Electromagnetic scattering by a smooth convex impedance cone

The problem of the diffraction of an electromagnetic planewave by a convex cone of arbitrary smooth cross-section withimpedance (Leontovich) boundary conditions is studied. The vectorproblem is reduced to that for the Debye potentials. By meansof Kontorovich–Lebedev integrals, two spectral functionsare introduced and the corresponding boundary value problemis formulated. The spectral functions for the potentials arefound to satisfy the Helmholtz equations on the unit sphereand to be coupled through non-traditional boundary conditionsof the impedance type with shifts on the spectral variable.The use of the Green theorem permits us to establish an integralformulation of the boundary value problem for the spectral functions.The formal asymptotic solution of the problem is then givenfor the case of a narrow cone. For this, two different methodsare given: a method of perturbation applied to the spectralintegral equations and an adaptation of the method of matchingthe asymptotic series in spectral domain. Both methods leadto the same closed-form result for the leading term of the scatteringdiagram asymptotics.     

Diffraction of a plane electromagnetic wave obliquely incident upon the edge of a wedge with a dielectric coating

The diffraction of a plane electromagnetic wave obliquely incident upon the edge of a coated wedge is considered. The generalized impedance boundary conditions (GIBC's) on the wedge's faces are used to simulate the effect of the coatings. To insure the well-posedness of the problem, special contact conditions (CC's) on the edge are additionally imposed. By using Sommerfeld integrals, the problem is reduced to a system of coupled functional equations, which is solved by the perturbation method. It is shown that, for a certain range of the angles of oblique incidence, the solution can be represented in the form of convergent series that are Neumann series for linear equations with contracting operators. Nonuniform asymptotics of the wave field for regions outside a neighborhood of the edge of the wedge are constructed. Bibliography: 9 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 218, 1994, pp. 72–95. Translated by L. G. Vardapetyan and M. A. Lyalinov.     

Wave-field asymptotics in the problem of diffraction by a planar junction of thin layers and boundary-contact conditions

The problem of diffraction by a planar junction of thin layers covering a perfectly conducting substratum is considered, and its asymptotic solution is constructed. The wave field in the vicinity of the junction of the layers is described by a function of the boundary layer. Based on the asymptotics obtained, the generalized impedance boundary condition, which simulates thin layers, and the contact conditions are derived. The uniqueness of the solution of a model problem is discussed. Bibliography: 6 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 230, 1995, pp. 157–171. Original Translated by M. A. Lyalinov.     

Focusing problem and the asymptotics of the spectral function of the Laplace-Beltrami operator. II

In the paper a formal high-frequency solution of the problem of a point source of oscillations near a reflecting boundary is constructed. The boundary is geodesically concave which makes it possible to use methods developed in diffraction theory by V. A. Fock and J. B. Keller. On the basis of the formal solution of the problem of a point source of oscillations, it is possible to construct the asymptotics of the spectral functions of the Laplace-Beltrami operator.Translated fromZapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 89, pp. 14–53, 1979.     

Asymptotics of Brownian integrals and pressure: Bose-Einstein statistics

The paper studies the asymptotics of the Brownian integrals with paths restricted to a bounded domain of ? ^v , when the domain is dilated to infinity. The framework is that of the Bose-Einstein statistics with paths observed within random time intervals which are integer multiplies of some fixed β > 0. The three first terms of the asymptotics are found explicitly via the functional integrals. In the case of a gas of interacting Brownian loops an expression for the volume term of the asymptotics of the log-partition function is found and the correction term is proved to by order be the boundary area of the domain.     

Asymptotics of unitary and orthogonal matrix integrals

In this paper, we prove that in small parameter regions, arbitrary unitary matrix integrals converge in the large N limit and match their formal expansion. Secondly we give a combinatorial model for our matrix integral asymptotics and investigate examples related to free probability and the HCIZ integral. Our convergence result also leads us to new results of smoothness of microstates. We finally generalize our approach to integrals over the orthogonal group.     

Spectral properties of a Sturm-Liouville type differential operator with a retarding argument

Spectral properties of a differential operator of Sturm-Liouville type are studied in the case of retarding argument with different boundary conditions. The asymptotics of solutions to the corresponding differential equation is studied in the case of a summable potential. An asymptotics of eigenvalues and an asymptotics of eigenfunctions of the differential operator are calculated for each considered case.     

Random surface growth with a wall and Plancherel measures for O (∞)

We consider a Markov evolution of lozenge tilings of a quarter‐plane and study its asymptotics at large times. One of the boundary rays serves as a reflecting wall. We observe frozen and liquid regions, prove convergence of the local correlations to translation‐invariant Gibbs measures in the liquid region, and obtain new discrete Jacobi and symmetric Pearcey determinantal point processes near the wall. The model can be viewed as the one‐parameter family of Plancherel measures for the infinite‐dimensional orthogonal group, and we use this interpretation to derive the determinantal formula for the correlation functions at any finite‐time moment. © 2010 Wiley Periodicals, Inc.     

Asymptotics of solutions near crack tips for Poisson equation with inequality type boundary conditions

The Poisson equation in two-dimensional case for a nonsmooth domain is considered. The geometrical domain has a cut (crack) where inequality type boundary conditions are imposed. A behavior of the solution near the crack tips is analyzed. In particular, estimates for the second derivatives in a weighted Sobolev space are obtained and asymptotics of the solution near crack tips is established.      

Computation of corrections to the fock asymptotics for the wave field near a circular cylinder and a sphere

Corrections are computed to the asymptotics of V. A. Fock for diffraction fields in the region of semishadow near surfaces of constant curvature. Numerical computations are carried out to demonstrate the effectiveness of the corrections found in the computation of fields.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 104, pp. 102–110, 1981.The present work was carried out during the author's stay at LOMI. The author is grateful to V. M. Babich for the opportunity of the visit, for attention to the work, and for discussion of the results in his seminars.     

Trapped-mode and gap-band effects in elastic waveguides with obstacles

Traveling wave propagation in elastic waveguides with obstacles in the form of cracks, voids, inclusions or surface irregularities is considered. The investigation is focused on the trapped–mode phenomena featured by the time–averaged harmonic wave energy localization near the obstacles in the form of energy vortices. The latter results, in particular, in narrow gap bands in the frequency plots of transmission coefficients. The study is carried out using analytically based computer models relying on wave expressions in terms of path Fourier integrals, Green's matrices for the laminate structures, and asymptotics for body and traveling waves derived from those integrals. The connection between the resonance effects and natural frequencies (spectral points of the related boundary value problems) in the complex frequency plane is analyzed as well. Examples of spectral points touching the real axis in the course of varying crack size are presented. The eigenforms associated with such discrete spectral points lying in a continuous spectrum depict strong wave energy localization. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Diffraction of short waves by a segment

The shortwave asymptotics of the Green function for a segment is investigated in the case of the Neumann boundary condition. In the shadow zone and the light zone terms describing the diffracted waves issuing from the end points of the segment are separated out from the solution in the form of a contour integral. The corresponding single integrals are then reduced to expressions coinciding with the formulas of the geometric theory of diffraction. It is here found, that the primary diffracted waves are described by series of residues having the same order with respect to the large parameter of the problem; for the series describing multiple diffracted waves it suffices to restrict attention to a single residue.     

Asymptotic model of fields in a thin-walled structure with crack-like defects

The transition from two-dimensional (2D) wave propagation throughthe square periodic structure in anti-plane shear time-harmoniccase to a discretised model of a 2D lattice with masses connectedby springs is considered. A model of a defect in the middlepart of the thin-walled bridges is presented. As a first partof the asymptotic model, the effective transmission conditionin the vicinity of the transverse cut of the thin-walled bridgesis discussed. Then, a boundary layer determining the asymptoticexpansion of the field near the tip of the crack is constructed.Stress intensity factors are evaluated for deep cracks in thejunction regions. The corresponding boundary layer analysisis non-trivial and has not been attempted elsewhere.     

Problem of focusing and the asymptotics of the spectral function of the Laplace-Beltrami operator. I

The connection between the asymptotics of the spectral function and the formal shortwave expansion of the solution of the problem of the asymptotics of the Green function near a geodesically concave boundary of a two-dimensional surface is considered in the paper.     

Sharp Growth Estimates for Modified Poisson Integrals in a Half Space

For continuous boundary data, including data of polynomial growth, modified Poisson integrals are used to write solutions to the half space Dirichlet and Neumann problems in Rⁿ. Pointwise growth estimates for these integrals are given and the estimates are proved sharp in a strong sense. For decaying data, a new type of modified Poisson integral is introduced and used to develop asymptotic expansions for solutions of these half space problems.     

Short-wavelength grazing scattering of a plane wave by a smooth periodic boundary. II. Diffraction by an infinite periodic boundary

The short-wavelength asymptotic behavior of the field near a reflecting boundary (the Fock zone and the neighborhood of the limit ray) is constructed for the problem of the diffraction of a plane wave by a smooth periodic boundary.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Seklova AN SSSR, Vol. 173, pp. 60–86, 1990.     

A simple inequality for the variance of the number of zeros of a differentiable Gaussian stationary process

The variance of the number of zeros of a Gaussian differentiable stationary process in a finite time interval can be represented by a single integral of a sophisticated function having singularities in the vicinity of zero, which complicates computer calculations. In this paper, for a wide class of correlation functions, an inequality estimating this variance in simpler terms is proved. Two of three considered examples demonstrate the limits of the effectiveness of the obtained inequality by comparison with special processes earlier established by the author for which the variance is calculated by formulas without integrals. In the two subsequent cases, the inequality is used for the asymptotic estimation of the variance of the number of zeros in a small time interval and, in the last one, in addition to this asymptotics, the upper and lower bounds for the most widely used analytic process in all time intervals.     

Asymptotics of two-dimensional integrals depending singularly on a small parameter

The asymptotics is constructed for some two-dimensional integrals depending singularly on a small parameter. The construction algorithm is described and two examples are given.     

Instantons via breaking geometric symmetry in hyperbolic traps

Using geometrical and algebraic ideas, we study tunnel eigenvalue asymptotics and tunnel bilocalization of eigenstates for certain class of operators (quantum Hamiltonians) including the case of Penning traps, well known in physical literature. For general hyperbolic traps with geometric asymmetry, we study resonance regimes which produce hyperbolic type algebras of integrals of motion. Such algebras have polynomial (non-Lie) commutation relations with creation-annihilation structure. Over this algebra, the trap asymmetry (higher-order anharmonic terms near the equilibrium) determines a pendulum-like Hamiltonian in action-angle coordinates. The symmetry breaking term generates a tunneling pseudoparticle (closed instanton). We study the instanton action and the corresponding spectral splitting.