Derivation and application of extended parabolic wave theories. II. Path integral representations

The n‐dimensional reduced scalar Helmholtz equation for a transversely inhomogeneous medium is naturally related to parabolic propagation models through (1) the n‐dimensional extended parabolic (Weyl pseudodifferential) equation and (2) an imbedding in an (n+1)‐dimensional parabolic (Schrodinger) equation. The first relationship provides the basis for the parabolic‐based Hamiltonian phase space path integral representation of the half‐space propagator. The second relationship provides the basis for the elliptic‐based path integral representations associated with Feynman and Fradkin, Feynman and Garrod, and Feynman and DeWitt‐Morette. Exact and approximate path integral constructions are derived for the homogeneous and transversely inhomogeneous cases corresponding to both narrow‐ and wide‐angle extended parabolic wave theories. The path integrals allow for a global perspective of the transition from elliptic to parabolic wave theory in addition to providing a unifying framework for dynamical approximation...

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