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Medet Nursultanov

Publications and source records attributed to Medet Nursultanov.

At least 19 recordsLinked to original sources
arXiv2026

Inverse Scattering for Single Photons in Quantum Optics

We study inverse problems for a time-harmonic one-photon model describing the interaction of a single photon with a medium of stationary two-level atoms. After time-harmonic reduction, the unknown compactly supported atomic density appears as a frequency-dependent potential in a scattering equation for the half Laplacian. We prove high-frequency uniqueness results for three types of intensity data: source-driven measurements, renormalized far-field intensity measurements, and phaseless far-field measurements obtained from coherent superpositions of incident plane waves. In each case, the corresponding data, given at all sufficiently large frequencies, determine the atomic density uniquely; the source-driven result requires a geometric visibility condition on the source and observation sets.

math.AP↗
arXiv2026

Negative eigenvalue estimates for polyharmonic Schrödinger operators with measure-potentials: the subcritical case

We study spectral estimates for polyharmonic Schrödinger operators $-Δ^l-μ$ in the subcritical regime $2l<\mathbf{N}$. The measure potential $μ$ is assumed to satisfy a capacitary smallness condition which guarantees that the corresponding operator is semibounded and self-adjoint. With such a measure $μ$ we associate an Otelbaev function, which reflects both the local concentration and the spatial distribution of the potential. In terms of this function, we obtain two-sided estimates for the distribution function of the negative eigenvalues, and derive a sufficient condition and a necessary condition for the discreteness of the negative spectrum. As an application, we establish two-sided estimates of Lieb-Thirring-type, improving the classical ones.

math.SP↗
arXiv2026

Introduction to inverse problems for hyperbolic PDEs

There are two main approaches to solve inverse coefficient determination problems for wave equations: the Boundary Control method and an approach based on geometric optics. These notes focus on the Boundary Control method, but we will have a brief look at the geometric optics as well.

math.AP↗
arXiv2026

An inverse problem for semilinear wave equations on metric tree graphs

We study the inverse problem for a semilinear wave equation on metric tree graphs. From the Dirichlet-to-Neumann map defined at all but one of the boundary vertices, we recover unknown connectivity of the graph, lengths of the edges, the time-independent potential and the time-dependent coefficient of the nonlinear term of the equation.

math.AP↗
arXiv2026

Quantum field theory and inverse problems: Imaging with Entangled Photons

We consider the quantum field theory for a scalar model of the electromagnetic field interacting with a system of two-level atoms. In this setting, we show that it is possible to uniquely determine the density of atoms from measurements of the source to solution map for a system of nonlocal partial differential equations, which describe the scattering of a two-photon state from the atoms. The required measurements involve correlating the outputs of a point detector with an integrating detector, thereby exploiting information about the entanglement of the photons.

math.AP↗
arXiv2024

Negative eigenvalue estimates for the 1D Schr{ö}dinger operator with measure-potential

We investigate the negative part of the spectrum of the operator $-\partial^2 - μ$ on $L^2(\mathbb R)$, where a locally finite Radon measure $μ\geq 0$ is serving as a potential. We obtain estimates for the eigenvalue counting function, for individual eigenvalues and estimates of the Lieb-Thirring type. A crucial tool for our estimates is Otelbaev's function, a certain average of the measure potential $μ$, which is used both in the proofs and the formulation of most of the results.

math.SP↗
arXiv2023

The heat kernel on curvilinear polygonal domains in surfaces

We construct the heat kernel on curvilinear polygonal domains in arbitrary surfaces for Dirichlet, Neumann, and Robin boundary conditions as well as mixed problems, including those of Zaremba type. We compute the short time asymptotic expansion of the heat trace and apply this expansion to demonstrate a collection of results showing that corners are spectral invariants.

math.AP↗
arXiv2023

Eigenvalue Variations of the Neumann Laplace Operator Due to Perturbed Boundary Conditions

This work considers the Neumann eigenvalue problem for the weighted Laplacian on a Riemannian manifold $(M,g,\partial M)$ under the singular perturbation. This perturbation involves the imposition of vanishing Dirichlet boundary conditions on a small portion of the boundary. We derive a sharp asymptotic of the perturbed eigenvalues, as the Dirichlet part shrinks to a point $x^*\in \partial M$, in terms of the spectral parameters of the unperturbed system. This asymptotic demonstrates the impact of the geometric properties of the manifold at a specific point $x^*$. Furthermore, it becomes evident that the shape of the Dirichlet region holds significance as it impacts the first terms of the asymptotic. A crucial part of this work is the construction of the singularity structure of the restricted Neumann Green's function which may be of independent interest. We employ a fusion of layer potential techniques and pseudo-differential operators during this work.

math.AP↗
arXiv2023

Disjoint data inverse problem on manifolds with quantum chaos bounds

We consider the inverse problem to determine a smooth compact Riemannian manifold $(M,g)$ from a restriction of the source-to-solution operator, $Λ_{\mathcal{S,R}}$, for the wave equation on the manifold. Here, $\mathcal{S}$ and $\mathcal{R}$ are open sets on $M$, and $Λ_{\mathcal{S,R}}$ represents the measurements of waves produced by smooth sources supported on $\mathcal{S}$ and observed on $\mathcal{R}$. We emphasise that $\overline{\mathcal{S}}$ and $\overline{\mathcal{R}}$ could be disjoint. We demonstrate that $Λ_{\mathcal{S,R}}$ determines the manifold $(M,g)$ uniquely under the following spectral bound condition for the set $\mathcal{S}$: There exists a constant $C>0$ such that any normalized eigenfunction $ϕ_k$ of the Laplace-Beltrami operator on $(M,g)$ satisfies \begin{equation*} 1\leq C\|ϕ_k\|_{L^2(\mathcal{S})}. \end{equation*} We note that, for the Anosov surface, this spectral bound condition is fulfilled for any non-empty open subset $\mathcal{S}$. Our approach is based on the paper [18] and the spectral bound condition above is an analogue of the Hassell-Tao condition there.

math.AP↗
arXiv2022

$L_p\rightarrow L_q$ boundedness of Fourier multipliers

We investigate the $L_p \mapsto L_q$ boundedness of the Fourier multipliers. We obtain sufficient conditions, namely, we derive Hormander and Lizorkin type theorems. We also obtain the necessary conditions. For $M$-generalized monotone functions, we obtain a criteria for boundedness of the corresponding Fourier multipliers.

math.FA↗
arXiv2022

The narrow capture problem on general Riemannian surfaces

In this article, we study the narrow capture problem on a Riemannian 2-manifold. This involves the derivation of the mean first passage (sojourn) time of a surface-bound ion modelled as a Brownian particle. We use a layer potential argument in conjunction with microlocal analysis in order to derive the leading order singularity as well as the O(1) term of the mean first passage time and the associated spatial average.

math.PR↗
arXiv2021

The strength of diversity: mathematical proof that collections of variable individuals are robust in the face of challenges

Can one demonstrate quantitative effects of diversity within a system comprised of distinct individuals on the performance of the system as a whole? Assuming that individuals can be different, we develop a model to interpolate between individual-level interactions and collective-level ramifications. Rooted in theoretical mathematics, the model is not constrained to any specific context. Potential applications include research, education, sports, politics, ecology, agriculture, algorithms, and finance. Our first main contribution is a game theoretic framework for further analysis of the internal composition of an ensemble of individuals and the repercussions for the ensemble as a whole in competition with others. The second main contribution is the rigorous identification of all equilibrium points and strategies. These equilibria suggest a mechanistic underpinning for biological and physical systems to tend towards increasing complexity and entropy, because diversity imparts strength to a system in competition with others.

physics.soc-ph↗
arXiv2021

Narrow escape problem in the presence of the force field

This paper considers the narrow escape problem of a Brownian particle within a three-dimensional Riemannian manifold under the influence of the force field. We compute an asymptotic expansion of mean sojourn time for Brownian particles. As an auxiliary result, we obtain the singular structure for the restricted Neumann Green's function which may be of independent interest.

math.PR↗
arXiv2021

Spectral Theory for Sturm-Liouville operators with measure potentials through Otelbaev's function

We investigate the spectral properties of Sturm-Liouville operators with measure potentials. We obtain two-sided estimates for the spectral distribution function of the eigenvalues. As a corollary, we derive a criterion for the discreteness of the spectrum and a criterion for the membership of the resolvents to Schatten classes. We give two side estimates for the lower bound of the essential spectrum. Our main tool in achieving this is Otelbaev's function.

math.FA↗
arXiv2021

On the Mean First Arrival Time of Brownian Particles on Riemannian Manifolds

We use geometric microlocal methods to compute an asymptotic expansion of mean first arrival time for Brownian particles on Riemannian manifolds. This approach provides a robust way to treat this problem, which has thus far been limited to very special geometries. This paper can be seen as the Riemannian 3-manifold version of the planar result of \cite{ammari} and thus enable us to see the full effect of the local extrinsic boundary geometry on the mean arrival time of the Brownian particles. Our approach also connects this question to some of the recent progress on boundary rigidity and integral geometry [23,20].

math.PR↗
arXiv2021

Biodiversity of marine microbes is safeguarded by phenotypic variability in ecological traits

Why, contrary to theoretical predictions, do marine microbe communities harbor tremendous phenotypic heterogeneity? How can so many marine microbe species competing in the same niche coexist? We discovered a unifying explanation for both phenomena by investigating a non-cooperative game that interpolates between individual-level competitions and species-level outcomes. We identified all equilibrium strategies of the game. These strategies are characterized by maximal phenotypic heterogeneity. They are also neutral towards each other in the sense that an unlimited number of species can co-exist while competing according to the equilibrium strategies. Whereas prior theory predicts that natural selection would minimize trait variation around an optimum value, here we obtained a rigorous mathematical proof that species with maximally variable traits are those that endure. This discrepancy may reflect a disparity between predictions from models developed for larger organisms in contrast to our microbe-centric model. Rigorous mathematics proves that phenotypic heterogeneity is itself a mechanistic underpinning of microbial diversity. This discovery has fundamental ramifications for microbial ecology and may represent an adaptive reservoir sheltering biodiversity in changing environmental conditions.

q-bio.PE↗