclidean and Real hyperbolic spaces

In this article, we extend a result of L. Loomis and W. Rudin, regarding boundary behavior of positive harmonic functions on the upper half space $\R_+^{n+1}$. We show that similar results remain valid for more general approximate identities. We apply this result to prove a result regarding boundary behavior of nonnegative eigenfunctions of the Laplace-Beltrami operator on real hyperbolic space $\mathbb H^n$. We shall also prove a generalization of a result regarding large time behavior of solution of the heat equation proved in \cite{Re}. We use this result to prove a result regarding asymptotic behavior of certain eigenfunctions of the Laplace-Beltrami operator on real hyperbolic space $\mathbb H^n$.

math.CA↗
arXiv2019

Sputtered MoRe SQUID-on-tip for high-field magnetic and thermal nanoimaging

Scanning nanoscale superconducting quantum interference devices (SQUIDs) are gaining interest as highly sensitive microscopic magnetic and thermal characterization tools of quantum and topological states of matter and devices. Here we introduce a novel technique of collimated differential-pressure magnetron sputtering for versatile self aligned fabrication of SQUID on tip (SOT) nanodevices, which cannot be produced by conventional sputtering methods due to their diffusive, rather than the required directional point-source, deposition. The new technique provides access to a broad range of superconducting materials and alloys beyond the elemental superconductors employed in the existing thermal deposition methods, opening the route to greatly enhanced SOT characteristics and functionalities. Utilizing this method, we have developed MoRe SOT devices with sub-50 nm diameter, magnetic flux sensitivity of 1.2 $μΦ_0/Hz^{1/2}$ up to 3 T at 4.2 K, and thermal sensitivity better than 4 $μK/Hz^{1/2}$ up to 5 T, about five times higher than any previous report, paving the way to nanoscale imaging of magnetic and spintronic phenomena and of dissipation mechanisms in previously inaccessible quantum states of matter.

cond-mat.mes-hall↗
arXiv2017

Hanbury-Brown and Twiss exchange effects in a four-terminal tunnel junction

We investigate the current-current correlations in a four-terminal Al-AlOx-Al tunnel junction where shot noise dominates. We demonstrate that cross-correlations in the presence of two biasing sources of the Hanbury-Brown and Twiss type are much stronger (approximately twice) than an incoherent sum of correlations generated by single sources. The difference is due to voltage fluctuations of the central island that give rise to current-current correlations in the four contacts of the junction. Our measurements are in close agreement with results obtained using a simple theoretical model based on the theory of shot noise in multi-terminal conductors, generalized here to arbitrary contacts.

cond-mat.mes-hall↗
arXiv2016

Nanoscale thermal imaging of dissipation in quantum systems

Energy dissipation is a fundamental process governing the dynamics of physical, chemical, and biological systems. It is also one of the main characteristics distinguishing quantum and classical phenomena. In condensed matter physics, in particular, scattering mechanisms, loss of quantum information, or breakdown of topological protection are deeply rooted in the intricate details of how and where the dissipation occurs. Despite its vital importance the microscopic behavior of a system is usually not formulated in terms of dissipation because the latter is not a readily measureable quantity on the microscale. Although nanoscale thermometry is gaining much recent interest, the existing thermal imaging methods lack the necessary sensitivity and are unsuitable for low temperature operation required for study of quantum systems. Here we report a superconducting quantum interference nano-thermometer device with sub 50 nm diameter that resides at the apex of a sharp pipette and provides scanning cryogenic thermal sensing with four orders of magnitude improved thermal sensitivity of below 1 μK/Hz1/2. The non-contact non-invasive thermometry allows thermal imaging of very low nanoscale energy dissipation down to the fundamental Landauer limit of 40 fW for continuous readout of a single qubit at 1 GHz at 4.2 K. These advances enable observation of dissipation due to single electron charging of individual quantum dots in carbon nanotubes and reveal a novel dissipation mechanism due to resonant localized states in hBN encapsulated graphene, opening the door to direct imaging of nanoscale dissipation processes in quantum matter.

cond-mat.mes-hall↗