In the last paragraph of this last paper of Klaas Landsman, you can read:

Finally, let me note that this was a winner's (or "whig") history, full of hero-worship: following in the footsteps of Hilbert, von Neumann established the link between quantum theory and functional analysis that has lasted. Moreover, partly through von Neumann's own contributions (which are on a par with those of Bohr, Einstein, and Schrodinger), the precision that functional analysis has brought to quantum theory has greatly benefited the foundational debate. However, it is simultaneously a loser's history: starting with Dirac and continuing with Feynman, until the present day physicists have managed to bring quantum theory forward in utter (and, in my view, arrogant) disregard for the relevant mathematical literature. As such,

functional analysis has so far failed to make any real contribution to quantum theory as a branch of physics(as opposed to mathematics), and in this respect its role seems to have been limited to something like classical music or other parts of human culture that adorn life but do not change the economy or save the planet. On the other hand, like General Relativity, perhaps the intellectual development reviewed in this paper is one of those human achievements that make the planet worth saving.

To balance this interesting debate, if there actually exists real reasons to disagree with above bolded sentence of Klaas Landsman, let me ask the following:

What are the real contributions of functional analysis to quantum theory as a branch of physics?

Here "real" should be understood in the sense underlying the above paragraph.

This question was asked on physics.stackexchange and on PhysicsOverflow.

real physicsis (e. g. "something for which physics Nobel prize can be awarded" or "predicting or explaining experimentally observable phenomena") as well as some common notion of what areal contribution of functional analysiswould be. At least, "realness" defines a rather obvious order in both aspects, and hence this question is as good as asking for the strongest known version of a theorem. I would very much like to see answers of the form "a (non-trivial) theorem A informed prediction of phenomenon B", if possible. $\endgroup$theoremon classification of representations of Poincaré group was proven with the purpose of classifying particles one can expect to discover. The behavior of Fermi-Pasta-Ulam chain was a complete mystery before KAMtheoremcame along. Spin-statisticstheorem... Indextheorem... Also, manytheoremsjust underscore and organize general features ofcalculations, so I do not agree with your countraposing the two. $\endgroup$13more comments