On Exponential Asymptotics for Nonseparable Wave Equations I: Complex Geometrical Optics and Connection

Abstract

Kelley’s Geometrical Optics method opens an efficient approach to the study of waves when the variables in the wave equation cannot be separated. The trajectories of the Hamilton-Jacobi equation are then needed for access to the wave structure. When several disjoint potential wells are present, account of the interaction between them requires consideration of trajectories and tori in complex phase space and of the unfamiliar connectivity of such complex toroids. It is shown how connection theory across wells and barriers can be reformulated in a manner that is simple and flexible and assures the rigorous reliability needed for exponentially precise approximation to functionals of the Hamiltonian operator in several dimensions. To make this difficult and unfamiliar subject more accessible, its discussion is preceded by a simple example of a double well on a strip domain in two dimensions. Parts II and III will apply the results to the quantitative resolution of degeneracy of discrete and continuous spectra, respectively, for genuine partial differential equations.