The precise mathematical investigation of various natural phenomena is an old and difficult problem. For the special case of self-adjoint problems in mechanics and physics, the Fourier method of approximating exact solutions by elementary solutions has been used successfully for the last 200 years, and has been especially powerfully applied thanks to Hilbert's classical results. One can find this approach in many mathematical physics textbooks.

This book is the first monograph to treat systematically the general non-self-adjoint case, including all the questions connected with the completeness of elementary solutions of mathematical physics problems. In particular, the completeness problem of eigenvectors and associated vectors (root vectors) of unbounded polynomial operator pencils, and the coercive solvability and completeness of root functions of boundary value problems for both ordinary and partial differential equations are investigated.

The author deals mainly with bounded domains having smooth boundaries, but elliptic boundary value problems in tube domains, i.e. in non-smooth domains, are also considered.