Exact constructions of square-root Helmholtz operator symbols: The focusing quadratic profile

Operator symbols play a pivotal role in both the exact, well-posed, one-way reformulation of solving the (elliptic) Helmholtz equation and the construction of the generalized Bremmer coupling series. The inverse square-root and square-root Helmholtz operator symbols are the initial quantities of interest in both formulations, in addition to providing the theoretical framework for the development and implementation of the ‘parabolic equation’ (PE) method in wave propagation modeling. Exact, standard (left) and Weyl symbol constructions are presented for both the inverse square-root and square-root Helmholtz operators in the case of the focusing quadratic profile in one transverse spatial dimension, extending (and, ultimately, unifying) the previously published corresponding results for the defocusing quadratic case [J. Math. Phys. 33 (5), 1887-1914 (1992)]. Both (i) spectral (modal) summation representations and (ii) contour-integral representations, exploiting the underlying periodicity of the associated, quantum mechanical, harmonic oscillator problem, are derived, and, ultimately, related through the propagating and nonpropagating contributions to the operator symbol. Highand low-frequency, asymptotic operator symbol expansions are given along with the exact symbol representations for the corresponding operator rational approximations which provide the basis for the practical computational realization of the PE method. Moreover, while the focusing quadratic profile is, in some respects, nonphysical, the corresponding Helmholtz operator symbols, nevertheless, establish canonical symbol features for more general profiles containing locally-quadratic wells.