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. These equations are form invariant to the free Schrödinger equations and to the Schrödinger equations with power-law space-dependent potentials. Our new type of solutions are important because we found a new realm of solutions which lie between the solutions of the classical regular diffusion equation and the usual quantum mechanical solutions of the Schrödinger equation. As the solution method, we applied the the self-similar Ansatz, which reduces the original partial differential equation (PDE) to an ordinary differential equation (ODE) which can be solved analytically. The self-similar Ansatz couples the spatial and temporal variables together instead of the usual separation which has to be used in ordinary quantum mechanics for time-independent Hamiltonian. For the complex diffusion equation—without any additional source term—the solutions are the Kummer’s M and Kummer’s U functions. For some parameter values we found L2 integrability, as in the Cartesian case. We interpret that this property can be a “quantum mechanical heritage” and can be a far relation to ordinary quantum mechanics. Therefore, in this sense, our solutions might have quantum mechanical interest in the future. For the complex reaction–diffusion-type equation we derived the Whittaker M and Whittaker W functions as solutions. These solutions have no L2 integrability at all. All derived solutions have complex quadratic arguments. These kind of analytic solutions are new and cannot be found in the existing scientific literature. Finally, the role of the complex angular momentum was investigated as well.