Analytic Solutions of the Complex Diffusion Equation with Possible Quantum Mechanical Relevance

Abstract

In our latest studies with the help of the self-similar Ansatz we derived new type of 1
 solutions for the regular diffusion equation which are much more complex than the well-known 2
 Gaussian and error functions. These solutions contain additional Kummer’s M and Kummer’s U 3
 functions with quadratic argument. In the present treaties we perform an analogous analysis for the 4
 regular diffusion equation which has a complex diffusion coefficient. Formally, it is equivalent to 5
 the free Schrödinger equation however it is far from being trivial how the solutions can be given 6
 quantum mechanical interpretation. We investigate both the one dimensional Cartesian and the 7
 spherical symmetric equations. We find solutions which fulfill the L2 integrability criteria therefore 8
 might have quantum mechanical relevance in the future.