Eigenfunction expansions associated with the Laplacian for certain domains with infinite boundaries. III
Charles Irwin Goldstein · Transactions of the American Mathematical Society · 1969
In the present paper we shall consider the effect of small perturbations on the spectrum of the selfadjoint operator -A associated with certain homogeneous boundary conditions in a domain S. We shall then apply these results to scattering theory.In all of the cases considered, a spectral representation for the unperturbed operator is obtained with the aid of separation of variables.In § §2-4, S will represent a wedge shaped region in two dimensional Euclidean space (F2), and the unperturbed operator A0 will be given by -A acting on functions which vanish on S (the boundary of S). (Higher dimensional cones may be treated similarly.)We shall first perturb the operator A0 by altering a finite portion of the boundary of S. We assume that the perturbed domain Q, has a sufficiently smooth boundary Ù.The new operator A is defined in the same manner as A0 with S replaced by Q.We shall prove that A is unitarily equivalent to A0 by employing the method of distorted plane waves, first used to prove an expansion theorem by Ikebe [7].This method was also used to deal with the exterior problem for the Laplacian by Shenk [11], and Shizuta [13].The author [5](2), extended this method to treat certain domains with infinite boundaries.The domains in I were perturbed infinite cylinders.In §2, we consider the unperturbed operator A0.A complete, orthonormal set of generalized eigenfunctions W° are exhibited.Employing the functions W°, we immediately obtain a spectral representation for A0.The spectrum of A0 is absolutely continuous and the spectral multiplicity is infinite at each point in the spectrum of A0.This differs from the case of an infinite cylinder, for which it was proven in I that the spectral multiplicity is nonuniform and finite throughout the spectrum.(Note that it follows from the results of §2 that the operators associated with different wedges are unitarily equivalent to each other.)