Phase Space Factorization and Functional Integral Methods in Direct and Inverse Scattering — Symbol Analysis

Abstract

The analysis and fast, accurate numerical computation of the wave equations of classical physics are often quite difficult for rapidly changing, multidimensional environments extending over many wavelengths. This is particularly so for environments characterized by a refractive index field with a compact region of arbitrary (n-dimensional) variability superimposed upon a transversely inhomogeneous ((n-1)-dimensional) background profile. For such environments, the entire domain is in the scattering regime, with the subsequent absence of an “asymptotically free” region. While classical, macroscopic methods have resulted in direct wave field approximations, derivations of approximate wave equations, and discrete numerical approximations, mathematicians studying linear partial differential equations have developed a sophisticated, microscopic phase space analysis centered about the theory of pseudo-differential and Fourier integral operators. In conjunction with the global functional integral techniques pioneered by Wiener (Brownian motion) and Feynman (quantum mechanics), and so successfully applied today in quantum field theory and statistical physics, the n-dimensional classical physics propagators can be both represented explicitly and computed directly. The phase space, or microscopic, methods and path (functional) integral representations provide the appropriate framework to extend homogeneous Fourier methods to inhomogeneous environments, in addition to suggesting the basis for the formulation and solution of corresponding arbitrary-dimensional nonlinear inverse problems.