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Discrete Spectrum (discrete + spectrum)

Distribution by Scientific Domains

Mathematics and Statistics	80%

Selected Abstracts

On the asymptotic behaviour of the discrete spectrum in buckling problems for thin plates

MATHEMATICAL METHODS IN THE APPLIED SCIENCES, Issue 7 2006
Monique Dauge
Abstract We consider the buckling problem for a family of thin plates with thickness parameter ,. This involves finding the least positive multiple , of the load that makes the plate buckle, a value that can be expressed in terms of an eigenvalue problem involving a non-compact operator. We show that under certain assumptions on the load, we have , = ,,(,2). This guarantees that provided the plate is thin enough, this minimum value can be numerically approximated without the spectral pollution that is possible due to the presence of the non-compact operator. We provide numerical computations illustrating some of our theoretical results. Copyright © 2005 John Wiley & Sons, Ltd. [source]

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

MATHEMATICAL METHODS IN THE APPLIED SCIENCES, Issue 3 2004
A. V. Balakrishnan
Abstract 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. [source]

Kinks and rotations in long Josephson junctions

MATHEMATICAL METHODS IN THE APPLIED SCIENCES, Issue 15 2001
Wolfgang Hauck
Abstract Kinks and rotations are studied in long Josephson junctions for small and large surface losses. Geometric singular perturbation theory is used to prove existence for small surface losses, while numerical continuation is necessary to handle large surface losses. A survey of the system behaviour in terms of dissipation parameters and bias current is given. Linear orbital stability for kinks is proved for small surface losses by calculating the spectrum of the linearized problem. The spectrum is split into essential spectrum and discrete spectrum. For the determination of the discrete spectrum, robustness of exponential dichotomies is used. Puiseux series together with perturbation theory for linear operators are an essential tool. In a final step, a smooth Evans function together with geometric singular perturbation theory is used to count eigenvalues. For kinks, non-linear orbital stability is shown. For this purpose, the asymptotic behaviour of a semigroup is given and the theory of centre and stable manifolds is applied. Copyright © 2001 John Wiley & Sons, Ltd. [source]

Deficiency indices and spectral theory of third order differential operators on the half line

MATHEMATISCHE NACHRICHTEN, Issue 12-13 2005
Horst Behncke
Abstract We investigate the spectral theory of a general third order formally symmetric differential expression of the form acting in the Hilbert space ,2w(a ,,). A Kummer,Liouville transformation is introduced which produces a differential operator unitarily equivalent to L . By means of the Kummer,Liouville transformation and asymptotic integration, the asymptotic solutions of L [y ] = zy are found. From the asymptotic integration, the deficiency indices are found for the minimal operator associated with L . For a class of operators with deficiency index (2, 2), it is further proved that almost all selfadjoint extensions of the minimal operator have a discrete spectrum which is necessarily unbounded below. There are however also operators with continuous spectrum. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim) [source]

Interface and confined polar optical phonons in spherical ZnO quantum dots with wurtzite crystal structure

PHYSICA STATUS SOLIDI (C) - CURRENT TOPICS IN SOLID STATE PHYSICS, Issue 11 2004
Vladimir A. Fonoberov
Abstract We derive analytically the interface and confined polar optical-phonon modes for spherical quantum dots with wurtzite crystal structure. While the frequency of confined optical phonons in zincblende nanocrystals is equal to that of the bulk crystal phonons, the confined polar optical phonons in wurtzite nanocrystals are shown to have a discrete spectrum of frequencies different from those in bulk crystal. The calculated frequencies of confined polar optical phonons in wurtzite ZnO quantum dots are found to be in excellent agreement with experimental resonant Raman scattering data. The derived analytical expression for phonon modes can facilitate interpretation of experimental data obtained for ZnO quantum dots. (© 2004 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim) [source]