Infinite propagation speed for the Schrödinger equation

Question

I've seen many articles making reference to the property of the infinite propagation speed for the solution of the linear Schrödinger equation; but i can't find a book giving a 'good' definition or a clear theorem. So is there a book, article,... explaining this notion?

Answer 1

To get what is meant by this informal statement, consider the 1d heat equation $$ u_{t}(t,x) =u_{xx}(t,x) \text{ on }(0,\infty)\times\mathbb{R}. $$ Its fundamental solution is $$ \Phi(t,x)=\frac{1}{\sqrt{4\pi t}}\exp\left(-\frac{x^{2}}{4t}\right). $$ Denote by $\delta$ the Dirac delta "function". Informally, the fundamental solution is the "solution" to the heat equation when $u(0,x)=\delta(x)$. Informally, $\delta(x)$ is zero everywhere but the origin (i.e., all the information is at the origin). However, $x\mapsto\Phi(t,x)$ is positive everywhere for any $t>0$! (i.e., the information has spread everywhere)

In short, the "information" that was gathered at the origin at the initial time $t=0$ has spread to the whole real line. This is what is meant by infinite speed of information propagation. I assume the authors of the articles you have been reading are referring to a similar phenomenon.

If you would like a more formal background on the topic, I suggest reading about fundamental solutions and distributions.