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.