Properties of eigenvalues and spectral singularities for impulsive quadratic pencil of difference operators

In this paper, we investigate the spectral analysis of impulsive quadratic pencil of difference operators. We first present a boundary value problem consisting one interior impulsive point on the whole axis corresponding to the above mentioned operator. After introducing the solutions of impulsive quadratic pencil of difference equation, we obtain the asymptotic equation of the function related to the Wronskian of these solutions to be helpful for further works, then we determine resolvent operator and continuous spectrum. Finally, we provide sufficient conditions guarenteeing finiteness of eigenvalues and spectral singularities by means of uniqueness theorems of analytic functions. The main aim of this paper is demonstrating the impulsive quadratic pencil of difference operator is of finite number of eigenvalues and spectral singularities with finite multiplicities which is an uninvestigated problem proposed in the literature.     

EIGENFUNCTION EXPANSION FOR STURM-LIOUVILLE PROBLEMS WITH TRANSMISSION CONDITIONS AT ONE INTERIOR POINT

The purpose of this article is to extend some spectral properties of regular SturmLiouville problems to the special type discontinuous boundaxy-value problem,which consists of a Sturm-Liouville equation together with eigenparameter-dependent boundary conditions and two supplementary transmission conditions.We construct the resolvent operator and Green's function and prove theorems about expansions in terms of eigenfunctions in modified Hilbert space L_2[a,b].     

Asymptotic and spectral properties of operator‐valued functions generated by aircraft wing model

The present paper is devoted to the asymptotic and spectral analysis of an aircraft wing model in a subsonic air flow. The model is governed by a system of two coupled integro‐differential equations and a two parameter family of boundary conditions modelling the action of the self‐straining actuators. The differential parts of the above equations form a coupled linear hyperbolic system; the integral parts are of the convolution type. The system of equations of motion is equivalent to a single operator evolution–convolution equation in the energy space. The Laplace transform of the solution of this equation can be represented in terms of the so‐called generalized resolvent operator, which is an operator‐valued function of the spectral parameter. More precisely, the generalized resolvent is a finite‐meromorphic function on the complex plane having a branch‐cut along the negative real semi‐axis. Its poles are precisely the aeroelastic modes and the residues at these poles are the projectors on the generalized eigenspaces. The dynamics generator of the differential part of the system has been systematically studied in a series of works by the second author. This generator is a non‐selfadjoint operator in the energy space with a purely discrete spectrum. In the aforementioned series of papers, it has been shown that the set of aeroelastic modes is asymptotically close to the spectrum of the dynamics generator, that this spectrum consists of two branches, and a precise spectral asymptotics with respect to the eigenvalue number has been derived. The asymptotical approximations for the mode shapes have also been obtained. It has also been proven that the set of the generalized eigenvectors of the dynamics generator forms a Riesz basis in the energy space. In the present paper, we consider the entire integro‐differential system which governs the model. Namely, we investigate the properties of the integral convolution‐type part of the original system. We show, in particular, that the set of poles of the adjoint generalized resolvent is asymptotically close to the discrete spectrum of the operator that is adjoint to the dynamics generator corresponding to the differential part. The results of this paper will be important for the reconstruction of the solution of the original initial boundary‐value problem from its Laplace transform and for the analysis of the flutter phenomenon in the forthcoming work. Copyright © 2004 John Wiley & Sons, Ltd.     

B-Fredholm Properties of Closed Invertible Operators

In this paper, we study B-Fredholm spectral properties of an invertible closed linear operator in relation with the B-Fredholm spectral properties of its bounded inverse. Precisely, for such operator, we characterize its B-Fredholm spectrum and other related spectra in terms of the corresponding spectra of its bounded inverse. As an application, we show that every normal operator with nonempty resolvent set, in particular self-adjoint Schrödinger operators, satisfies generalized Weyl’s theorem.

     

N-group neutron transport theory: A criticality problem in slab geometry

The steady-state equation for N-group neutron transport in slab geometry is written as an integral equation. A spectral analysis is made of the integral operator and related to the criticality problem. The method depends on a representation for the resolvent kernel for a subcritical slab and on analytic continuation in a complex parameter to characterize eigenvalues in terms of singularities of the resolvent. The analytic continuation is based on a bifurcation analysis of some nonlinear matrix integral equations whose solutions provide a matrix Wiener-Hopf factorization of the Fourier transform of the kernel of the transport operator.     

Asymptotic analysis of aircraft wing model in subsonic air flow

This paper is the first in a series of several works devotedto the asymptotic and spectral analysis of an aircraft wingin a subsonic air flow. This model has been developed in theFlight Systems Research Center of UCLA and is presented in theworks of Balakrishnan. The model is governed by a system oftwo coupled integro-differential equations and a two parameterfamily of boundary conditions modelling the action of the self-strainingactuators. The unknown functions (the bending and the torsionangle) depend on time and one spatial variable. The differentialparts of the above equations form a coupled linear hyperbolicsystem; the integral parts are of convolution type. The systemof equations of motion is equivalent to a single operator evolution–convolutiontype equation in the state space of the system equipped withthe so-called energy metric. The Laplace transform of the solutionof this equation can be represented in terms of the so-calledgeneralized resolvent operator. The generalized resolvent operatoris an operator-valued function of the spectral parameter. Thisgeneralized resolvent operator is a finite meromorphic functiondefined on the complex plane having the branch cut along thenegative real semi-axis. The poles of the generalized resolventare precisely the aeroelastic modes, and the residues at thesepoles are the projectors on the generalized eigenspaces. Inthis paper, our main object of interest is the dynamics generatorof the differential parts of the system. It is a non-selfadjointoperator in the state space with a pure discrete spectrum. Inthe present paper, we show that the spectrum consists of twobranches, and we derive their precise spectral asymptotics.Based on these results, in the next paper we will derive theasymptotics of the aeroelastic modes and approximations forthe mode shapes.     

单调函数在具有可选服务的M/M/1排队模型研究中的应用

运用函数的导数与单调性之间的关系证明具有可选服务的M/M/1排队模型的主算子的豫解集研究中出现的三个不等式.由此推出,在虚轴上除了零外其它所有点都属于该模型的主算子的豫解集.     

SchröDinger operators - geometric estimates in terms of the occupation time

The difference of Schrödinger and Dirichlet semigroups is expressed in terms of the Laplace transform of the Brownian motion occupation time. This implies quantitative upper and lower bounds for the operator norms of the corresponding resolvent differences. One spectral theoretical consequence is an estimate for the eigenfunction for a Schrödinger operator in a ball where the potential is given as a cone indicator function.     

Spectral analysis of nonselfadjoint Schrdinger problem with eigenparameter in the boundary condition

In this paper we consider the nonselfadjoint (dissipative) Schrodinger boundary value problem in the limit-circle case with an eigenparameter in the boundary condition. Since the boundary conditions are nonselfadjoint, the approach is based on the use of the maximal dissipative operator, and the spectral analysis of this operator is adequate for the boundary value problem. We construct a selfadjoint dilation of the maximal dissipative operator and its incoming and outgoing spectral representations, which make it possible to determine the scattering matrix of the dilation. We construct a functional model of the maximal dissipative operator and define its characteristic function in terms of solutions of the corresponding Schrodinger equation. Theorems on the completeness of the system of eigenvectors and the associated vectors of the maximal dissipative operator and the Schrodinger boundary value problem are given.     

On the eigenfunction expansion of the Laplace–Beltrami operator in hyperbolic space

We describe the spectral projection of the Laplace–Beltrami operator in n-dimensional hyperbolic space by studying its resolvent as an analytic operator-valued function and applying the technique of contour integration. As a result an integral formula is established for the associated Legendre function     

Local spectral theory for normal operators in Krein spaces

Sign type spectra are an important tool in the investigation of spectral properties of selfadjoint operators in Krein spaces. It is our aim to show that also sign type spectra for normal operators in Krein spaces provide insight in the spectral nature of the operator: If the real part and the imaginary part of a normal operator in a Krein space have real spectra only and if the growth of the resolvent of the imaginary part (close to the real axis) is of finite order, then the normal operator possesses a local spectral function defined for Borel subsets of the spectrum which belong to positive (negative) type spectrum. Moreover, the restriction of the normal operator to the spectral subspace corresponding to such a Borel subset is a normal operator in some Hilbert space. In particular, if the spectrum consists entirely out of positive and negative type spectrum, then the operator is similar to a normal operator in some Hilbert space. We use this result to show the existence of operator roots of a class of quadratic operator polynomials with normal coefficients.     

Resolvent Operator and Self-Adjointness of Sturm–Liouville Operators with a Finite Number of Transmission Conditions

In this paper, we consider a Sturm–Liouville operator with eigenparameter-dependent boundary conditions and transmission conditions at a finite number of interior points. We introduce a Hilbert space formulation such that the problem under consideration can be interpreted as an eigenvalue problem for a suitable self-adjoint linear operator. We construct Green function of the problem and resolvent operator. We establish the self-adjointness of the discontinuous Sturm–Liouville operator.     

Analyse spectrale et singularités d'un problème de transmission non coercif

This Note is devoted to the spectral analysis of an unbounded operator associated with a non-coercive transmission problem. Using an integral equation method, we show that, if the interface is regular, this operator is selfadjoint and has compact resolvent. If the interface has a corner, the study of the singularities using Mellin transform allows us to derive a necessary and sufficient condition on the contrast between the two media for selfadjointness. If the operator is not selfadjoint, a characterization of its selfadjoint extensions is given.     

Spectral analysis of nonselfadjoint Schrödinger problem with eigenparameter in the boundary condition

In this paper we consider the nonselfadjoint (dissipative) Schrödinger boundary value problem in the limit-circle case with an eigenparameter in the boundary condition. Since the boundary conditions are nonselfadjoint, the approach is based on the use of the maximal dissipative operator, and the spectral analysis of this operator is adequate for the boundary value problem. We construct a selfadjoint dilation of the maximal dissipative operator and its incoming and outgoing spectral representations, which make it possible to determine the scattering matrix of the dilation. We construct a functional model of the maximal dissipative operator and define its characteristic function in terms of solutions of the corresponding Schrödinger equation. Theorems on the completeness of the system of eigenvectors and the associated vectors of the maximal dissipative operator and the Schrödinger boundary value problem are given.     

Spectral analysis of nonselfadjoint Schr(o)dinger problem with eigenparameter in the boundary condition

In this paper we consider the nonselfadjoint (dissipative) Schr(o)dinger boundary value problem in the limit-circle case with an eigenparameter in the boundary condition. Since the boundary conditions are nonselfadjoint, the approach is based on the use of the maximal dissipative operator,and the spectral analysis of this operator is adequate for the boundary value problem. We construct a selfadjoint dilation of the maximal dissipative operator and its incoming and outgoing spectral representations, which make it possible to determine the scattering matrix of the dilation. We construct a functional model of the maximal dissipative operator and define its characteristic function in terms of solutions of the corresponding Schr(o)dinger equation. Theorems on the completeness of the system of eigenvectors and the associated vectors of the maximal dissipative operator and the Schr(o)dinger boundary value problem are given.     

A Characterization of Periodic Resolvent Operators

We characterize in terms of spectral properties those resolvent operator families which are periodic. This result extends and summarizes analogous results for C₀-semigroups and strongly continuous cosine operator functions.     

Asymptotic Representations for Root Vectors of Nonselfadjoint Operators and Pencils Generated by an Aircraft Wing Model in Subsonic Air Flow

This paper is the second in a series of several works devoted to the asymptotic and spectral analysis of an aircraft wing in a subsonic air flow. This model has been developed in the Flight Systems Research Center of UCLA and is presented in the works by A. V. Balakrishnan. The model is governed by a system of two coupled integrodifferential equations and a two parameter family of boundary conditions modeling the action of the self-straining actuators. The differential parts of the above equations form a coupled linear hyperbolic system; the integral parts are of the convolution type. The system of equations of motion is equivalent to a single operator evolution-convolution equation in the energy space. The Laplace transform of the solution of this equation can be represented in terms of the so-called generalized resolvent operator, which is an operator-valued function of the spectral parameter. This generalized resolvent operator is a finite-meromorphic function on the complex plane having the branch cut along the negative real semi-axis. Its poles are precisely the aeroelastic modes and the residues at these poles are the projectors on the generalized eigenspaces. In the first paper and in the present one, our main object of interest is the dynamics generator of the differential parts of the system. It is a nonselfadjoint operator in the energy space with a purely discrete spectrum. In the first paper, we have shown that the spectrum consists of two branches and have derived their precise spectral asymptotics. In the present paper, we derive the asymptotical approximations for the mode shapes. Based on the asymptotical results of these first two papers, in the next paper, we will discuss the geometric properties of the mode shapes such as minimality, completeness, and the Riesz basis property in the energy space.     

A new concept for block operator matrices:the quadratic numerical range

In this paper a new concept for 2×2-block operator matrices – the quadratic numerical range – is studied. The main results are a spectral inclusion theorem, an estimate of the resolvent in terms of the quadratic numerical range, factorization theorems for the Schur complements, and a theorem about angular operator representations of spectral invariant subspaces which implies e.g. the existence of solutions of the corresponding Riccati equations and a block diagonalization. All results are new in the operator as well as in the matrix case.     

Solution of the Cauchy problem for the Boiti-Leon-Pempinelli equation

The Cauchy problem for the (2+1)-dimensional nonlinear Boiti-Leon-Pempinelli (BLP) equation is studied within the framework of the inverse problem method. Evolution equations generated by the system of BLP equations under study are derived for the resolvent, Jost solutions, and scattering data for the two-dimensional Klein-Gordon differential operator with variable coefficients. Additional conditions on the scattering data that ensure the stability of the solutions to the Cauchy problem are revealed. A recurrence procedure is suggested for constructing the polynomial integrals of motion and the generating function for these integrals in terms of the spectral data.Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 109, No. 2, pp. 163–174, November, 1996.