Phase Space and Functional Integral Methods in Direct and Inverse Scattering

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)-dimensionaI) background profile. For such environments, the entire domain is in the scattering regime, with the subsequent absence of an “asymptotically free”, or homogeneous, region. For the most part, classical, “macroscopic” methods have resulted in direct wave field approximations (perturbation theory, ray-theory asymptotics, transmutation theory, modal analysis, hybrid ray-mode methods), derivations of approximate wave equations (scaling analysis, field-splitting techniques, formal operator expansions, approximation theory), and discrete numerical approximations (finite differences, finite elements, spectral methods). In the last several decades, however, mathematicians studying linear partial differential equations have developed, in the language of physicists, a sophisticated, “microscopic” phase space analysis (pseudo-differential and Fourier integral operators) [1]. 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.KeywordsPhase SpacePseudodifferential OperatorFourier Integral OperatorSymbol AnalysisPath Integral RepresentationThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.