Pencils of waveguide type and related extremal problems

The variational properties of the spectra of a class of quadratic pencils are investigated. These operator pencils are not strongly damped, which is expressed in a considerable manner in its properties. The obtained results are fundamental in the investigation of two-parameter pencils of waveguide type, which model pencils arising in the theory of regular waveguides. The considerable difficulties, arising at the investigation of pencils of waveguide type, are explained by the fact that they do not generate Rayleigh systems in the entire space, but only on certain of its nonconvex homogeneous sets. These sets occur here as the sets of the admissible vectors of the corresponding extremal problems.Translated from Problemy Matematicheskogo Analiza, No. 11, pp. 80–96, 1990.     

On the spectrum of a weak class of operator pencils of waveguide type

The paper introduces a new class of two parameter non‐overdamped operator pencils arising from evolution equations. We investigate spectral properties, including variational principles for “interior” points of the spectrum. Examples leading to pencils of the new class are given. (© 2006 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Scattering in a Forked-Shaped Waveguide

We consider wave scattering in a forked-shaped waveguide which consists of two finite and one half-infinite intervals having one common vertex. We describe the spectrum of the direct scattering problem and introduce an analogue of the Jost function. In case of the potential which is identically equal to zero on the half-infinite interval, the problem is reduced to a problem of the Regge type. For this case, using Hermite-Biehler classes, we give sharp results on the asymptotic behavior of resonances, that is, the corresponding eigenvalues of the Regge-type problem. For the inverse problem, we obtain sufficient conditions for a function to be the S-function of the scattering problem on the forked-shaped graph with zero potential on the half-infinite edge, and present an algorithm that allows to recover potentials on the finite edges from the corresponding Jost function. It is shown that the solution of the inverse problem is not unique. Some related general results in the spectral theory of operator pencils are also given. This work was supported by the grant UM1-2567-OD-03 from the Civil Research and Development Foundation (CRDF). YL was partially supported by the NSF grants 0338743, 0354339 and 0754705, by the Research Board and Research Council of the University of Missouri, and by the EU Marie Curie “Transfer of Knowledge” program.     

On spectral properties of multiparameter polynomial matrices

Spectral problems for multiparameter polynomial matrices are considered. The notions of the spectrum (including those of its finite, infinite, regular, and singular parts), of the analytic multiplicity of a point of the spectrum, of bases of null-spaces, of Jordan s-semilattices of vectors and of generating vectors, and of the geometric and complete geometric multiplicities of a point of the spectrum are introduced. The properties of the above characteristics are described. A method for linearizing a polynomial matrix (with respect to one or several parameters) by passing to the accompanying pencils is suggested. The interrelations between spectral characteristics of a polynomial matrix and those of the accompanying pencils are established. Bibliography: 12 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 229, 1995, pp. 284–321. Translated by V. B. Khazanov.     

Interpolation lattices in several variables

Principal lattices are classical simplicial configurations of nodes suitable for multivariate polynomial interpolation in n dimensions. A principal lattice can be described as the set of intersection points of n + 1 pencils of parallel hyperplanes. Using a projective point of view, Lee and Phillips extended this situation to n + 1 linear pencils of hyperplanes. In two recent papers, two of us have introduced generalized principal lattices in the plane using cubic pencils. In this paper we analyze the problem in n dimensions, considering polynomial, exponential and trigonometric pencils, which can be combined in different ways to obtain generalized principal lattices.We also consider the case of coincident pencils. An error formula for generalized principal lattices is discussed. Partially supported by the Spanish Research Grant BFM2003-03510, by Gobierno de Aragón and Fondo Social Europeo.     

Cremona transformations and the conundrum of dimensionality and signature of macro-spacetime

The issue of dimensionality and signature of the observed universe is analysed. Neither of the two properties follows from first principles of physics, save for a remarkably fruitful Cantorian fractal spacetime approach pursued by El Naschie, Nottale and Ord. In the present paper, the author's theory of pencil-generated spacetime(s) is invoked to provide a clue. This theory identifies spatial coordinates with pencils of lines and the time dimension with a specific pencil of conics. Already its primitive form, where all pencils lie in one and the same projective plane, implies an intricate connection between the observed multiplicity of spatial coordinates and the (very) existence of the arrow of time. A qualitatively new insight into the matter is acquired, if these pencils are not constrained to be coplanar and are identified with the pencils of fundamental elements of a Cremona transformation in a projective space. The correct dimensionality of space (3) and time (1) is found to be uniquely tied to the so-called quadro-cubic Cremona transformations – the simplest non-trivial, non-symmetrical Cremona transformations in a projective space of three dimensions. Moreover, these transformations also uniquely specify the type of a pencil of fundamental conics, i.e. the global structure of the time dimension. Some physical and psychological implications of these findings are mentioned, and a relationship with the Cantorian model is briefly discussed.     

Inverse spectral problems for differential pencils with boundary conditions dependent on the spectral parameter

In this paper, we discuss two inverse problems for differential pencils with boundary conditions dependent on the spectral parameter. We will prove the Hochstadt–Lieberman type theorem of 1 – 3 except for arbitrary one eigenvalue and the Borg type theorem of 1 – 3 except for at most arbitrary two eigenvalues, respectively. The new results are generalizations of the related results. Copyright © 2016 John Wiley & Sons, Ltd.     

Determination of differential pencils with spectral parameter dependent boundary conditions from interior spectral data

Second‐order differential pencils L(p,q,h₀,h₁,H₀,H₁) on a finite interval with spectral parameter dependent boundary conditions are considered. We prove the following: (i) a set of values of eigenfunctions at the mid‐point of the interval [0,π] and one full spectrum suffice to determine differential pencils L(p,q,h₀,h₁,H₀,H₁); and (ii) some information on eigenfunctions at some an internal point and parts of two spectra suffice to determine differential pencils L(p,q,h₀,h₁,H₀,H₁). Copyright © 2013 The Authors. Mathematical Methods in the Applied Sciences published by John Wiley & Sons, Ltd.     

Spectral sequences for quadratic pencils and the inverse spectral problem for the damped wave equation

We give necessary conditions for a sequence of complex numbers closed under complex conjugation to be the spectrum of the weakly damped wave operator. These restrictions are a consequence of some new results on spectral sequences for Hermitian quadratic pencils which are based on Weyl's and Mirsky's classical eigenvalue inequalities. In the case of finite-dimensional weakly damped pencils our conditions are both necessary and sufficient. We also obtain some conditions for overdamped pencils of degree m, and show that some of the inequalities that have to be satisfied in the weakly damped case are now reversed.     

The Schrödinger operator in a periodic waveguide on a plane and quasiconformal mappings

We consider the magnetic Schrödinger operator with a variable metric in a two-dimensional simply connected periodic waveguide. All the coefficients are assumed to be periodic along the waveguide. We investigate the Dirichlet and Neumann boundary problems, as well as the boundary problem of the third type. Under wide conditions on the boundary of the waveguide providing a band structure of the spectrum, we prove the absolute continuity of the spectrum. Bibliography: 16 titles.Dedicated to Academician O. A. Ladyzhenskaya on the occasion of her jubilee__________Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 295, 2003, pp. 204–243.     

The resolvent of a periodic optical waveguide the continuous spectrum

We consider a two-dimensional optical waveguide with periodic media interface. We study the resolvent of the waveguide in a neighborhood of a purely imaginary point of the spectral parameter. We prove that the resolvent exists on the subspace of functions orthogonal ina certain sense to the singular functions of the continuous spectrum. Bibliography: 7 titles. Translated fromProblemy Matematicheskogo Analiza, No. 13, 1992, pp. 79–89.     

On the parametric eigenvalue behavior of matrix pencils under rank one perturbations

We study the eigenvalues of rank one perturbations of regular matrix pencils depending linearly on a complex parameter. We prove properties of the corresponding eigenvalue sets including a convergence result as the parameter tends to infinity and an eigenvalue interlacing property for real valued pencils having real eigenvalues only. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Methods for solving spectral problems for multiparameter matrix pencils

Methods for solving the partial eigenproblem for multiparameter regular pencils of real matrices, which allow one to improve given approximations of an eigenvector and the associated point of the spectrum (both finite and infinite) are suggested. Ways of extending the methods to complex matrices, polynomial matrices, and coupled multiparameter problems are indicated. Bibliography: 10 titles.__________Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 296, 2003, pp. 139–168.     

Invariant subspaces of a lightguide with periodic boundary

A waveguide operator is defined. It is proved that its spectrum coincides with the spectrum of a lightguide. The classification of singular points of the continuous spectrum is given. Invariant subspaces of the waveguide operator are distinguished that are related to an interval of the continuous spectrum without singular points. Bibliography: 9 titles. Translated fromProblemy Matematicheskogo Analiza, No. 14, 1995, pp. 51–62.     

Asymptotic behavior of spectral gaps in a regularly perturbed periodic waveguide

Since the spectrum of a periodic waveguide is the union of a countable family of closed bounded segments (spectral bands), it can contain opened spectral gaps, i.e., intervals in the real positive semi-axis that are free of the spectrum but have both endpoints in it. A cylindrical waveguide has an intact spectrum that is a closed ray. We consider a small periodic perturbation of the waveguide wall, and, by means of an asymptotic analysis of the eigenvalues in the model problem on the periodicity cell, we show how a spectral gap opens when the cylindrical waveguide converts into a periodic one. Indeed, a cylindrical waveguide can be interpreted as a periodic one with an arbitrary period, but all its spectral bands touch each other. A periodic perturbation of the waveguide wall provides the splitting of the band edges. This effect is known in the physical literature for waveguides of different shapes, and, in this paper, we provide a rigorous mathematical proof of the effect. Several variants of the edge splitting (alone and coupled, simple and multiple knots) are examined. Explicit formulas are obtained for a plane waveguide.     

To solving spectral problems for <Emphasis Type="Italic">q</Emphasis>-parameter polynomial matrices

The method of hereditary pencils, originally suggested by the author for solving spectral problems for two-parameter matrices (pencils of matrices), is extended to the case of q-parameter, q ≥ 2, polynomial matrices. Algorithms for computing points of the finite regular and singular spectra of a q-parameter polynomial matrix and their theoretical justification are presented. Bibliography: 2 titles.     

Analysis of parameterized quadratic eigenvalue problems in computational acoustics with homotopic deviation theory

This paper analyzes a family of parameterized quadratic eigenvalue problems from acoustics in the framework of homotopic deviation theory. Our specific application is the acoustic wave equation (in 1D and 2D) where the boundary conditions are partly pressure release (homogeneous Dirichlet) and partly impedance, with a complex impedance parameter ζ. The admittance t = 1/ζ is the classical homotopy parameter. In particular, we study the spectrum when t → ∞. We show that in the limit part of the eigenvalues remain bounded and converge to the so‐called kernel points. We also show that there exist the so‐called critical points that correspond to frequencies for which no finite value of the admittance can cause a resonance. Finally, the physical interpretation that the impedance condition is transformed into a pressure release condition when |t| → ∞ enables us to give the kernel points in closed form as eigenvalues of the discrete Dirichlet problem. Copyright © 2006 John Wiley & Sons, Ltd.     

Orthogonality of eigenwaves in a semi-open elastic waveguide

We consider a two-layer elastic waveguide structure such that its one layer is unbounded in the lateral direction, and the standard boundary conditions are stated on the boundary of another one. We study several kinds of eigenwaves for this structure. In general, they have a complex-valued longitudinal propagation constant (the spectral parameter). On the base of the Green formula we introduce the scalar product of two waves and prove that the system of eigenwaves of the semiopen elastic waveguide is orthogonal. We construct families of waves which belong to discrete and continuous parts of the spectrum.     

Basis property of eigenfunctions of nonselfadjoint operator pencils generated by the equation of nonhomogeneous damped string

We consider a class of nonselfadjoint quadratic operator pencils generated by the equation, which governs the vibrations of a string with nonconstant bounded density subject to viscous damping with a nonconstant damping coefficient. These pencils depend on a complex parameterh, which enters the boundary conditions. Depending on the values ofh, the eigenvalues of the above pencils may describe the resonances in the scattering of elastic waves on an infinite string or the eingenmodes of a finite string. We obtain the 7asymptotic representations for these eigenvalues. Assuming that the proper multiplicity of each eigenvalue is equal to one, we prove that the eigenfunctions of these pencils form Riesz bases in the weightedL ²-space, whose weight function is exactly the density of the string. The general case of multiple eigenvalues will be treated in another paper, based on the results of the present work.     

On spectra of a certain class of quadratic operator pencils with one-dimensional linear part

We consider a class of quadratic operator pencils that occur in many problems of physics. The part of such a pencil linear with respect to the spectral parameter describes viscous friction in problems of small vibrations of strings and beams. Patterns in the location of eigenvalues of such pencils are established. If viscous friction (damping) is pointwise, then the operator in the linear part of the pencil is one-dimensional. For this case, rules in the location of purely imaginary eigenvalues are found. Published in Ukrains’kyi Matematychnyi Zhurnal, Vol. 59, No. 5, pp. 702–716, May, 2007.