arXiv subjects
Publications and source records attributed to B. Vainberg.
The Anderson Hamiltonian $H_0=-Δ+V(x,ω)$ is considered, where $V$ is a random potential of Bernoulli type. The operator $H_0$ is perturbed by a non-random, continuous potential $-w(x) \leq 0$, decaying at infinity. It will be shown that the borderline between finitely, and infinitely many negative eigenvalues of the perturbed operator, is achieved with a decay of the potential $-w(x)$ as $O(\ln^{-2/d} |x|)$.
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.
We consider a model for the distribution of a long homopolymer with a zero-range potential at the origin in $\mathbb{R}^3$. The distribution can be obtained as a limit of Gibbs distributions corresponding to properly normalized potentials concentrated in small neighborhoods of the origin as the size of the neighborhoods tends to zero. The distribution depends on the length $T$ of the polymer and a parameter $γ$ that corresponds, roughly speaking, to the difference between the inverse temperature in our model and the critical value of the inverse temperature. At the critical point $γ_{cr} = 0$ the transition occurs from the globular phase (positive recurrent behavior of the polymer, $γ> 0$) to the extended phase (Brownian type behavior, $γ< 0$). The main result of the paper is a detailed analysis of the behavior of the polymer when $γ$ is near $γ_{cr}$. Our approach is based on analyzing the semigroups generated by the self-adjoint extensions $\mathcal{L}_γ$ of the Laplacian on $C_0^\infty(\mathbb{R}^3 \setminus \{0\})$ parametrized by $γ$, which are related to the distribution of the polymer. The main technical tool of the paper is the explicit formula for the resolvent of the operator $\mathcal{L}_γ$.
We consider the model for the distribution of a long homopolymer in a potential field. The typical shape of the polymer depends on the temperature parameter. We show that at a critical value of the temperature the transition occurs from a globular to an extended phase. For various values of the temperature, including those at or near the critical value, we consider the limiting behavior of the polymer when its size tends to infinity.
The paper contains a simplified and improved version of the results obtained by the authors earlier. Wave propagation is discussed in a network of branched thin wave guides when the thickness vanishes and the wave guides shrink to a one dimensional graph. It is shown that asymptotically one can describe the propagating waves, the spectrum and the resolvent in terms of solutions of ordinary differential equations on the limiting graph. The vertices of the graph correspond to junctions of the wave guides. In order to determine the solutions of the ODE on the graph uniquely, one needs to know the gluing conditions (GC) on the vertices of the graph. Unlike other publications on this topic, we consider the situation when the spectral parameter is greater than the threshold, i.e., the propagation of waves is possible in cylindrical parts of the network. We show that the GC in this case can be expressed in terms of the scattering matrices related to individual junctions. The results are extended to the values of the spectral parameter below the threshold and around it.
The paper is withdrawn by the authors and replaced be an improved and extended version arxiv: 0812.2968
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.
Small diameter asymptotics is obtained for scattering solutions in a network of thin fibers. The asymptotics is expressed in terms of solutions of related problems on the limiting quantum graph. We calculate the Lagrangian gluing conditions at vertices for the problems 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 for each individual junction of the network.
We establish the long time soliton asymptotics for the translation invariant nonlinear system consisting of the Klein-Gordon equation coupled to a charged relativistic particle. The coupled system has a six dimensional invariant manifold of the soliton solutions. We show that in the large time approximation any finite energy solution, with the initial state close to the solitary manifold, is a sum of a soliton and a dispersive wave which is a solution of the free Klein-Gordon equation. It is assumed that the charge density satisfies the Wiener condition which is a version of the ``Fermi Golden Rule''. The proof is based on an extension of the general strategy introduced by Soffer and Weinstein, Buslaev and Perelman, and others: symplectic projection in Hilbert space onto the solitary manifold, modulation equations for the parameters of the projection, and decay of the transversal component.
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).
The Cauchy problem for two dimensional difference wave operators is considered with potentials and initial data supported in a bounded region. The large time asymptotic behavior of solutions is obtained. In contrast to the continuous case (when the problem in the Euclidian space is considered, not on the lattice) the resolvent of the corresponding stationary problem has singularities on the continuous spectrum, and they contribute to the asymptotics.
We consider wave propagation and scattering governed by one dimensional Schrodinger operators with truncated periodic potentials. The propagation of wave packets with narrow frequency supports is studied. The goal is to describe potentials for which the group velocity (for periodic problem) is small and the transmission coefficient for the truncated potential is not too small, i.e. to find media where a slowing down of the wave packets coexists with a transparency.