Wave Function Solutions by Transformation from the Helmholtz to Laplacian Operator.

The Helmholtz, Schr¨odinger and Klein-Gordon equations all have a similar form (for constant wavelength) and have applications in optics, quantum mechanics and relativistic quantum mechanics, respectively. Central to these applications is the theory of barrier and potential scattering,which, through application of the Green’s function method yields transcendental equations for the scattered wave function thereby requiring approximation methods to be employed. This paper reports on a new approach to solving this problem which is based on transforming the Helmholtz operator to the Laplacian operator and applying a Green’s function solution to the Poisson equation. This approach yields an exact forward and inverse scattering solution subject to a fundamental condition, whose physical basis is briefly explored. It also provides a series solution that is not predicated on a convergence condition.