On negative eigenvalues of low-dimensional Schr\"{o}dinger operators

The paper concerns upper and lower estimates for the number of negative eigenvalues of one- and two-dimensional Schr\"{o}dinger 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 Schr\"{o}dinger operators are also studied.Comment: Added reference

Radiation Conditions for the Difference Schr\"{o}dinger Operators

The problem of determining a unique solution of the Schr\"{o}dinger equation $\left(\Delta+q-\lambda\right) \psi=f$ on the lattice $\mathbb{Z}^{d}$ is considered, where $\Delta$ 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 $Sp(\Delta)=[-2d,2d]$ 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 $\lambda \in[-2d,2d]\backslash S_{0},$ 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 $\lambda.

Laplace Operator in Networks of Thin Fibers: Spectrum Near the Threshold

Our talk at Lisbon SAMP conference was based mainly on our recent results (published in Comm. Math. Phys.) on small diameter asymptotics for solutions of the Helmgoltz equation in networks of thin fibers. The present paper contains a detailed review of these results under some assumptions which make them much more transparent. It also contains several new theorems on the structure of the spectrum near the threshold. small diameter asymptotics of the resolvent, and solutions of the evolution equation

Global limit theorems on the convergence of multidimensional random walks to stable processes

Symmetric heavily tailed random walks on $Z^d, d\geq 1,$ are considered. Under appropriate regularity conditions on the tails of the jump distributions, global (i.e., uniform in $x,t, |x|+t\to\infty,$) asymptotic behavior of the transition probability $p(t,0,x)$ is obtained. The examples indicate that the regularity conditions are essential.Comment: A misprint is corrected in the main Theorem 1.

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

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.Comment: A mistake in the statement of Theorem 2.5 was correcte

On the negative spectrum of the hierarchical Schr\"{o}dinger operator

This paper is devoted to the spectral theory of the Schr\"{o}dinger 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

Wave propagation in periodic networks of thin fibers

We will discuss a one-dimensional approximation for the problem of wave propagation in networks of thin fibers. The main objective here is to describe the boundary (gluing) conditions at branching points of the limiting one-dimensional graph. The results will be applied to Mach-Zehnder interferometers on chips and to periodic chains of the interferometers. The latter allows us to find parameters which guarantee the transparency and slowing down of wave packets

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.Comment: Misprints are correcte

Transition from a network of thin fibers to the quantum graph: an explicitly solvable model

We consider an explicitly solvable model (formulated in the Riemannian geometry terms) for a stationary wave process in a specific thin domain with the Dirichlet boundary conditions on the boundary of the domain. The transition from the solutions of the scattering problem to the solutions of a problem on the limiting quantum graph is studied. We calculate the Lagrangian gluing conditions at vertices for the problem on the limiting graph. If the frequency of the incident wave is above the bottom of the absolutely continuous spectrum, the gluing conditions are formulated in terms of the scattering data of a problem in a neighborhood of each vertex. Near the bottom of the absolutely continuous spectrum the wave propagation is generically suppressed, and the gluing condition is degenerate (any solution of the limiting problem is zero at each vertex)