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B. Vainberg

Publications and source records attributed to B. Vainberg.

At least 19 recordsLinked to original sources
arXiv2023

Negative eigenvalues of non-local Schrödinger operators with sign-changing potentials

Simon's results on the negative spectrum of recurrent Schrödinger operators ($d=1,2$) are extended to a wider class of potentials and to non-local operators. An example of $L^1-$potental is constructed for which the essential spectrum of two-dimensional Schrödinger operator covers the whole axis. Some counterexamples are provided for transient operators ($d\geq3$) showing that the assumptions on the potential for the validity of the Cwikel-Lieb-Rozenblum estimate can't be improved significantly.

math.SP↗
arXiv2021

Global limit theorem for parabolic equations with a potential

We obtain the asymptotics, as $t + |x| \rightarrow \infty$, of the fundamental solution to the heat equation with a compactly supported potential. It is assumed that the corresponding stationary operator has at least one positive eigenvalue. Two regions with different types of behavior are distinguished: inside a certain conical surface in the $(t,x)$ space, the asymptotics is determined by the principal eigenvalue and the corresponding eigenfunction; outside of the conical surface, the main term of the asymptotics is a product of a bounded function and the fundamental solution of the unperturbed operator, with the contribution from the potential becoming negligible if $|x|/t \rightarrow \infty$. A formula for the global asymptotics, as $t + |x| \rightarrow \infty$, of the solution in the entire half-space $t > 0$ is provided. In probabilistic terms, the result describes the asymptotics of the density of particles in a branching diffusion with compactly supported branching and killing potentials.

math.AP↗
arXiv2020

On the Near-Critical Behavior of Continuous Polymers

The aim of this paper is to investigate the distribution of a continuous polymer in the presence of an attractive finitely supported potential. The most intricate behavior can be observed if we simultaneously and independently vary two parameters: the temperature, which approaches the critical value, and the length of the polymer chain, which tends to infinity. We describe how the typical size of the polymer depends on the two parameters.

math-ph↗
arXiv2020

The Radius of a Polymer at a Near-Critical Temperature

We consider a mean-field model of a polymer with a spherically-symmetric finitely supported potential. We describe how the typical size of the polymer depends on the two parameters: the temperature, which approaches the critical value, and the length of the polymer chain, which goes to infinity.

math-ph↗
arXiv2019

Population dynamics with moderate tails of the underlying random walk

Symmetric random walks in $R^d$ and $Z^d$ are considered. It is assumed that the jump distribution density has moderate tails, i.e., several density moments are finite, including the second one. The global (for all $x$ and $t$) asymptotic behavior at infinity of the transition probability (fundamental solution of the corresponding parabolic convolution operator) is found. Front propagation of ecological waves in the corresponding population dynamics models is described.

math.PR↗
arXiv2019

On critical value of the coupling constant in exterior elliptic problems

We consider exterior elliptic problems with coefficients stabilizing at infinity and study the critical value $β_{cr}$ of the coupling constant (the coefficient at the potential) that separates operators with a discrete spectrum and those without it. The dependence of $β_{cr}$ on the boundary condition and on the distance between the boundary and the support of the potential is described. The discrete spectrum of a non-symmetric operator with the FKW boundary condition (that appears in diffusion processes with traps) is also investigated.

math-ph↗
arXiv2018

Radiation Conditions for the Difference Schrödinger Operators

The problem of determining a unique solution of the Schrödinger equation $\left(Δ+q-λ\right) ψ=f$ on the lattice $\mathbb{Z}^{d}$ is considered, where $Δ$ is the difference Laplacian and both $f$ and $q$ have finite supports$.$ It is shown that there is an exceptional set $S_{0}$ of points on $Sp(Δ)=[-2d,2d]$ for which the limiting absorption principle fails, even for unperturbed operator ($q(x)=0$). This exceptional set consists of the points $\left\{ \pm4n\right\} $ when $d$ is even and $\left\{ \pm2(2n+1)\right\} $ when $d$ is odd. For all values of $λ\in[-2d,2d]\backslash S_{0},$ the radiation conditions are found which single out the same solutions of the problem as the ones determined by the limiting absorption principle. These solutions are combinations of several waves propagating with different frequencies, and the number of waves depends on the value of $λ.$

math-ph↗
arXiv2017

Spectral analysis of non-local Schrödinger operators

We study spectral properties of convolution operators $\mathcal L$ and their perturbations $H=\mathcal L+v(x)$ by compactly supported potentials. Results are applied to determine the front propagation of a population density governed by operator $H$ with a compactly supported initial density provided that $H$ has positive eigenvalues. If there is no positive spectrum, then the stabilization of the population density is proved.

math.SP↗
arXiv2016

Intermittency for branching walks with heavy tails

Branching random walks on multidimensional lattice with heavy tails and a constant branching rate are considered. It is shown that under these conditions (heavy tails and constant rate), the front propagates exponentially fast, but the particles inside of the front are distributed very non-uniformly. The particles exhibit intermittent behavior in a large part of the region behind the front (i.e., the particles are concentrated only in very sparse spots there). The zone of non-intermittency (were particles are distributed relatively uniformly) extends with a power rate. This rate is found.

math.PR↗
arXiv2013

On mathematical foundation of the Brownian motor theory

The paper contains mathematical justification of basic facts concerning the Brownian motor theory. The homogenization theorems are proved for the Brownian motion in periodic tubes with a constant drift. The study is based on an application of the Bloch decomposition. The effective drift and effective diffusivity are expressed in terms of the principal eigenvalue of the Bloch spectral problem on the cell of periodicity as well as in terms of the harmonic coordinate and the density of the invariant measure. We apply the formulas for the effective parameters to study the motion in periodic tubes with nearly separated dead zones.

math-ph↗
arXiv2012

On general Cwikel-Lieb-Rozenblum and Lieb-Thirring inequalities

These classical inequalities allow one to estimate the number of negative eigenvalues and the sums $S_γ=\sum |λ_i|^γ$ for a wide class of Schrödinger operators. We provide a detailed proof of these inequalities for operators on functions in metric spaces using the classical Lieb approach based on the Kac-Feynman formula. The main goal of the paper is a new set of examples which include perturbations of the Anderson operator, operators on free, nilpotent and solvable groups, operators on quantum graphs, Markov processes with independent increments. The study of the examples requires an exact estimate of the kernel of the corresponding parabolic semigroup on the diagonal. In some cases the kernel decays exponentially as $t\to \infty $. This allows us to consider very slow decaying potentials and obtain some results that are precise in the logarithmical scale.

math-ph↗
arXiv2012

On the negative spectrum of the hierarchical Schrödinger operator

This paper is devoted to the spectral theory of the Schrödinger operator on the simplest fractal: Dyson's hierarchical lattice. An explicit description of the spectrum, eigenfunctions, resolvent and parabolic kernel are provided for the unperturbed operator, i.e., for the Dyson hierarchical Laplacian. Positive spectrum is studied for the perturbations of the hierarchical Laplacian. Since the spectral dimension of the operator under consideration can be an arbitrary positive number, the model allows a continuous phase transition from recurrent to transient underlying Markov process. This transition is also studied in the paper.

math-ph↗
arXiv2012

Non-Random Perturbations of the Anderson Hamiltonian in the 1-D case

Recently (see Molchanov & Vainberg 2011), two of the authors applied the Lieb method to the study of the negative spectrum for particular operators of the form $H=H_0-W$. Here, $H_0$ is the generator of the positive stochastic (or sub-stochastic) semigroup, $W(x) \geq 0$ and $W(x) \to 0$ as $x \to \infty$ on some phase space $X$. They used the general results in several "exotic" situations, among them the Anderson Hamiltonian $H_0$. In the 1-d case, the subject of the present paper, we will prove similar but more precise results.

math-ph↗
arXiv2012

Bargmann type estimates of the counting function for general Schrödinger operators

The paper concerns upper and lower estimates for the number of negative eigenvalues of one- and two-dimensional Schrödinger operators and more general operators with the spectral dimensions $d\leq 2$. The classical Cwikel-Lieb-Rosenblum (CLR) upper estimates require the corresponding Markov process to be transient, and therefore the dimension to be greater than two. We obtain CLR estimates in low dimensions by transforming the underlying recurrent process into a transient one using partial annihilation. As a result, the estimates for the number of negative eigenvalues are not translation invariant and contain Bargmann type terms. The general theorems are illustrated by analysis of several classes of the Schrödinger type operators (on the Riemannian manifolds, lattices, fractals, etc.). We provide estimates from below which prove that the results obtained are sharp. Lieb-Thirring estimates for the low-dimensional Schrödinger operators are also studied.

math-ph↗
arXiv2011

On negative eigenvalues of low-dimensional Schrödinger operators

The paper concerns upper and lower estimates for the number of negative eigenvalues of one- and two-dimensional Schrödinger operators and more general operators with the spectral dimensions $d\leq 2$. The classical Cwikel-Lieb-Rosenblum (CLR) upper estimates require the corresponding Markov process to be transient, and therefore the dimension to be greater than two. We obtain CLR estimates in low dimensions by transforming the underlying recurrent process into a transient one using partial annihilation. As a result, the estimates for the number of negative eigenvalues are not translation invariant and contain Bargmann type terms. We show that a classical form of CLR estimates can not be valid for operators with recurrent underlying Markov processes. We provide estimates from below which prove that the obtained results are sharp. Lieb-Thirring estimates for the low-dimensional Schrödinger operators are also studied.

math-ph↗
arXiv2010

High Frequency Scattering by a Classically Invisible Body

We consider a polyhedron with zero classical resistance, i.e., a polyhedron invisible to an observer viewing only the paths of geometrical optics rays. The corresponding problem of scattering of plane waves by the polyhedron is studied. The quasiclassical approximation is obtained and justified in the case of impedance boundary conditions with a non zero absorbing part. It is shown that the total momentum transmitted to the obstacle vanishes when the frequency $k$ goes to infinity, and that the total cross section oscillates at high frequencies. When the impedance $λ_0$ is real (i. e., there is no absorption), it is shown that there exists a sequence of frequencies $k_n$ such that the averages in the impedance of the total cross section over shrinking intervals around $λ_0 $ go to zero as $k_n \to \infty$.

math-ph↗