EIGENFUNCTION EXPANSIONS ASSOCIATED WITH THE LAPLACIAN FOR CERTAIN DOMAINS WITH INFINITE BOUNDARIES.
Charles Irwin Goldstein · OSTI OAI (U.S. Department of Energy Office of Scientific and Technical Information) · 1967
The purpose of this dissertation is to prove an expansion theorem for the negative Laplacian -Δ acting on functions with zero boundary conditions in a perturbed semi-infinite cylinder in N-dimensional Euclidean space (N ≥ 2). The case of an unperturbed cylinder is discussed first. The spectral multiplicity is completely determined and a complete, orthogonal set of general eigenfunction is explicitly given. The finite end of the cylinder is then subjected to certain perturbations. Two sets of generalized eigenfunctions are constructed for the perturbed cylinder. They are constructed using the method of limiting absorption. The behavior of these eigenfunctions at infinity is described by means of a radiation condition analogous to the Sommerfeld radiation condition for the exterior problem. These functions are shown to form a complete, orthogonal set of generalized eigenfunctions for certain types of perturbed cylinders. Finally, it is shown for a wider class of domains that there can be at most a finite number of linearly independent generalized eigenfunctions associated with each real number.