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Alexander Nazarov

Publications and source records attributed to Alexander Nazarov.

17 recordsLinked to original sources
arXiv2026

Functional a posteriori estimates for the fractional Laplacian problem

The paper is concerned with a posteriori estimates for approximations of boundary value problems generated by the spectral fractional Laplace operator. The derivation is based upon the Stinga--Torrea extension, which generalizes the Caffarelli--Silvestre extension and transfers the corresponding nonlocal problem in a bounded domain to a local problem of higher dimensionality. A posteriori estimates are first derived for this local problem. Two-sided error bounds for the original problem follow from them. The estimates are fully computable and contain no conditions and constants depending on a method or mesh used to compute an approximation. They are valid for any energy admissible approximation of the extended problem.

math.AP↗
arXiv2025

Modulation transfer spectroscopy of $^7$Li D1 line

We present the first implementation of modulation transfer spectroscopy (MTS) on the D$1$ line of $^7$Li, carried out in a compact heat-pipe vapor cell with cold windows. By varying the pump and probe intensities and polarization configurations, we systematically map the MTS error-signal amplitude, effective linewidth, and slope for the ground-state crossover resonance $F=1\times2\rightarrow F'=2$. We observe Rabi-induced power broadening and identify optimal conditions for laser frequency stabilization via tightly focused beams with total power below 1 mW. The lin$\perp$lin polarization configuration yields a sharp, symmetric, high-contrast error signal with a maximal slope. Our findings establish the MTS spectrum of the $^7$Li D1 line transitions as a reliable frequency reference for quantum technology and ultracold atom experiments, particularly in scenarios where frequency stabilization must be achieved with low laser optical power.

physics.atom-ph↗
arXiv2024

Ultrafast and highly collimated radially polarized photons from a colloidal quantum dot in a hybrid nanoantenna at room-temperature

To harness the potential of radially polarized single photons in applications such as high-dimensional quantum key distribution (HD-QKD) and quantum communication, we demonstrate an on-chip, room-temperature device, which generates highly directional radially polarized photons at very high rates. The photons are emitted from a giant CdSe/CdS colloidal quantum dot (gQD) accurately positioned at the tip of a metal nanocone centered inside a hybrid metal-dielectric bullseye antenna. We show that due to the large and selective Purcell enhancement specifically for the out-of-plane optical dipole of the gQD, the emitted photons can have a very high degree of radial polarization (>93%), based on a quantitative metric. Our study emphasizes the importance of accurate gQD positioning for optimal radial polarization purity through extensive experiments and simulations, which contribute to the fundamental understanding of radial polarization in nanostructured devices and pave the way for implementation of such systems in practical applications using structured quantum light.

physics.optics↗
arXiv2023

Room Temperature Fiber-Coupled single-photon devices based on Colloidal Quantum Dots and SiV centers in Back Excited Nanoantennas

We demonstrate an important step towards on chip integration of single photon sources operating at room temperature fiber coupling of a directional quantum emitter with back-excitation. Directionality is achieved with a hybrid metal-dielectric bullseye antenna, while back-excitation is permitted by placement of the emitter at or in a sub-wavelength hole positioned at the bullseye center. Overall, the unique design enables a direct laser excitation from the back of the on-chip device and very efficient coupling of the highly collimated photon emission to either low numerical aperture (NA) free space optics or directly to an optical fiber from the front. To show the versatility of the concept, we fabricate devices containing either a colloidal quantum dot or a silicon-vacancy center containing nanodiamond, which are accurately coupled to the nano-antenna using two different nano-positioning methods. Both back-excited devices display front collection efficiencies of about 70 % at NAs as low as 0.5. Moreover, the combination of back-excitation with forward low-NA directionality enables direct coupling of the emitted photons into a proximal optical fiber without the need of any coupling optics, thereby facilitating and greatly simplifying future integration.

physics.optics↗
arXiv2019

A comparison theorem for nonsmooth nonlinear operators

We prove a comparison theorem for super- and sub-solutions with non-vanishing gradients to semilinear PDEs provided a nonlinearity $f$ is $L^p$ function with $p > 1$. The proof is based on a strong maximum principle for solutions of divergence type elliptic equations with VMO leading coefficients and with lower order coefficients from a Kato class. An application to estimation of periodic water waves profiles is given.

math.AP↗
arXiv2018

$L_2$-Small Deviations for Weighted Stationary Processes

We find logarithmic asymptotics of $L_2$-small deviation probabilities for weighted stationary Gaussian processes (both for real and complex-valued) having power-type discrete or continuous spectrum. As in the recent work by Hong, Lifshits and Nazarov, our results are based on the spectral theory of pseudo-differential operators developed by Birman and Solomyak.

math.PR↗
arXiv2016

Stability by linear approximation for time scale dynamical systems

We study systems on time scales that are generalizations of classical differential or difference equations. In this paper we consider linear systems and their small nonlinear perturbations. In terms of time scales and of eigenvalues of matrices we formulate conditions, sufficient for stability by linear approximation. We demonstrate that those conditions are close to necessary ones. For non-periodic time scales we use techniques of central upper Lyapunov exponents (a common tool of the theory of linear ODEs) to study stability of solutions. Also, time scale versions of the famous Chetaev's theorem on conditional instability are proved.

math.DS↗
arXiv2016

Small Deviations in $L_2$-norm for Gaussian Dependent Sequences

Let $U=(U_k)_{k\in\mathbb{Z}}$ be a centered Gaussian stationary sequence satisfying some minor regularity condition. We study the asymptotic behavior of its weighted $\ell_2$-norm small deviation probabilities. It is shown that \[ \ln \mathbb{P}\left( \sum_{k\in\mathbb{Z}} d_k^2 U_k^2 \leq \varepsilon^2\right) \sim - M \varepsilon^{-\frac{2}{2p-1}}, \qquad \textrm{ as } \varepsilon\to 0, \] whenever \[ d_k\sim d_{\pm} |k|^{-p}\quad \textrm{for some } p>\frac{1}{2} \, , \quad k\to \pm\infty, \] using the arguments based on the spectral theory of pseudo-differential operators by M. Birman and M. Solomyak. The constant $M$ reflects the dependence structure of $U$ in a non-trivial way, and marks the difference with the well-studied case of the i.i.d. sequences.

math.PR↗
arXiv2011

The Dirichlet problem for non-divergence parabolic equations with discontinuous in time coefficients in a wedge

We consider the Dirichlet problem in a wedge for parabolic equation whose coefficients are measurable function of t. We obtain coercive estimates in weighted $L_{p,q}$-spaces. The concept of "critical exponent" introduced in the paper plays here the crucial role. Various important properties of the critical exponent are proved. We give applications to the Dirichlet problem for linear and quasi-linear non-divergence parabolic equations with discontinuous in time coefficients in cylinders $Ω\times(0,T)$, where $Ω$ is a bounded domain with an edge or with a conical point.

math.AP↗
arXiv2010

An extremal problem with applications to testing multivariate independence

Some problems of statistics can be reduced to extremal problems of minimizing functionals of smooth functions defined on the cube $[0,1]^m$, $m\geq 2$. In this paper, we study a class of extremal problems that is closely connected to the problem of testing multivariate independence. By solving the extremal problem, we provide a unified approach to establishing weak convergence for a wide class of empirical processes which emerge in connection with testing independence. The use of our result is also illustrated by describing the domain of local asymptotic optimality of some nonparametric tests of independence.

math.PR↗
arXiv2009

On asymptotic efficiency of multivariate version of Spearman's rho

A multivariate version of Spearman's rho for testing independence is considered. Its asymptotic efficiency is calculated under a general distribution model specified by the dependence function. The efficiency comparison study that involves other multivariate Spearman-type test statistics is made. Conditions for Pitman optimality of the test are established. Examples that illustrate the quality of the multivariate Spearman's test are included.

math.PR↗