Mathematically Exact Non-Square-Integrable Solutions in Schrödinger-Equivalent Diffusion Dynamics

We analyze the spherically symmetric complex diffusion and special type of the complex reaction-diffusion equations with the self-similar Ansatz. These equations are form invariant to the free Schrödinger equations and to Schrödinger equations with power-law space dependent potentials. The self-similar Ansatz couples the spatial and temporal variables together instead of the usual separation, therefore new type of solutions can be derived. For both cases analytic solutions are presented which are the Kummer's and the Whittaker functions with complex quadratic arguments. The results are analyzed in depth. In the second case the role of the complex angular momenta is investigated as well.