
Russian Journal of Mathematical Physics [SJR: 0.858] [HI: 24] [0 followers] Follow Hybrid journal (It can contain Open Access articles) ISSN (Print) 15318621  ISSN (Online) 10619208 Published by SpringerVerlag [2345 journals] 
 On the distribution of energy of localized solutions of the Schrödinger
equation that propagate along symmetric quantum graphs Authors: A. I. Allilueva; A. I. Shafarevich
Pages: 139  147
Abstract: The theory of differential equations and differential operators on geometric graphs is actively developing in recent decades. One of the directions is devoted to the study of the socalled Gaussian packets, i.e., localized asymptotic solutions of the nonstationary Schrödinger equation. An interesting feature of such solutions is in their close connection with problems of analytic number theory and, in particular, with estimates for the number of integer points in polyhedra and the number of integer solutions of linear inequalities. At the same time, from the point of view of applications to quantum mechanics, it is natural to raise the question of the energy distribution of such solutions along the graph (in other words, the probabilities of finding a quantum particle in some area). Seemingly, this question is very complicated for general graphs, because the energy distribution is much more sensitive to the type of boundary conditions and to the initial state than the asymptotics of the number of localized functions. A similar problem is to describe the energy distribution of a solution of the wave equation on a geometric graph. For infinite regular trees, this question was studied in the paper [10, 11]; at the same time, the general case is practically unstudied. The main observation of the present paper is that the situation is considerably simplified if we consider strongly localized asymptotic solutions; in this case, a general unitary operator describing the scattering at a vertex is replaced by the operator of reflection from the subspace. In the simplest situations, this circumstance makes it possible to obtain comparatively simple formulas for the energy distribution along the edges.
PubDate: 20170401
DOI: 10.1134/s1061920817020017
Issue No: Vol. 24, No. 2 (2017)
 Authors: A. I. Allilueva; A. I. Shafarevich
 Stabilizer of a function in the gage group
 Authors: V. K. Beloshapka
Pages: 148  152
Abstract: It is proved that, for the dimension d of the stabilizer of an analytic function z(x, y) in the gage pseudogroup G = {z(x, y) → c(z(a(x), b(y))}, there are precisely four possibilities: (1) d = ∞ and the complexity of z is zero, (2) d = 3 and the complexity of z is equal to one, (3) d = 1 and z is equivalent the function r(x + y) − x of complexity two, (4) d = 0 in all remaining cases.
PubDate: 20170401
DOI: 10.1134/s1061920817020029
Issue No: Vol. 24, No. 2 (2017)
 Authors: V. K. Beloshapka
 Topical problems of the theory of Transcendental numbers: Development of
approaches to their solution in the works of Yu. V. Nesterenko Authors: V. G. Chirskii
Pages: 153  171
Abstract: The present paper is a survey of a part of the theory devoted to certain problems concerning the algebraic independence of the values of analytic functions, to quantitative results on estimates for the measure of transcendence or the measure of algebraic independence of numbers, to functional analogs of these results on the algebraic independence of solutions of algebraic differential equations, and estimates for the multiplicities of zeros for polynomials in these solutions, which play an important role in the proof of numerical results. This choice is related to the fact that, in December 2016, the head of the Department of Number Theory of Moscow State University, Corresponding Member of the RAS Yu.V. Nesterenko, who did a lot to develop these directions of the theory Transcendental numbers and whose works are marked by many awards, became seventy. He is a laureate of the Markov RAS Prize, 2006, of the Ostrovsky international prize, 1997, of the Hardy–Ramanujan Society Prize, 1997, and the Alexander von Humboldt Prize, 2003. Since the article is dedicated to the 70th anniversary of the birth of Yurii Valentinovich, we preface the scientific part with a brief biography.
PubDate: 20170401
DOI: 10.1134/s1061920817020030
Issue No: Vol. 24, No. 2 (2017)
 Authors: V. G. Chirskii
 On the asymptotical normality of statistical solutions for wave equations
coupled to a particle Authors: T. V. Dudnikova
Pages: 172  194
Abstract: We consider a linear Hamiltonian system consisting of a classical particle and a scalar field describing by the wave or Klein–Gordon equations with variable coefficients. The initial data of the system are supposed to be a random function which has some mixing properties. We study the distribution μ t of the random solution at time moments t ∈ R. The main result is the convergence of μ t to a Gaussian probability measure as t→∞. The application to the case of Gibbs initial measures is given.
PubDate: 20170401
DOI: 10.1134/s1061920817020042
Issue No: Vol. 24, No. 2 (2017)
 Authors: T. V. Dudnikova
 On an inverse spectral problem
 Authors: Yu. V. Egorov
Pages: 195  206
Abstract: We study an inverse spectral problem for the Sturm–Liouville problem. Such a problem arises in the study of micro and nanorods.
PubDate: 20170401
DOI: 10.1134/s1061920817020054
Issue No: Vol. 24, No. 2 (2017)
 Authors: Yu. V. Egorov
 Generalized optical theorem to a multipole source excitation in the
scattering theory Authors: Yu. A. Eremin; A. G. Sveshnikov
Pages: 207  215
Abstract: In the present paper, the Optical Theorem is generalized to the case of a penetrable obstacle excited by a multipole of arbitrary order in the presence of a transparent substrate. This generalization allows one to test computer modules when wave scattering by lossless penetrable obstacle is considered. Besides, it enables one to evaluate the absorption crosssection by subtracting the scattering crosssection from the extinction crosssection. This seems to be important because, in this particular case, the far field does not involve a Sommerfeld integral.
PubDate: 20170401
DOI: 10.1134/s1061920817020066
Issue No: Vol. 24, No. 2 (2017)
 Authors: Yu. A. Eremin; A. G. Sveshnikov
 Some features of bending of a rod under a strong longitudinal compression
 Authors: A. A. Ershov; B. I. Suleimanov
Pages: 216  233
Abstract: Typical processes of rod bending under strong longitudinal compression are studied. The corresponding dynamic equation of bending is considered as a perturbation of the twodimensional Laplace equation. It is established that, for these processes, the expantion of domains of rapid increase of bending begins in small neighborhoods of singularity points of solutions of the limiting Laplace equation. The initial stages of these increases are described using the Hardy integral.
PubDate: 20170401
DOI: 10.1134/s1061920817020078
Issue No: Vol. 24, No. 2 (2017)
 Authors: A. A. Ershov; B. I. Suleimanov
 Measures on the Hilbert space of a quantum system
 Authors: A. Yu. Khrennikov; O. G. Smolyanov
Pages: 234  240
Abstract: The paper is the first in a series of papers on the use of measures and generalized measures in quantum theory. In particular, a survey of the proofs of equivalence of various definitions of the density operator is presented. The exposition is of algebraic nature, and analytic assumptions are usually omitted.
PubDate: 20170401
DOI: 10.1134/s106192081702008x
Issue No: Vol. 24, No. 2 (2017)
 Authors: A. Yu. Khrennikov; O. G. Smolyanov
 Degenerate Laplace transform and degenerate gamma function
 Authors: T. Kim; D. S. Kim
Pages: 241  248
Abstract: In this paper, we introduce the degenerate Laplace transform and degenerate gamma function and investigate some of their properties. From our investigation, we derive some interesting formulas related to the degenerate Laplace transform and degenerate gamma function.
PubDate: 20170401
DOI: 10.1134/s1061920817020091
Issue No: Vol. 24, No. 2 (2017)
 Authors: T. Kim; D. S. Kim
 Topological phase transitions in the theory of partitions of integers
 Authors: V. P. Maslov
Pages: 249  260
Abstract: In the paper, the problem of partitioning a natural number into summands is considered in connection with the BKT (Berezinskii–Kosterlitz–Thouless) phase transition and its two critical points. As examples, the passage from superfluid state to normal state and from a celllike vortical state to turbulent state are considered.
PubDate: 20170401
DOI: 10.1134/s1061920817020108
Issue No: Vol. 24, No. 2 (2017)
 Authors: V. P. Maslov
 Remark concerning Maslov’s theorem on homomorphisms of topological
groups Authors: A. I. Shtern
Pages: 261  262
Abstract: Maslov’s theorem on finally continuous sequences of homomorphisms of topological groups is presented for more general passages to the limit.
PubDate: 20170401
DOI: 10.1134/s106192081702011x
Issue No: Vol. 24, No. 2 (2017)
 Authors: A. I. Shtern
 Singular sets of surfaces
 Authors: I. G. Tsar’kov
Pages: 263  271
Abstract: Sets of values of the metric projection for an approximatively compact subset of Hilbert space are studied. The results obtained in this way are used to study the geometry of hypersurfaces in ℝ n and their singular sets.
PubDate: 20170401
DOI: 10.1134/s1061920817020121
Issue No: Vol. 24, No. 2 (2017)
 Authors: I. G. Tsar’kov
 Feynman formulas for semigroups generated by an iterated Laplace operator
 Authors: M. S. Buzinov
Pages: 272  277
Abstract: In the present paper, we find representations of a oneparameter semigroup generated by a finite sum of iterated Laplace operators and an additive perturbation (the potential). Such semigroups and the evolution equations corresponding to them find applications in the field of physics, chemistry, biology, and pattern recognition. The representations mentioned above are obtained in the form of Feynman formulas, i.e., in the form of a limit of multiple integrals as the multiplicity tends to infinity. The term “Feynman formula” was proposed by Smolyanov. Smolyanov’s approach uses Chernoff’s theorems. A simple form of representations thus obtained enables one to use them for numerical modeling the dynamics of the evolution system as a method for the approximation of solutions of equations. The problems considered in this note can be treated using the approach suggested by Remizov (see also the monograph of Smolyanov and Shavgulidze on path integrals). The representations (of semigroups) obtained in this way are more complicated than those given by the Feynman formulas; however, it is possible to bypass some analytical difficulties.
PubDate: 20170401
DOI: 10.1134/s1061920817020133
Issue No: Vol. 24, No. 2 (2017)
 Authors: M. S. Buzinov
 Erratum to “A new class of Abelian theorems for the
Mehler–Fock transforms” Authors: H. M. Srivastava; B. J. González; E. R. Negrín
Pages: 278  278
PubDate: 20170401
DOI: 10.1134/s1061920817020145
Issue No: Vol. 24, No. 2 (2017)
 Authors: H. M. Srivastava; B. J. González; E. R. Negrín
 Doubledeck structure of the boundary layer in the problem of flow in an
axially symmetric pipe with small irregularities on the wall for large
Reynolds numbers Authors: V. G. Danilov; R. K. Gaydukov
Pages: 1  18
Abstract: The problem of flow of a viscous incompressible fluid in an axially symmetric pipe with small irregularities on the wall is considered. An asymptotic solution of the problem with the doubledeck structure of the boundary layer and the unperturbed flow in the environment (the “core flow”) is obtained. The results of flow numerical simulation in the thin and “thick” boundary layers are given.
PubDate: 20170101
DOI: 10.1134/s1061920817010010
Issue No: Vol. 24, No. 1 (2017)
 Authors: V. G. Danilov; R. K. Gaydukov
 Tsallis p , q deformed Touchard polynomials and Stirling numbers
 Authors: O. Herscovici; T. Mansour
Pages: 37  50
Abstract: In this paper, we develop and investigate a new twoparametrized deformation of the Touchard polynomials, based on the definition of the NEXT qexponential function of Tsallis. We obtain new generalizations of the Stirling numbers of the second kind and of the binomial coefficients and represent two new statistics for the set partitions.
PubDate: 20170101
DOI: 10.1134/s1061920817010034
Issue No: Vol. 24, No. 1 (2017)
 Authors: O. Herscovici; T. Mansour
 On λBell polynomials associated with umbral calculus
 Authors: T. Kim; D. S. Kim
Pages: 69  78
Abstract: In this paper, we introduce some new λBell polynomials and Bell polynomials of the second kind and investigate properties of these polynomials. Using our investigation, we derive some new identities for the two kinds of λBell polynomials arising from umbral calculus.
PubDate: 20170101
DOI: 10.1134/s1061920817010058
Issue No: Vol. 24, No. 1 (2017)
 Authors: T. Kim; D. S. Kim
 Differentialgeometric structures associated with Lagrangians
corresponding to scalar physical fields Authors: A. K. Rybnikov
Pages: 111  121
Abstract: The paper is devoted to the investigation, using the method of Cartan–Laptev, of the differentialgeometric structure associated with a Lagrangian L, depending on a function z of the variables t, x 1,...,x n and its partial derivatives. Lagrangians of this kind are considered in theoretical physics (in field theory). Here t is interpreted as time, and x 1,...,x n as spatial variables. The state of the field is characterized by a function z(t, x 1,..., x n ) (a field function) satisfying the Euler equation, which corresponds to the variational problem for the action integral. In the present paper, the variables z(t, x 1,..., x n are regarded as adapted local coordinates of a bundle of general type M with ndimensional fibers and 1dimensional base (here the variable t is simultaneously a local coordinate on the base). If we agree to call t time, and a typical fiber an ndimensional space, then M can be called the spatiotemporal bundle manifold. We consider the variables t, x 1,...,x n , z (i.e., the variables t, x 1,...,x n with the added variable z) as adapted local coordinates in the bundle H over the fibered base M. The Lagrangian L, which is a coefficient in the differential form of the variational action integral in the integrand, is a relative invariant given on the manifold J 1 H (the manifold of 1jets of the bundle H). In the present paper, we construct a tensor with components Λ00, Λ0i , Λ ij (Λ ij = Λ ji ) which is generated by the fundamental object of the structure associated with the Lagrangian. This tensor is an invariant (with respect to admissible transformations the variables t, x 1,...,x n , z) analog of the energymomentum tensor of the classical theory of physical fields. We construct an invariant I, a vector G i , and a bivalent tensor G jk generated by the Lagrangian. We also construct a relative invariant of E (in the paper, we call it the Euler relative invariant) such that the equation E = 0 is an invariant form of the Euler equation for the variational action integral. For this reason, a nonvariational interpretation of the Euler equation becomes possible. Moreover, we construct a connection in the principal bundle with base J 2 H (the variety of 2jets of the bundle H) and with the structure group GL(n) generated by the structure associated with the Lagrangian.
PubDate: 20170101
DOI: 10.1134/s1061920817010083
Issue No: Vol. 24, No. 1 (2017)
 Authors: A. K. Rybnikov
 A new class of Abelian theorems for the Mehler–Fock transforms
 Authors: H. M. Srivastava; B. J. González; E. R. Negrín
Pages: 124  126
Abstract: The main object of this paper is to derive several new Abelian theorems for the Mehler–Fock transforms. The results presented here are compared with those given earlier by R. S. Pathak and R. N. Pandey [Math. Soc. 3 (1987), 91–95]. Some applications and particular cases are also considered.
PubDate: 20170101
DOI: 10.1134/s1061920817010101
Issue No: Vol. 24, No. 1 (2017)
 Authors: H. M. Srivastava; B. J. González; E. R. Negrín
 Propagation of a linear wave created by a spatially localized perturbation
in a regular lattice and punctured Lagrangian manifolds Authors: S. Yu. Dobrokhotov; V. E. Nazaikinskii
Pages: 127  133
Abstract: The following results are obtained for the Cauchy problem with localized initial data for the crystal lattice vibration equations with continuous and discrete time: (i) the asymptotics of the solution is determined by Lagrangian manifolds with singularities (“punctured” Lagrangian manifolds); (ii) Maslov’s canonical operator is defined on such manifolds as a modification of a new representation recently obtained for the canonical operator by the present authors together with A. I. Shafarevich (Dokl. Ross. Akad. Nauk 46 (6), 641–644 (2016)); (iii) the projection of the Lagrangian manifold onto the configuration plane specifies a bounded oscillation region, whose boundary (which is naturally referred to as the leading edge front) is determined by the Hamiltonians corresponding to the limit wave equations; (iv) the leading edge front is a special caustic, which possibly contains stronger focal points. These observations, together with earlier results, lead to efficient formulas for the wave field in a neighborhood of the leading edge front.
PubDate: 20170101
DOI: 10.1134/s1061920817010113
Issue No: Vol. 24, No. 1 (2017)
 Authors: S. Yu. Dobrokhotov; V. E. Nazaikinskii