Spectral properties of matrix polynomials in the max algebra

We consider the spectral properties of matrix polynomials over the max algebra. In particular, we show how the Perron-Frobenius theorem for the max algebra extends to such polynomials and illustrate the relevance of this for multistep difference equations in the max algebra. We also present a number of inequalities for the largest max eigenvalue of a matrix polynomial.     

Analytic properties of the scattering matrix of many particle systems

We generalize the previous results on a meromorphic continuation of the scattering matrix for N particle systems. Our proofs seem to be much simpler than those given before.Dedicated to the dear memory of david Milman     

Spectral properties of the linear multiple scattering operator in L1-Banach lattices

In this article the spectrum of the linear transport operator for a scattering system is studied. Because only positive solutions of the corresponding transport equation are physically meaningful, the concepts of Banach lattices and positive operators are used. The spectrum contains information about the asymptotic behavior for large times of the transport system.     

Spectral decomposition of a matrix using the generalized sign matrix

An algorithm for spectral decomposition is presented which does not require knowledge of eigenvalues and eigenvectors. A set of eigenprojectors are defined which covers the entire spectrum of a matrix, and special attention is given to the projection on the zero eigenvalue. Some useful applications are discussed in the paper.     

Spectral analysis of the fourth moment matrix

The fourth moment of a random vector is a matrix whose elements are all moments of order four which can be obtained from the random vector itself. In this paper, we give a lower bound for its dominant eigenvalue and show that its eigenvectors corresponding to positive eigenvalues are vectorized symmetric matrices. Fourth moments of standardized and exchangeable random vectors are examined in more detail.     

Asymptotic behavior of the spectrum of the scattering matrix

For a fixed value of the spectral parameter (energy) one derives the asymptotic behavior with respect to a large index of the eigenvalue of the scattering matrix. The role of the unperturbed operator is played by a pseudodifferential operator (PDO) with constant coefficients in L²(^m). The perturbed operator is obtained by the addition of a differential expression whose coefficients behave asymptotically at infinity as homogeneous functions of order. One considers also perturbations of the type medium perturbation. On the initial PDO and on the perturbations one imposes certain restrictions. The problem reduces to the investigation of the asymptotics of the spectrum of some PDO of order on a surface of constant energy. Typical examples are: different variants of the Schrodinger equations, the Dirac system, the Maxwell system.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 110, pp. 3–29, 1981.     

Spectral properties of the twisted bi-Laplacian

The eigenvalues and eigenfunctions of the twisted bi-Laplacian are studied in the context of analytic number theory. The essential self-adjointness and the global hypoellipticity in terms of a new family of Sobolev spaces are also studied.     

Note on the distribution of poles of the scattering matrix

Trapping obstacles, having at least one isolated multiple reflecting trapping ray, are considered. It is proved that for every ε > 0 there are poles λ_j of the scattering matrix such that $\text{0 <}\text{Im}\text{λ}{}_{\mathrm{j}}\text{< ε}\text{Log}\text{|λ}{}_{\mathrm{j}}\text{¦}$.     

The scattering matrix and the differences of spectral projections

In the scattering theory framework, we point out a connectionbetween the spectrum of the scattering matrix of two operatorsand the spectrum of the difference of spectral projections ofthese operators.     

四分Cantor测度的谱性质

自仿测度μM,D谱性质的研究始于四分Cantor测度μ4(即M=4,D={0,2}的情形).在长期从事谱集研究的基础上,Jorgensen和Pedersen在1998年首次发现μ4是一个具有谱性质的分形测度,其谱Λ(M,S)与和谐对(M~(-1)D,S)密切相关,其中S={0,1}.近年来的研究表明,对于某些奇数l,数乘集合lΛ(M,S)也是测度μ4的谱.这使得测度μ4的一些谱具有较强的稀疏性.本文重点对具有上述性质的奇数l进行讨论.利用数论中同余关系和有限群中元素的阶的性质,得到当l分别为素数、素数幂和素数乘积时,lΛ(M,S)为谱的判别依据,改进推广Dutkay等人的工作.     

Spectral properties of the generalized Friedrichs model

We consider the self-adjoint generalized Friedrichs model with small values of the “coupling parameter.” In this case, we completely investigate the spectrum of the model and the structure of its eigenvectors (both ordinary and generalized). The constructions we use are based on an analysis of the resolvent of the Friedrichs operator and on the corresponding scattering theory.     

Computation of waveguide scattering matrix near thresholds

A waveguide occupies infinite strip with one or several narrows on a two-dimensional (2D) plane and is governed by the Helmholtz equation with Dirichlet boundary condition. On the waveguide continuous spectrum, which coincides with a half-axis, a scattering matrix is defined. At each point of the continuous spectrum this matrix has finite size, which changes at thresholds. The thresholds form a sequence of positive numbers increasing to infinity. Approximate calculation of the scattering matrix in a threshold vicinity requires special treatment. We discuss and compare two methods of numerical approximation to the scattering matrix near a threshold.     

Spectral properties of the neutron transport equation

The general equation describing the steady-state flow through a porous column is λu ? D_xA(D_x?(u) + G(u)) = f, where λ is a nonnegative constant. In this paper existence, uniqueness and comparison results for solutions to the Dirichlet and mixed boundary value problems associated with this equation are proven. The existence of a weak solution to the evolution problems associated with the equation u_t = D_x(D_x?(u) + G(u)) are deduced.