Differential and Integral Equations

Multiple completeness of root vectors of a system of operator pencils and its applications

Sasun Yakubov

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Abstract

The role and importance of the Fourier method for investigation of mathematical physics problems is well known. By this method we can investigate problems for which the corresponding spectral problem is self-adjoint. By the Fourier method a solution $u(t,x)$ to the problem is written in the form of the series $u(t,x)=\sum^\infty_{k=1} C_k u_k(t,x),$ where $u_k(t,x)$ are elementary solutions of the considered problem. In the self-adjoint case the Hilbert theory of self-adjoint operators with a compact resolvent gives us information about the existence of elementary solutions and whether enough elementary solutions exist to write $u(t,x)$ in this series form. In the case when the principal part of the corresponding spectral problem is non-self-adjoint the existence of elementary solutions and convergence of the series are already problems. And this, in turn, leads to the problem of completeness of root functions. The completeness problem of the root vectors of a system of operator pencils and the root functions of elliptic boundary value problems is a subject of this paper.

Article information

Source
Differential Integral Equations, Volume 10, Number 4 (1997), 649-686.

Dates
First available in Project Euclid: 1 May 2013

Permanent link to this document
https://projecteuclid.org/euclid.die/1367438636

Mathematical Reviews number (MathSciNet)
MR1741767

Zentralblatt MATH identifier
0889.35070

Subjects
Primary: 47A56: Functions whose values are linear operators (operator and matrix valued functions, etc., including analytic and meromorphic ones)
Secondary: 35P10: Completeness of eigenfunctions, eigenfunction expansions 47F05: Partial differential operators [See also 35Pxx, 58Jxx] (should also be assigned at least one other classification number in section 47)

Citation

Yakubov, Sasun. Multiple completeness of root vectors of a system of operator pencils and its applications. Differential Integral Equations 10 (1997), no. 4, 649--686. https://projecteuclid.org/euclid.die/1367438636


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