TITLE: Connection between solution for Schrödinger equation and solution for heat equation
QUESTION [12 upvotes]: 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?

REPLY [2 votes]: Let me add some more comment on Mugnolo's answer. It is rather standard to study the heat group with complex time $t$ provided $\Re t>0$. The question of what happens as $t$ tends to the imaginary axis is a natural one and is in the mind of most people working on the subject I guess. It all depends on what you want to prove, of course. If you are interested in proving some 'typically Schroedinger' properties, like e.g. Strichartz estimates, using only information from the heat kernel, then there are not many results. A recent and rather interesting result is this paper. Let me also mention that the connection with the wave equation (more precisely, with the finite speed of propagation property for the wave flow) is a well known and useful one.