Abstract: Abstract
We consider the Schrödinger operator with a periodic potential p plus a compactly supported potential q on the real line. We assume that both p and q have m ⩾ 0 derivatives. For generic p, the essential spectrum of the operator has an infinite sequence of open gaps. We determine the asymptotics of the S-matrix at high energy. PubDate: 2014-03-01

Abstract: Abstract
Let A be a densely defined symmetric operator and let {Ã′, Ã} be an ordered pair of proper extensions of A such that their resolvent difference is of trace class. We study the perturbation determinant ΔÃ′/Ã(·) of the singular pair {Ã′, Ã} by using the boundary triplet approach. We show that, under additional mild assumptions on {Ã′, Ã, the perturbation determinant ΔÃ′/Ã(·) is the ratio of two ordinary determinants involving the Weyl function and boundary operators. In particular, if the deficiency indices of A are finite, then we obtain ΔÃ′/Ã(z) = det (B′ - M(z))/det (B - M (z)), z ∈ ρ(Ã), where M(·) stands for the Weyl function and B′ and B for the boundary operators corresponding to Ã′ and Ã with respect to a chosen boundary triplet Π. The results are applied to ordinary differential operators and to second-order elliptic operators. PubDate: 2014-03-01

Abstract: Abstract
We show that the fourth-order nonlinear ODE which controls the pole dynamics in the general solution of equation P
I
2
compatible with the KdV equation exhibits two remarkable properties: (1) it governs the isomonodromy deformations of a 2 × 2 matrix linear ODE with polynomial coefficients, and (2) it does not possess the Painlevé property. We also study the properties of the Riemann-Hilbert problem associated to this ODE and find its large-t asymptotic solution for physically interesting initial data. PubDate: 2014-03-01

Abstract: Abstract
In the article, the relationship between the mesoscopic picture and the microscopic and macroscopic pictures is considered in the light of the work of V. E. Panin and his school. The coincidence of isochores and isotherms obtained by applying the undistinguishing statistics of objectively distinguishable objects with those of the Van-der-Waals model allows us to specify the heat capacity C
V
corresponding to real gases. The difference between Gentile statistics and parastatistical models in the mesoscopic case is indicated. PubDate: 2014-03-01

Abstract: Abstract
As in the first part (J. Brüning, S.Yu. Dobrokhotov, D.S. Minenkov, 2011), we construct a family of special solutions of the Dirichlet problem for the Laplace equation in a domain with fast changing boundary. Using these solutions, we construct an analytic model of cold field electron emission from surfaces simulating arrays of vertically aligned nanotubes. Explicit analytic formulas lead to fast computations and also allow us to study the case of random arrays of tubes with stochastic distribution of parameters. We present these results and compare them with numerical approximations given in [1]. PubDate: 2014-03-01

Abstract: Abstract
In the paper, we study uniqueness problems for solutions of a boundary value problem for a polyharmonic equation in the exterior of a compact set and in a half-space under the assumption that the generalized solution of the problem in question admits a finite Dirichlet integral with a weight of the form x
a
. In dependence on the values of the parameter a, we prove uniqueness theorems and also present precise formulas to evaluate the dimension of the space of solutions of this problem in the exterior of a compact set and in a half-space. PubDate: 2014-03-01

Abstract: Abstract
In this paper, we investigate some identities for Laguerre polynomials involving Bernoulli and Euler polynomials derived from umbral calculus. PubDate: 2014-03-01

Abstract: Abstract
We consider abelian groups formed by simply connected closed oriented smooth 6-manifolds with given 2-dimensional homology and given 2-dimensional Stiefel-Whitney class. In particular, an effective presentation of these groups is given. PubDate: 2014-03-01

Abstract: Abstract
The paper continues the first part (Russ. J. Math. Phys. 20 (3), 360–373). Let Ω be a John domain, let Γ ⊂ ∂Ω be an h-set, and let g and υ be weights on Ω that are distance functions to the set Γ of special form. In the paper, sufficient conditions are obtained under which the Sobolev weighted class W
p,g
r
(Ω) is continuously embedded in the space L
q,v
(Ω). Moreover, bounds for the approximation of functions in W
p,g
r
(Ω) by polynomials of degree not exceeding r − 1 in L
q,v
(
$\tilde \Omega $
) are found, where
$\tilde \Omega $
is a subdomain generated by a subtree of the tree T defining the structure of Ω. PubDate: 2014-03-01

Abstract: Abstract
We continue the study of automatic continuity conditions for finite-dimensional representations of connected Lie groups. In particular, we claim that every locally bounded finite-dimensional representation of a connected Lie group is continuous on the commutator subgroup in the intrinsic Lie topology of the subgroup and continuous on the intersection of the commutator subgroup with the radical of the group in the original topology of the Lie group, thus correcting one of our previous results. PubDate: 2014-03-01

Abstract: Abstract
The paper is devoted to the role of critical nonlinearities in the framework of the theory of global solvability of nonlinear partial differential equations. In particular, a new approach to blow-up problems for solutions of nonlinear partial differential equations is suggested. PubDate: 2013-10-01

Abstract: Abstract
A theorem about two-parameter families of Schrödinger operators in proved; the potential is parameter dependent. PubDate: 2013-10-01

Abstract: Abstract
We study the inverse spectral problem for the Schrödinger operator H on the two-dimensional torus with even magnetic field B(x) and even electric potential V(x). Guillemin [11] proved that the spectrum of H determines B(x) and V(x). A simple proof of Guillemin’s results was given by the authors in [3]. In the present paper, we consider gauge equivalent classes of magnetic potentials and give conditions which imply that the gauge equivalence class and the spectrum of H determine the magnetic field and the electric potential. We also show that, generically, the spectrum and the magnetic field determine the “extended” gauge equivalence class of the magnetic potential. The proof is a modification of that in [3] with some corrections and clarifications. PubDate: 2013-10-01

Abstract: Abstract
We study the two-dimensional problem of propagation of linear water waves in deep water in the presence of a submerged body. Under some geometrical requirements, we derive an explicit bound for the solution depending on the domain and the functions on the right-hand side. PubDate: 2013-10-01

Abstract: Abstract
In the paper, two boundary value problems for the system of Poisson equations in three-dimensional domains are studied. PubDate: 2013-10-01

Abstract: Abstract
A phase transition of the first kind is a jump of a function, a phase transition of the second kind is a jump of its first derivative, a phase transition of the third kind, a jump of the second derivative. A phase transition from one statistic to another is very gradual, but finally it is as considerable as the phase transition of the first kind. However, we cannot introduce a clearly defined parameter to which this transition corresponds. This is due to the fact that the fluctuations near the critical point are huge, and this violates, in the vicinity of that point, the main law of equilibrium thermodynamics, which asserts that fluctuations are relatively small.
The paper describes the transition in the supercritical fluid region of equilibrium thermodynamics from parastatistics to mixed statistics, in which the Boltzmann statistics is realized for long-living clusters. In economics this corresponds to a negative nominal credit rate. Examples of this non-standard situation are presented. PubDate: 2013-10-01

Abstract: Abstract
We consider second-order differential-difference equations in bounded domains in the case where several degenerate difference operators enter the equation. The degeneration leads to the fact that the multiplicity of the zero eigenvalue for the corresponding differential-difference operator becomes infinite. Regularity of generalized solutions for such equations is known to fail in the interior of the domain. However, we prove that the projections of solutions onto the orthogonal complement to the kernel of the “leading” difference operator remain regular in certain subdomains which form a decomposition of the original domain. PubDate: 2013-10-01

Abstract: Abstract
A far field asymptotic formula is derived for the single integral obtained in the first part of the research and giving the height of tsunami in the framework of the hydrodynamical potential model with a special axially-symmetric source of piston type. Conditions are indicated for the variables (of the model under consideration) that can vary in wide intervals and for which the asymptotic formula is of high precision for the calculation of the wave profile and for the time history of the height of the tsunami. Using results of manyvariant computer-aided processing, we found domains of variables of the model in which the asymptotic formula has high precision for the computations of the wave profile and of time history of height not only the leading waves of tsunami but also of tailing waves. PubDate: 2013-10-01

Abstract: Abstract
We study the influence of disorder on propagation of waves in one-dimensional structures. Transmission properties of the process governed by the Schrödinger equation with the white noise potential can be expressed through the Lyapunov exponent γ which we determine explicitly as a function of the noise intensity σ and the frequency ω. We find uniform two-parameter asymptotic expressions for γ which allow us to evaluate γ for different relations between σ and ω. The value of the Lyapunov exponent is also obtained in the case of a short-range correlated noise, which is shown to be less than its white noise counterpart. PubDate: 2013-10-01