Michael E. Taylor

Potential and scattering theory on wildly perturbed domains

Abstract We study the potential, scattering, and spectral theory associated with boundary value problems for the Laplacian on domains which are perturbed in very irregular fashions. Of particular interest are problems in which a “thin” set is deleted and the behavior of the Laplace operator changes very little, and problems where many tiny domains are deleted. In the latter case the “clouds” of tiny obstacles may tend to disappear, to solidify, or to produce an intermediate effect, depending on the relative numbers and sizes of the tiny domains. These phenomena vary according to the specific boundary value problem and in many cases their behavior is contrary to crude intuitive guesses.

Scattering length and perturbations of -Δ by positive potentials

In [5], Kac and Luttinger gave an elegant connection between the scattering length of a positive potential and Brownian motion. The purpose of this paper is to develop this notion further into a tool for studying the effectiveness of such a potential, as a perturbation of --d. In the first section we define scattering length, prove Kac and Luttinger’s formula, and state a few simple properties of scattering length. In the next two sections we look for conditions on when a sequence vj of positive potentials has, in the limit, a negligible effect on --d and when it “solidifies” to a compact set K, leading to a Dirichlet problem ford on the complement of K. We also produce a two-sided bound for the lowest eigenvalue of --d + v on a bounded region Q, with Neumann conditions on ?X2, in terms of the scattering length of v. The fourth section treats an analog of a problem dealt with by Kac and Luttinger involving randomly placed potentials. In a sense this is intermediate between two special cases handled in Sections 2 and 3 and more delicate. The last section introduces several notions of regularity of a compact set, generalizing that of Stroock, and looks at the application to potential theory. For simplicity, we work on ifP only for rz > 3. Several problems investigated in this paper are analogous to problems involving obstacles investigated in [9].

Decaying states of perturbed wave equations

Abstract We study the solutions of perturbed wave equations that represent free wave motion outside some ball. When there are no trapped rays, it is shown that every solution whose total energy decays to zero must be smooth. This extends results of Rauch to the even-dimensional case and to systems having more than one sound speed. In these results, obstacles are not considered. We show that, even allowing obstacles, waves with compact spatial support cannot decay, assuming a unique continuation hypothesis. An example with obstacle is given where nonsmooth, compactly supported, decaying waves exist.

Essential Self-Adjointness of Powers of Generators of Hyperbolic Mixed Problems

Abstract The theory of hyperbolic mixed initial-boundary value problems is used to prove the essential self-adjointness of certain differential operators associated with formally symmetric boundary value problems.