# 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?

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 $p\ne 2$.

• Oh, I didn't know that theorem of Hormander, do you have a reference so I can read up on it? Apr 22 '16 at 15:33
• @WillieWong I haven't read Hörmander's original paper: In the book I quote the relevant result is Theorem 3.9.4, which is then said to be found in: L. Hörmander. Estimates for translation invariant operators in L^p spaces. Acta Math. 104 (1960), 93–139. Apr 22 '16 at 21:59
• Okay, I am quite familiar with that article by Hormander, and I see where my confusion was. Hormander did prove that the Schrodinger free propagator is not a bounded operator on $L^p$. This actually has a very simple argument: using the dispersion of the Schrodinger kernel we know that $\lim_{t\to\infty} \|e^{it\Delta} u\|_p = 0$ for any $p > 2$ (lemma 1.2 on p103); a rescaling argument however shows that the $L^p\to L^p$ norm of $e^{it\Delta}$ must be independent of $t$ if the operator were bounded (lemma 1.3 on p109), so we get a contradiction. ... Apr 25 '16 at 14:00
• ... So the original proof did not go through the heat semigroup. This is of course different from the proof given in the book you cited; though having just looked it up, there are some aspects of the proof which are similar. For example, estimate 3.62 is precisely the dispersive decay bound in the first step I outlined above. Apr 25 '16 at 14:04

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.

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.

• Sure. The point in the quoted paper is that the result are stated not only for the Laplacian (in which case the theory is standard indeed), but for general generators of analytic semigroups; and a precise characterization of analytic semigroups that extend to "boundary groups" along $i\mathbb R$ in terms of their asymptotic behavior at the boundary of their analyticity domain is provided. Apr 21 '16 at 6:38