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

Abstract

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 parameter h, which enters the boundary conditions. Depending on the values of h, the eigenvalues of the above pencils may describe the resonances in the scattering of elastic waves on an infinite string or the eigenmodes 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 weighted L2-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. 1.Introduction.In the present paper we develop the spectral analysis for a one-parameter family of nonselfadjoint quadratic operator pencils. These pencils appear in the study of the wave equation which describes a nonhomogeneous damped string. The main result of the work is the proof of the fact that the systems of the eigenvectors and associated vectors of the above mentioned pencils form unconditional bases in the corresponding Hilbert space. There exists an extensive literature on operator pencils (it can be traced, for example, through the references in the book [1]). However, known abstract results cannot be applied to the pencils we consider in this work.In the present paper we prove the above result in the case when there are no associated vectors, i.e. the proper multiplicity (see [1,2]) of each eigenvalue is equal to one. The proof of the main result in the general case will be given in another paper. This proof will be based on the results of the present work. We have considered the case of

Riesz Basis Property and Exponential Stability of Controlled Euler--Bernoulli Beam Equations with Variable Coefficients

“…This approach has been used in dealing with the beam equations with low order perturbation of variable coefficients (see [6], [7], and [8]). A similar adoption can also be found in the study of the string equations with variable coefficients, for which we refer to [9], [10], and [11] as well as the references therein.…”

“…The result greatly simplifies the procedure in establishing the Riesz basis property for systems described by discrete operators in a Hilbert space since the result eliminates the requirement of estimation of low eigenfunctions, which is rather difficult by other methods found in all previous papers [10], [11], [12].…”

Nonselfadjoint operators generated by the equation of a nonhomogeneous damped string

“…One of the main results of the above series of works (see [19,20,22,23]) is the fact that the operators A h defined in (1.11) and (1.16) provide a class of nontrivial examples of spectral operators. To explain this statement, we recall the definition of a spectral operator.…”

“…The first step of the aforementioned program (see [19,20]) consists in the derivation of the asymptotic formulas for the eigenvalues {λ n } n∈Z and eigenvectors {Φ n } n∈Z of the operator A h as |n| → ∞. (The spectrum of A h belongs to a strip in the complex plane parallel to the real axis.…”

Exponential Stabilisation of a Tree-Shaped Network of Strings With Variable Coefficients

“…Even in single-string cases, the proof is also complicated (see [17]). Herein, we mainly employ the asymptotic analysis technique (see [11]) to overcome this difficulty.…”