Point spectrum of elliptic operators in fibered half-cylinders and the related completeness problem

Abstract

We consider an elliptic system in a half-cylinder ℝ+ × ω with coefficients constant in the direction of the axis and not necessarily smooth. We take different boundary conditions on {0} × ω and Dirichlet condition on {0}×ω. This defines a self-adjoint operatorA D. The main result in this paper is thatA D does not have eigenvalues. This answers conjecture 1.6 raised in [3]. When ω is bounded, we use this result to prove the completeness of a part of the family of eigenvectors, and associated vectors, of a corresponding operator pencils.