Analytic solutions of the complex diffusion equation with aspects on quantum mechanics

In this paper, we derive and compare the solutions of the real and complex diffusion equations obtained by the self-similar solution to each other. Additionally, possible quantum mechanical aspects are analyzed as well. In the second part of the study, a complex reaction–diffusion equation is investigated and compared to the quantum mechanical solutions of the corresponding Schrödinger equation with a power-law-type potential. We show that the same parameter ratio emerges in the quantum mechanical and the self-similar solutions as well, which is a remarkable property. For both investigated equations, the complex diffusion and complex reaction–diffusion equations, we found some solutions for which the absolute value squared has a convergent finite numerical integral. This is not a rigorous [Formula: see text] integrability condition but a good hint for that, which is a key property in quantum mechanics.