Simulation of Wave Focusing in Cavities

Simulation results are presented for the focusing of shock waves in conical and wedge-shaped cavities. Results are reported for plane waves and explosion waves. The distribution of parameters in the focusing zone is found to depend on the specific conditions.     

Absolutely continuous spectrum of a nondissipative operator and the functional model. I

In the paper one investigates nondissipative operators L in a Hilbert space; for them one constructs a model representation similar to the Nagy-Foias model for dissipative operators. In this representation one succeeds to calculate the action of the resolvent of a nondissipative operator on selected subspaces. This allows us to relinquish the consideration of the -self-adjoint dilation of the operator, whose spectral representation involves considerable difficulties. Isolated results are new also for the dissipative case which is not excluded. In part I one considers the “triangular” factorization of the characteristic funcion of the operator L and one carries out the proof of the fundamental theorem which gives a formula for the calculation of (L-No)⁻¹(J_mN_oO T_mN_o<O) on selected subspaces. The applications of this theorem to the spectral analysis on the absolutely continuous spectrum and to the problems of linear similitude for nondissipative operators are considered in part II. Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Ihstituta im. V. A. Steklova AN SSSR, Vol. 65, pp. 90–102, 1976.     

Simplicity of eigenvalues in Anderson-type models

We show almost sure simplicity of eigenvalues for several models of Anderson-type random Schrödinger operators, extending methods introduced by Simon for the discrete Anderson model. These methods work throughout the spectrum and are not restricted to the localization regime. We establish general criteria for the simplicity of eigenvalues which can be interpreted as separately excluding the absence of local and global symmetries, respectively. The criteria are applied to Anderson models with matrix-valued potential as well as with single-site potentials supported on a finite box.     

Jacobi matrices with absolutely continuous spectrum

Let be a Jacobi matrix defined in as , where is a unilateral weighted shift with nonzero weights such that Define the seqences: If and , then has an absolutely continuous spectrum covering . Moreover, the asymptotics of the solution is also given.

     

Study of three-dimensional wave structure of nonstationary gas outflow from a planar sonic nozzle

Results are reported of an experimental study of the three-dimensional structure of nonstationary gas outflow from a planar nozzle. Outflow of a heated shock wave in a nitrogen tube at different moments of time from the start of outflow (0–1 msec) in two mutually perpendicular directions is considered. A scheme for reconstructing the flow at different outflow stages is proposed. The dimensions of the Riemann wave are found to oscillate.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 1, pp. 41–45, January–February, 1976.     

Unbounded Jacobi Matrices with a Few Gaps in the Essential Spectrum: Constructive Examples

We give explicit examples of unbounded Jacobi operators with a few gaps in their essential spectrum. More precisely a class of Jacobi matrices whose absolutely continuous spectrum fills any finite number of bounded intervals is considered. Their point spectrum accumulates to +?? and ???. The asymptotics of large eigenvalues is also found.     

The Inverse Resonance Problem for Jacobi Operators

It is proved in this paper that super-exponentially decaying,possibly non-selfadjoint perturbations of the free Jacobi operatorare uniquely determined by the location of all their eigenvaluesand resonances. 2000 Mathematics Subject Classification 39A70,34K29.     

Structure of the singularities of operator functions with a positive imaginary part

Leningrad State University. Translated from Funktsional'nyi Analiz i Ego Prilozheniya, Vol. 25, No. 4, pp. 1–13, October–December, 1991.     

Discrete spectrum in a critical coupling case of Jacobi matrices with spectral phase transitions by uniform asymptotic analysis

For a two-parameter family of Jacobi matrices exhibiting first-order spectral phase transitions, we prove discreteness of the spectrum in the positive real axis when the parameters are in one of the transition boundaries. To this end, we develop a method for obtaining uniform asymptotics, with respect to the spectral parameter, of the generalized eigenvectors. Our technique can be applied to a wide range of Jacobi matrices.     

Decay Bounds on Eigenfunctions and the Singular Spectrum of Unbounded Jacobi Matrices

Bounds on the exponential decay of generalized eigenfunctionsof bounded and unbounded selfadjoint Jacobi matrices in are established. Two cases are considered separatelyand lead to different results: (i) the case in which the spectralparameter lies in a general gap of the spectrum of the Jacobimatrix and (ii) the case of a lower semibounded Jacobi matrixwith values of the spectral parameter below the spectrum. Itis demonstrated by examples that both results are sharp. Weapply these results to obtain a "many barriers-type" criterionfor the existence of square-summable generalized eigenfunctionsof an unbounded Jacobi matrix at almost every value of the spectralparameter in suitable open sets. In particular, this leads toexamples of unbounded Jacobi matrices with a spectral mobilityedge, i.e., a transition from purely absolutely continuous spectrumto dense pure point spectrum.