Asymptotics of the discrete spectrum of a perturbed Hill operator

One considers a perturbed one-dimensional Hill operator. One gives a formula for the asymptotics of the discrete spectrum in the gap of the continuous spectrum and conditions under which this formula is valid.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 147, pp. 188–189, 1985.In conclusion the author expresses his deep gratitude to his scientific adviser M. Sh. Birman for the formulation of the problem and for his constant interest in this note.     

Spectral properties of a linearized model of magnetohydrodynamics

One considers the frequency spectrum of the linear oscillations of a perfectly conducting nonviscous plasma with a nonzero pressure. It is shown that the essential spectrum for each of the harmonic modes is not empty and can consist of at most two finite intervals (or points). The discrete spectrum contains an infinite number of eigenvalues, diverging to +. One has found the first term of the asymptotics of this sequence. One shows the absence of the accumulation of the eigenvalues towards the endpoints of one of the intervals of the essential spectrum and also the absence of the eigenvalues to the left of the essential spectrum.Translated from Problemy Matematicheskogo Analiza, No. 10, pp. 195–207, 1986.In conclusion the author expresses his deep gratitude to his scientific advisor M. Sh. Birman for his help and numerous discussions and to A. E. Lifshits for consultations and for his interest in this paper.     

Asymptotic behavior of the energy levels of a quantum particle in a homogeneous magnetic field,perturbed by a decreasing electric field. I

One investigates the bound statesE _n , situated to the left of the boundary of the continuous spectrum. Under the assumption that the potential is nonpositive and has at infinity a power type asymptotics, one computes the principal term of the asymptoticsE _n as n. One does not assume the axial symmetry of the potential of the electric field.Translated from Problemy Matematicheskogo Analiza, No. 9, pp. 67–84, 1984.In conclusion, the author expresses his deep gratitude to his scientific adviser M. Sh. Birman for the formulation of the problem and for his constant interest in the paper.     

Asymptotics of the spectra of variational problems on the solutions of a homogeneous elliptic equation,under the presence of a constraint on part of the boundary

One obtains the principal term of the asymptotics of the spectrum of the ratio of two differential quadratic forms, considered in a bounded domain on the solutions of a homogeneous, regularly elliptic equation, under the presence of additional conditions on part of the boundary. This condition has the form of a pseudodifferential constraint and it is imposed on the Dirichlet data of the solutions. On the leading symbol of the corresponding pseudodifferential operator on the boundary, one imposes a nonsingularity conditioa.Translated from Problemy Matematicheskogo Analiza, No. 9, pp. 84–97, 1984.In conclusion, the author expresses her gratitude to her scientific adviser M. Sh. Birman for his assistance and also to M. Z. Solomyak for his interest in the paper and for a series of useful consultations.     

Negative spectrum of the Schrödinger operational equation

The negative part of the spectrum of Schrödinger's equation with a completely continuous operational potential is studied. Several conditions are obtained for the discreteness and boundedness of the negative spectrum, already known for the scalar and matrix cases.Translated from Matematicheskie Zametki, Vol. 7, No. 6, pp. 753–763, June, 1970.In conclusion the author wishes to express his gratitude to M. Sh.Birman for his valuable advice and his help in this work.     

Equation describing Hilbert's modular surface

One forwards the idea of obtaining the equation of a modular algebraic surface with the aid of the Hirzebruch curves.Translated from Zapiski Nuachnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 129, pp. 177–181, 1983.The author expresses his gratitude to A. I. Vinogradov for his interest in this paper.     

Finite-zone solutions in terms of automorphic functions the Kadomtsev-petviashvili equation

One obtains an expression for the finite-zone solutions of the Kadomtsev-Petviashvili equation in terms of the Poincaré theta-series.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 129, pp. 5–16, 1983.The author expresses his sincere gratitude to A. R. Its for useful discussions and for his constant interest in the paper.     

Invariance principle for functions of stationarily connected Gaussian variables

Let be a stationary Gaussian sequence,, One proves an invariance principle for. One obtains also a representation of the limiting process in the form of a stochastic integral.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 97, pp. 32–44, 1980.The author expresses his deep gratitude to Yu. A. Davydov for useful discussions.     

Spectral measure of the transition operator and harmonic functions connected with random walks on discrete groups

One considers a random walk on a countable group, determined by the measure. For amenable groups one gives an estimate of the spectral measure of the transition operator of the random walk in terms of the growth of the Følner sets, which allows us to find a lower bound for the probability of returning to the identity of inn steps. One gives an example of this estimate for the group. In the second part of the paper one obtains a lower bound for the entropy in terms of the oscillation of a nontrivial bounded -harmonic function on.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 97, pp. 102–109, 1980.In conclusion, the author expresses his deep gratitude to this scientific advisor A. M. Vershik for his constant interest and help.     

Radiation from an open resonator formed by axisymmetric,weakly curved mirrors

One considers the eigenoscillations of a resonator formed by axisymmetric, weakly curved mirrors. The constructed eigenfunctions satisfy the Dirichlet condition on the mirrors and the Sommerfeld radiation condition at infinity.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 128, pp. 158–165, 1983.In conclusion, the author wishes to express his sincere gratitude to his scientific advisor V. F. Lazutkin for the formulation of the problem and for his constant interest in the paper.     

A probabilistic remark about the Abel-Goncharov interpolation problem

One considers probability distributions related to the Poisson distribution. It is shown that the boundary problems for the Poisson process admit an interpretation in terms of the well-known Abel-Goncharov interpolation problem.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 56, pp. 5–11, 1976.In conclusion the author expresses his gratitude to N. A. Sapogov for his interest in this paper.     

What does the spectrum of an endomorphism of the disc-algebra look like?

One gives a theorem regarding the spectrum of an operator acting in a pair of Banach spaces, from which there follows the Kamowitz-Scheinberg theorem on the spectrum of an automorphism of a semisimple commutative Banach algebra. One describes some general properties of endomorphisms of uniform algebras. For the disc-algebra one gives a compactness criterion for the sum of weighted endomorphisms, one indicates the case of stability of the spectrum with respect to the perturbations of the permution, one presents new examples of spectra of endomorphisms, one describes completely the spectra of weighted endomorphisms, corresponding to an arbitrary linear-fractional mapping of the disc.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 126, pp. 55–68, 1983.     

On the discrete spectrum of a perturbed periodic Schrödinger operator

A perturbed periodic Schrödinger operator is considered. Conditions on the perturbation are obtained, under which the discrete spectrum in the gap is finite or, on the contrary, is infinite. In the latter case there are given formulas for the asymptotic behavior of the eigenvalues with respect to the index and conditions under which these formulas hold. The results are formulated in terms of a model problem.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 190, pp. 157–162, 1991.In conclusion the author expresses his gratitude to his scientific adviser M. Sh. Birman.     

Fisher's method of scoring in the problem of frequency estimation

One considers the application of Fisher's method of scoring to the solving of the problem on the estimation of the frequency of a sinusoidal signal in a mixture with white Gaussian noise. One assumes that as estimates one can use only functions from a finite number of linear functionals of the observed process. In such a situation the method of scoring is used twice: at the beginning for the construction of the consistent estimates and then for obtaining asymptotically efficient estimates.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Institute im, V. A. Steklova AN SSSR, Vol. 108, pp. 22–44, 1981.In conclusion, the author expresses his gratitude to R. Z. Khas'minskii for his constant attention to this paper and to E. M. Gabidulin for the useful discussions of the results.     

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.     

Analogues of the Kesten inequality for concentration functions and their refinements

One considers the Kawata and the Kunisawa concentration functions. For these concentration functions one proves the analogues of Kesten's inequality.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 97, pp. 15–21, 1980.In conclusion, the author expresses his gratitude to V. V. Petrov for suggesting this topic and for his constant interest in the paper.     

Detection of gross errors in observation results with the aid of the Chauvenet test

One gives the asymptotic distribution of the number N of gross observations, rejected by the Chauvenet method, for random variables with distribution function containing an unknown parameter, when as the estimator of the unknown parameter one makes use of the maximum likelihood estimator.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 153, pp. 153–159, 1986.In conclusion, I express my gratitude to I. A. Ibragimov for his interest in the problem and for his remarks.     

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.     

Singular points of the continuous spectrum of a two-dimensional periodic waveguide

One investigates the eigenfunctions of the continuous spectrum of a waveguide in the neighborhood of a special singular point of the continuous spectrum. One proves the existence of a complete collection of eigenfunctions of the continuous spectrum in the neighborhood of such a point.Translated from Problemy Matematicheskogo Analiza, No. 10, pp. 116–123, 1986.     

Asymptotic behavior of the solutions of the Helmholtz equation in the caustic shadow domain. I

One makes use of the complex ray method in order to construct the uniform asymptotics of the wave field in the shadow zone beyond the caustic for the Helmholtz equation with an analytic refraction index. The complex eikonal is obtained as a result of the analytic continuation of the eikonal equation into the two-dimensional complex coordinate space. One considers a special example.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 128, pp. 172–185, 1983.The author expresses his gratitude to V. S. Buldyrev for the discussion of the results.