Eigenfunction expansions for non self adjoint operators and separation of variables

Abstract

We consider an unbounded operator A in a Hilbert space H. We do not assume self adjointness but we assume that the inverse of A belongs to a Carleman class ℒ_p(H) with a small enough p. In addition we assume that the resolvent of A has minimal growth along the negative real axis. Then we show that the solution u of the abstract elliptic problem

$$u''(t) - A u(t) = 0, t > 0$$

with the boundary condition u(0)=x may be expanded in a series of generalized eigenfunctions of A for large values of t.

This is applied to operators arising from two point boundary value problems and leads to explicit separation of the variables for the solution of elliptic boundary value problems in polygons near the vertices.

Examples are the biharmonic (hence elasticity system) in a polygon with the usual boundary conditions even in the case of fractures. Other examples are the Stokes equations and the Stokes-Beltrami equation in special geometries. This research has been mainly motivated by various papers of D.Joseph (especially ref. [7],[8] below). The main results are presented in the short note [5].