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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?

Michael Hardy
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J.Darmody
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1 Answers1

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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.

parsiad
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  • Thanks for your awnser! Indeed, i see this theorem for the heat equation, and the proof uses complex analysis.; but i find nothing regarding the Schrodinger equation. – J.Darmody Jul 18 '16 at 18:52
  • If you are considering a Schrodinger equation with a diffusive term, I assume it is the same idea. The equations in blue on https://en.wikipedia.org/wiki/Schr%C3%B6dinger_equation#Equation indeed seem to have diffusion. – parsiad Jul 18 '16 at 18:57
  • I just considering the linear equation. – J.Darmody Jul 18 '16 at 19:08
  • If I recall correctly from linear PDEs, the wave equation has this property as well in certain dimensions. Anyways, what I said is most likely what the authors mean. Someone has also given a good definition equivalent to the treatment above here: http://math.stackexchange.com/questions/1159797/finite-and-infinite-speed-of-propagation-for-wave-and-heat-equation – parsiad Jul 18 '16 at 19:19
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    Couldn't you reach a simpler definition without distributions for the initial condition, but rather with a function with compact support like the bump function? https://math.stackexchange.com/questions/1283783/an-example-of-an-infinitely-differentiable-function-with-compact-support/1283797#1283797 – Ciro Santilli OurBigBook.com Sep 02 '18 at 06:55
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    @CiroSantilli新疆改造中心六四事件法轮功: That's a good observation, yes. – parsiad Sep 03 '18 at 15:59