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

We consider an elliptic system in a half-cylinder IR+ x omega with coeffici ents constant in the direction of the axis and not necessarily smooth. We t ake different boundary conditions on IR+ x partial derivative omega and Dir ichlet condition on (0) x omega. This defines a self-adjoint operator A(D). The main result in this paper is that AD does not have eigenvalues. This a nswers conjecture 1.6 raised in [3]. When omega is bounded, we use this res ult to prove the completeness of a part of the family of eigenvectors, and associated vectors, of a corresponding operator pencils.