Connection between solution for Schrödinger equation and solution for heat equation 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?
 A: I suggest you take a look to Section 3.9 in the nice book by Arendt-Batty-Hieber-Neubrander, where this issue is studied in detail in a rather abstract setting. 
Summarising: just like many more parabolic equations - in particular those where a self-adjoint dissipative operator appears -, the heat equation is governed by a semigroup that can be extended to a holomorphic function mapping the open set $\{z\in \mathbb C:Re\ z>0\}$ to the space of bounded linear operators on the relevant Banach space $L^2(\mathbb R^d)$. Now, this function happens to have "boundary values" on the imaginary axis and - guess what - its boundary values (i.e., $\lim_{z\to is} e^{z\Delta}$ for certain $s\in \mathbb R$) are exactly the corresponding values of the unitary group governing the Schrödinger equation (i.e., $e^{is\Delta}$). By a Theorem due to Hörmander, this constructions cannot be extended to $L^p(\mathbb R^d)$ for $p\ne 2$.
A: I'm not sure if this is what you want, and might be too elementary, but there is a rather neat connection in terms of Brownian motion, which in my opinion is a very natural way to approach both heat and Schroedinger equations. By the Feynman Kac formula the solution to both equations is given by a fancy expectation of Brownian motion. There are physical interpretations of each. 
In the Schroedinger equation, the expectation of Brownian motion running in imaginary time is called Feynman path integration, and was the subject of his Ph.D. thesis. 
In the heat equation, there are arguments by Einstein and Smoluchowski, showing why Brownian motion might show up in the heat equation.
Overall, there is a fairly wide class of PDEs that have probabilistic solutions. All of these equations are connected in some way to Brownian motion. This should not be surprising! The central limit theorem guarantees that when there are a lot of independent interactions, a normal distribution should come out.
I did an undergrad thesis on this approach. Of course this is an undergrad senior thesis, so take it as that. :)
In general if you just search for "probabilistic representations of solutions to PDEs" you will get a lot of fairly modern, more exciting papers.
A: 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.
