Asymptotic behavior of the spectrum of combination scattering at Stokes phonons

For a class of polynomial quantum Hamiltonians used in models of combination scattering in quantum optics, we obtain the asymptotic behavior of the spectrum for large occupation numbers in the secondary quantization representation. Hamiltonians of this class can be diagonalized using a special system of polynomials determined by recurrence relations with coefficients depending on a parameter (occupation number). For this system of polynomials, we determine the asymptotic behavior a discrete measure with respect to which they are orthogonal. The obtained limit measures are interpreted as equilibrium measures in extremum problems for a logarithmic potential in an external field and with constraints on the measure. We illustrate the general case with an exactly solvable example where the Hamiltonian can be diagonalized by the canonical Bogoliubov transformation and the special orthogonal polynomials degenerate into the Krawtchouk classical discrete polynomials.     

Asymptotic behavior of the spectrum of differential equations

This survey is devoted to an exposition of results on the asymptotics of the discrete spectrum of self-adjoint differential operators, mainly partial differential operators.Translated from Itogi Nauki i Tekhniki, Matematicheskii Analiz, Vol. 14, pp. 5–58, 1977.     

Asymptotic behavior of the spectrum of the Maxwell operator

For the distribution functions of the positive and the negative eigenvalues of the operator in a domain with a smooth boundary, one obtains the asymptotic formula N^±()=(3²)^–1 mes ·³+0(²). Under additional assumptions on the properties of the geodesic billiard in , one shows that N±()= (3²)^–1 mes ·³+0(²).Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 127, pp. 169–180, 1983.     

Asymptotic behavior of the spectrum of a pseudodifferential operator with periodic bicharacteristics

Let _j be the eigenvalues of a positive elliptic pseudodifferential operator of order m > 0 on a closed compact d-dimensional C-manifold and let N()=#{j:_j^m}. It is shown that for each > 0 we have

Asymptotic behavior of the Fourier coefficients for the problem of scattering on contours tano

In this paper the analytic properties of the wave function in the region bounded by the scattering body are studied. This problem is of major importance for the development of the analytic side of the method of separation of variables in problems with noncoordinate boundaries. New asymptotic estimates for the Fourier coefficients of the scattered field are obtained. They indicate the analyticity of the wave function in the physical plane for all Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 62, pp. 27–38, 1976.In conclusion, we wish to thank V. M. Babich, M. M. Popov, and S. Yu. Slavyanov for making valuable suggestions regarding the topic treated here which contributed to the completion of this paper.     

Asymptotic behavior of the spectrum of variational problems on the solutions of elliptic systems

One computes the principal term of the asymptotics of the spectrum of variational problems of the form B[u]/A[u] considered on the solutions of elliptic systems Lu=0 on a compact manifold with boundary. Previously, the corresponding result has been obtained by the authors under the additional condition of the existence of an elliptic boundary problem for the operator L. Now this restriction is removed.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 115, pp. 23–39, 1982.     

Asymptotic behavior of the spectrum of certain problems,connected with the oscillation of fluids

One obtains the principal term of the asymptotics of the spectrum in a series of problems of the theory of small oscillations of fluids, filling partially a container. First one discusses a certain problem of general character, the nonlocal Steklov type problem. The considered problems are connected with the oscillations of a capillary ideal (or capillary stratified) fluid, of a heavy viscous fluid, of a capillary viscous fluid, and with the oscillations of a fluid in container with an elastic bottom.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 152, pp. 158–164, 1986.The author is deeply grateful to her scientific advisor M. Sh. Birman. The author is grateful also to N. D. Kopachevskii for his interest in the paper.     

Asymptotic behavior of singular solitons and the inverse scattering method for solving boundary value problems

We solve the boundary value problems for the sine-Gordon equation and the elliptic Toda lattice in the framework of the inverse scattering method. This approach allows incorporating the singular soliton solutions of the spiral vortex form. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 124, No. 2, pp. 279–291, August, 2000.     

Asymptotic completeness in optical scattering

In this paper time-dependent methods are developed in order to prove the existence of the wave operators and asymptotic completeness in the scattering theory of electromagnetic waves in inhomogeneous media. The medium has a localized perturbation of the dielectric susceptibility. There are no regularity assumptions about the dielectric susceptibility. It is shown directly that the range of the wave operator is the complete continuous subspace, and not just the absolutely continuous subspace of the generator of the time evolution describing the perturbed wave propagation.     

Rigidity of the scattering length spectrum

In this paper we consider properties of obstacles satisfying some non-degeneracy conditions that can be recovered from the scattering length spectrum (SLS). Clearly the latter tells us whether the obstacle K is trapping or non-trapping. If the set of trapped points is relatively small, then the SLS also determines the volume of the obstacle, the number of its connected components, and whether its boundary is convex everywhere or it has non-trivial concavities. Under the additional assumption that the curvature of the obstacle does not vanish of infinite order, it is proved that from the SLS one can recover certain information about the number of reflection points of any simply reflecting ray in the exterior of the obstacle. Finally, for some special classes of obstacles (e.g. star-shaped ones), it is shown that the SLS completely determines the obstacle. Received: 2 March 1999 / Revised version: 16 January 2001 / Published online: 5 September 2002     

Asymptotic behavior of the Cahn-Hilliard-Oono equation

Our aim in this article is to study the asymptotic behavior, in terms of finite-dimensional attractors, of the Cahn-Hilliard-Oono equation. This equation differs from the usual Cahn-Hilliard equation by the presence of a term of the form \( \epsilon u,\ \epsilon >0\), which takes into account long-ranged interactions. In particular, we prove the existence of a robust family of exponential attractors as \(\epsilon\) goes to \(0\).     

Asymptotic behavior of the Toeplitz determinant

One considers the asymptotic behavior of the Toeplitz determinant for a nonnegative function. One proves that under certain conditions on the function the determinant admits the asymptotic representation where Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 97, pp. 22–31, 1980.In conclusion I express my gratitude to V. N. Solev for the formulation of the problem and for his help in the preparation of the paper.     

Asymptotic behavior of waves

The asymptotic conjugation relation is established for all ƒL²(Rⁿ) under mild assumptions on and g, where denotes Fourier multiplication. The asymptotic estimate for finite energy solutions u of the wave equation is deduced from (*), along with generalizations to a class of first-order symmetric hyperbolic systems of partial differential equations that are homogeneous and constant coefficient, and a weakened version for the Klein-Gordon equation. Also deduced from (*) is the fact that for a free Schrödinger particle the probability of being in the set tA at time t tends to the probability that the velocity is in A as t → ±∞.     

Asymptotic behavior of the prediction error

Let {X_n} _– be a random process, stationary in the broad sense, with spectral density f() satisfying the singularity condition: · We denote _n ² the mean square prediction error at the prediction of _o by linear forms in X_–1, ... , X_–n. In the paper one investigates the rate of decrease of _n to zero.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 130, pp. 11–24, 1983.In conclusion, the author wishes to express his gratitude to I. A. Ibragimov for his constant interest and help.