Resonance regimes of scattering by small bodies with impedance boundary conditions

The paper concerns scattering of plane waves by a bounded obstacle with complex valued impedance boundary conditions. We study the spectrum of the Neumann-to-Dirichlet operator for small wave numbers and long wave asymptotic behavior of the solutions of the scattering problem. The study includes the case when $k=0$ is an eigenvalue or a resonance. The transformation from the impedance to the Dirichlet boundary condition as impedance grows is described. A relation between poles and zeroes of the scattering matrix in the non-self adjoint case is established. The results are applied to a problem of scattering by an obstacle with a springy coating. The paper describes the dependence of the impedance on the properties of the material, that is on forces due to the deviation of the boundary of the obstacle from the equilibrium position

On scattering of solitons for the Klein-Gordon equation coupled to a particle

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.Comment: 47 pages, 2 figure

Inverse problems for Schrodinger equations with Yang-Mills potentials in domains with obstacles and the Aharonov-Bohm effect

We study the inverse boundary value problems for the Schr\"{o}dinger equations with Yang-Mills potentials in a bounded domain $\Omega_0\subset\R^n$ containing finite number of smooth obstacles $\Omega_j,1\leq j \leq r$. We prove that the Dirichlet-to-Neumann operator on $\partial\Omega_0$ determines the gauge equivalence class of the Yang-Mills potentials. We also prove that the metric tensor can be recovered up to a diffeomorphism that is identity on $\partial\Omega_0$.Comment: 15 page

High orders of the perturbation theory for hydrogen atom in magnetic field

The states of hydrogen atom with principal quantum number $n\le3$ and zero magnetic quantum number in constant homogeneous magnetic field ${\cal H}$ are considered. The coefficients of energy eigenvalues expansion up to 75th order in powers of ${\cal H}^2$ are obtained for these states. The series for energy eigenvalues and wave functions are summed up to ${\cal H}$ values of the order of atomic magnetic field. The calculations are based on generalization of the moment method, which may be used in other cases of the hydrogen atom perturbation by a polynomial in coordinates potential.Comment: 16 pages, LaTeX, 6 figures (ps, eps

On the structure of eigenfunctions corresponding to embedded eigenvalues of locally perturbed periodic graph operators

The article is devoted to the following question. Consider a periodic self-adjoint difference (differential) operator on a graph (quantum graph) G with a co-compact free action of the integer lattice Z^n. It is known that a local perturbation of the operator might embed an eigenvalue into the continuous spectrum (a feature uncommon for periodic elliptic operators of second order). In all known constructions of such examples, the corresponding eigenfunction is compactly supported. One wonders whether this must always be the case. The paper answers this question affirmatively. What is more surprising, one can estimate that the eigenmode must be localized not far away from the perturbation (in a neighborhood of the perturbation's support, the width of the neighborhood determined by the unperturbed operator only). The validity of this result requires the condition of irreducibility of the Fermi (Floquet) surface of the periodic operator, which is expected to be satisfied for instance for periodic Schroedinger operators.Comment: Submitted for publicatio

Bounds on positive interior transmission eigenvalues

The paper contains lower bounds on the counting function of the positive eigenvalues of the interior transmission problem when the latter is elliptic. In particular, these bounds justify the existence of an infinite set of interior transmission eigenvalues and provide asymptotic estimates from above on the counting function for the large values of the wave number. They also lead to certain important upper estimates on the first few interior transmission eigenvalues. We consider the classical transmission problem as well as the case when the inhomogeneous medium contains an obstacle.Comment: We corrected inaccuracies cost by the wrong sign in the Green formula (17). In particular, the sign in the definition of \sigma was change

Replica analysis of the generalized p-spin interaction glass model

We investigate stability of replica symmetry breaking solutions in generalized $p$-spin models. It is shown that the kind of the transition to the one-step replica symmetry breaking state depends not only on the presence or absence of the reflection symmetry of the generalized "spin"-operators $\hat{U}$ but on the number of interacting operators and their individual characteristics.Comment: 14 pages, 1 figur

Electrocaloric Response of the Dense Ferroelectric Nanocomposites

Using the Landau-Ginzburg-Devonshire approach and effective media models, we calculated the spontaneous polarization, dielectric, pyroelectric, and electrocaloric properties of BaTiO$_3$ core-shell nanoparticles. We predict that the synergy of size effects and Vegard stresses can significantly improve the electrocaloric cooling (2- 7 times) of the BaTiO$_3$ nanoparticles with diameters (10-100) nm stretched by (1-3)% in comparison with a bulk BaTiO$_3$. To compare with the proposed and other known models, we measured the capacitance-voltage and current-voltage characteristics of the dense nanocomposites consisting of (28 -35) vol.% of the BaTiO$_3$ nanoparticles incorporated in the poly-vinyl-butyral and ethyl-cellulose polymers covered by Ag electrodes. We determined experimentally the effective dielectric permittivity and losses of the dense composites at room temperature. According to our analysis, to reach the maximal electrocaloric response of the core-shell ferroelectric nanoparticles incorporated in different polymers, the dense composites should be prepared with the nanoparticles volume ratio of more than 25 % and fillers with low heat mass and conductance, such as Ag nanoparticles, which facilitate the heat transfer from the ferroelectric nanoparticles to the polymer matrix. In general, the core-shell ferroelectric nanoparticles spontaneously stressed by elastic defects, such as oxygen vacancies or any other elastic dipoles, which create a strong chemical pressure, are relevant fillers for electrocaloric nanocomposites suitable for advanced applications as nano-coolers.Comment: 38 pages, including 10 figures and 2 appendixe

Stimulation of the fluctuation superconductivity by the PT-symmetry

We discuss fluctuations near the second order phase transition where the free energy has an additional non-Hermitian term. The spectrum of the fluctuations changes when the odd-parity potential amplitude exceeds the critical value corresponding to the PT-symmetry breakdown in the topological structure of the Hilbert space of the effective non-Hermitian Hamiltonian. We calculate the fluctuation contribution to the differential resistance of a superconducting weak link and find the manifestation of the PT-symmetry breaking in its temperature evolution. We successfully validate our theory by carrying out measurements of far from equilibrium transport in mesoscale-patterned superconducting wires.Comment: Phys. Rev. Lett 201

Dynamical Systems Gradient method for solving nonlinear equations with monotone operators

A version of the Dynamical Systems Gradient Method for solving ill-posed nonlinear monotone operator equations is studied in this paper. A discrepancy principle is proposed and justified. A numerical experiment was carried out with the new stopping rule. Numerical experiments show that the proposed stopping rule is efficient. Equations with monotone operators are of interest in many applications.Comment: 2 figure