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

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.