It's known, that if you write imaginary unit into a heat equation you'll get time-dependent Schrödinger equation. Recently one guy discovered a connection between solutions for these two equations (see this paper). Specifically, he found a way to obtain a solution for a Schrödinger equation from a solution for a heat equation (and even from computationally simpler objects related to а heat equation). The method is rather general and is for instance applicable to equations with arbitrary configurational space.
My question is who and in what papers suggested to study connections between Schrödinger and heat equations? Are there any results? Are these problem included in any sort of list of problems? I know about Wick rotation and Doss trick (see refernces in here). But what else exists in this field?