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Stanislav Molchanov

Publications and source records attributed to Stanislav Molchanov.

At least 19 recordsLinked to original sources
arXiv2026

On the Emergence of Discrete Spectrum for Weakly Disordered Schr\"odinger Operators

We investigate the spectral properties of the Anderson operator perturbed by a localized negative potential, \(-V\). Specifically, we analyze the random Schr\"odinger operator defined by \(H = -\Delta +\ve \sum_{n} \omega_n \chi_n - V\), where the unperturbed operator exhibits a disordered energy landscape. Our primary focus is to establish precise estimates on the number of negative eigenvalues (bound states) induced by the attractive perturbation. By analyzing the competition between Anderson localization and the binding capacity of the potential, we provide quantitative bounds on the discrete spectrum. These results offer new insights into how randomness enhances the eigenvalue bounds.

math-ph↗
arXiv2024

On the analytic extension of Random Riemann Zeta Functions for some probabilistic models of the primes

The first step in the formulation and study of the Riemann Hypothesis is the analytic continuation of the Riemann Zeta Function (RZF) in the full Complex Plane with a pole at $s=1$. In the current work, we study the analytic continuation of two random versions of RZF using, for $Re s>1$, the Euler representation of ZF in terms of the product of functions over primes. In the first case, we substitute in the Euler product pseudo-prime numbers from the famous Cramér Model. In the second case, we use pseudo-primes with local symmetries. We show that in the Cramér case analytic continuation is possible $\mathbb{P}$-a.s. for $Res>1/2$, but not through the critical line $Re s=1/2.$ In the second case, we show that the analytic continuation is possible in a larger domain. We also study for the Cramér pseudo-primes several problems from Additive Number Theory.

math.PR↗
arXiv2024

Brownian Motion on The Spider Like Quantum Graphs

The paper contains the probabilistic analysis of the Brownian motion on the simplest quantum graph, spider: a system of N-half axis connected only at the graph's origin by the simplest (so-called Kirchhoff's) gluing conditions. The limit theorems for the diffusion on such a graph, especially if $N \to \infty$ are significantly different from the classical case $N = 2$ (full axis). Additional results concern the properties of the spectral measure of the spider Laplacian and the corresponding generalized Fourier transforms. The continuation of the paper will contain the study of the spectrum for the class of Schrödinger operators on the spider graphs: Laplacian perturbed by unbounded potential and related phase transitions.

math-ph↗
arXiv2024

Spectral Analysis of Lattice Schrödinger-Type Operators Associated with the Nonstationary Anderson Model and Intermittency

The research explores a high irregularity, commonly referred to as intermittency, of the solution to the non-stationary parabolic Anderson problem: \begin{equation*} \frac{\partial u}{\partial t} = \varkappa \mathcal{L}u(t,x) + ξ_{t}(x)u(t,x) \end{equation*} with the initial condition \(u(0,x) \equiv 1\), where \((t,x) \in [0,\infty)\times \mathbb{Z}^d\). Here, \(\varkappa \mathcal{L}\) denotes a non-local Laplacian, and \(ξ_{t}(x)\) is a correlated white noise potential. The observed irregularity is intricately linked to the upper part of the spectrum of the multiparticle Schrödinger equations for the moment functions \(m_p(t,x_1,x_2,\cdots,x_p) = \langle u(t,x_1)u(t,x_2)\cdots u(t,x_p)\rangle\). In the first half of the paper, a weak form of intermittency is expressed through moment functions of order $p\geq 3$ and established for a wide class of operators $\varkappa \mathcal{L}$ with a positive-definite correlator $B=B(x))$ of the white noise. In the second half of the paper, the strong intermittency is studied. It relates to the existence of a positive eigenvalue for the lattice Schrödinger type operator with the potential $B$. This operator is associated with the second moment $m_2$. Now $B$ is not necessarily positive-definite, but $\sum B(x)\geq 0$.

math-ph↗
arXiv2023

Branching Random Walks in a Random Killing Environment with a Single Reproduction Source

We consider a continuous-time branching random walk on $\mathbb{Z}$ in a random non homogeneous environment. Particles can walk on the lattice points or disappear with random intensities. The process starts with one particle at initial time $t=0$. It can walk on the lattice points or disappear with a random intensity until it reach the point, where initial particle can split into two offspring. This lattice point we call reproduction source. The offspring of the initial particle evolve according to the same law, independently of each other and the entire prehistory. The aim of the paper is to study the conditions for the presence of exponential growth of the average number of particle at an every lattice point. For this purpose we investigate the spectrum of the random evolution operator of the average particle numbers. We derive the condition under which there is exponential growth with probability one. We also study the process under the violation of this condition and present the lower and upper estimates for the probability of exponential growth.

math.PR↗
arXiv2023

Non-stationary Lattice Anderson Model with Non-local Laplacian and Correlated White Noise

We study the non-stationary Anderson parabolic problem on the lattice $Z^d$, i.e., the equation \begin{equation}\label{andersonmodel} \begin{aligned} \frac{\partial u}{\partial t} &=\varkappa \mathcal{A}u(t,x)+ξ_{t}(x)u(t,x) u(0,x) &\equiv 1, \, (t,x) \in [0,\infty)\times Z^d. \end{aligned} \end{equation} Here $\mathcal{A}$ is non-local Laplacian, $ξ_t (x), \ t \geq 0, \ x \in Z^d$ is the family of the correlated white noises and $\varkappa >0$ is the diffusion coefficient. The changes of $\varkappa$ (large versus small) are responsible for the qualitative phase transition in the model. At the first step the analysis of the model is reduced to the solution of the stochastic differential equation(SDE) (in the standard Itô's form) on the weighted Hilbert space $l^2(Z^d,μ)$ with appropriate measure $μ$. The equations of first two moments of the solution $u(t,x)$ are derived and studied using the spectral analysis of the corresponding Schrödinger operators with special class of the positive definite potentials. The analysis reveals several bifurcations depending on the properties of the kernel of $\mathcal{A}$ and the correlation function in the potential.

math.PR↗
arXiv2020

One Explicitly Solvable Model For The Galton-Watson Processes In the Random Environment

In this paper, we study the Galton-Watson process in the random environment for the particular case when the number of the offsprings in each generation has the fractional linear generation function with random parameters. In this case, the distribution of $N_t$, the number of particles at the moment time $t=0,1,2,\cdots$ can be calculated explicitly. We present the classification of such processes and limit theorems of two types: quenched type which is for the fixed realization of the random environment and annealed type which includes the averaging over the environment.

math.PR↗
arXiv2020

Hierarchical Schrödinger-type operators: the case of potentials with local singularities

The goal of this paper is twofold. We prove that the operator $H=L+V$ , a perturbation of the Taibleson-Vladimirov multiplier $L=\mathfrak{D}^α$ by a potential $V(x)=b\left\Vert x\right\Vert ^{-α},$ $b\geq b_{\ast},$ is essentially self-adjoint and non-negative definite (the critical value $b_{\ast}$ depends on $α$ and will be specified later). While the operator $H$ is non-negative definite the potential $V(x)$ may well take negative values, e.g. $b_{\ast}<0$ for all $0<α<1$. The equation $Hu=v$ admiits a Green function $g_{H}(x,y)$, the integral kernel of the operator $H^{-1}$. We obtain sharp lower- and upper bounds on the ratio of the functions $g_{H}(x,y)$ and $g_{L}(x,y)$. Examples illustrate our exposition.

math.SP↗
arXiv2020

On the spectrum of the hierarchical Schrödinger type operators

The goal of this paper is the spectral analysis of the Schrödinger type operator $H=L+V$, the perturbation of the Taibleson-Vladimirov multiplier $L=\mathfrak{D}^α$ by a potential $V$. Assuming that $V$ belongs to a certain class of potentials we show that the discrete part of the spectrum of $H$ may contain negative energies, it also appears in the spectral gaps of $L$. We will split the spectrum of $H$ in two parts: high energy part containing eigenvalues which correspond to the eigenfunctions located on the support of the potential $V,$ and low energy part which lies in the spectrum of certain bounded Schrödinger-type operator acting on the Dyson hierarchical lattice. We pay special attention to the class of sparse potentials. In this case we obtain precise spectral asymptotics for $H$ provided the sequence of distances between locations tends to infinity fast enough. We also obtain certain results concerning localization theory for $H$ subject to (non-ergodic) random potential $V$. Examples illustrate our approach.

math.SP↗
arXiv2019

Structure of the particle population for a branching random walk with a critical reproduction law

We consider a continuous-time symmetric branching random walk on the $d$-dimensional lattice, $d\ge 1$, and assume that at the initial moment there is one particle at every lattice point. Moreover, we assume that the underlying random walk has a finite variance of jumps and the reproduction law is described by a critical Bienamye-Galton-Watson process at every lattice point. We study the structure of the particle subpopulation generated by the initial particle situated at a lattice point $x$. We answer why vanishing of the majority of subpopulations does not affect the convergence to the steady state and leads to clusterization for lattice dimensions $d=1$ and $d=2$.

math.PR↗
arXiv2018

Population Processes with Immigration

The paper contains the complete analysis of the Galton-Watson models with immigration, including the processes in the random environment, stationary or non-stationary ones. We also study the branching random walk on $Z^d$ with immigration and prove the existence of the limits for the first two correlation functions.

math.PR↗
arXiv2018

Steady states of lattice population models with immigration

We consider the time evolution of the lattice subcritical Galton-Watson model with immigration. We prove Carleman type estimation for the cumulants in the simple case (binary splitting) and show the existence of a steady state. We also present the formula of the limiting distribution in a particular solvable case.

math.PR↗
arXiv2018

On the spectrum of the hierarchical Schrödinger operator

The goal of this paper is the spectral analysis of the Schrödinger operator $H=L+V$ , the perturbation of the Taibleson-Vladimirov multiplier $L=\mathcal{D}^α$ by a potential $V$. Assuming that $V$ belonges to a class of fast decreasing potentials we show that the discrete part of the spectrum of $H$ may contain negative energies, it also appears in the spectral gaps of $L$. We will split the spectrum of $H$ in two parts: high energy part containing eigenvalues which correspond to the eigenfunctions located on the support of the potential $V,$ and low energy part which lies in the spectrum of certain bounded Schrödinger operator acting on the Dyson hierarchical lattice. The spectral asymptotics \ strictly depend on the transience versus recurrence properties of the underlying hierarchical random walk. In the transient case we will prove results in spirit of CLR theory, for the recurrent case we will provide Bargmann's type asymptotics.

math.FA↗
arXiv2018

Probabilistic approach to a cell growth model

We consider the time evolution of the supercritical Galton-Watson model of branching particles with extra parameter (mass). In the moment of the division the mass of the particle (which is growing linearly after the birth) is divided in random proportion between two offsprings (mitosis). Using the technique of moment equations we study asymptotic of the mass distribution of the particles. Mass distribution of the particles is the solution of the equation with linearly transformed argument: functional, functional-differential or integral. We derive several limit theorems describing the fluctuations of the density of the particles, first two moments of the total masses etc.

math.PR↗
arXiv2018

Boris R. Vainberg (on his 80th birthday)

Boris R. Vainberg was born on March 17, 1938, in Moscow. His father was a Lead Engineer in an aviation design institute. His mother was a homemaker. From early age, Boris was attracted to mathematics and spent much of his time at home and in school working through collections of practice problems for the Moscow Mathematical Olympiad. His first mathematical library consisted of the books he received as one of the prize-winners of these olympiads.

math.HO↗
arXiv2018

Limit Theorems for the Alloy-type Random Energy Model

In this paper, we consider limit laws for the model, which is a generalisation of the random energy model (REM) to the case when the energy levels have the mixture distribution. More precisely, the distribution of the energy levels is assumed to be a mixture of two normal distributions, one of which is standard normal, while the second has the mean \(\sqrt{n}a\) with some \(a\in \R,\) and the variance \(σ\ne 1\). The phase space \((a,σ) \subset \R \times \R_+\) is divided onto several domains, where after appropriate normalisation, the partition function converges in law to the stable distribution. These domains are separated by the critical surfaces, corresponding to transitions from the normal distribution to \(α-\)stable with \(α\in (1,2)\), after to 1-stable, and finally to \(α-\)stable with \(α\in (0,1).\) The corresponding phase diagram is the central result of this paper.

math.PR↗