Over in the Quantum Physics forums, we occasionally have threads involving rigged Hilbert space -- a.k.a. Gel'fand triple: ##\Omega \subset H \subset \Omega'## where ##H## is a Hilbert space, ##\Omega## a dense subspace thereof such that certain unbounded continuous-spectrum operators are well-defined everywhere thereon, and ##\Omega'## is its topological dual. A recent thread of this kind is: https://www.physicsforums.com/showthread.php?t=668013(adsbygoogle = window.adsbygoogle || []).push({});

With certain extensions of the meaning of "self-adjoint operator" to the space ##\Omega'##, some treatments of QM rely on the so-called nuclear spectral theorem (cf. Gelfand & Vilenkin vol 4) which basically assures us that the generalized eigenvectors of a self-adjoint operator ##A## in ##\Omega'## span ##\Omega##, and that operators of the form ##f(A)##, for analytic functions ##f##, make sense. This is basically a generalization of the usual spectral theorem for unbounded operators on Hilbert space to the distributional context of rigged Hilbert space.

I've always felt it unsatisfactory that arbitrary elements of the dual space ##\Omega'## are not also covered by such a theorem. Recently, David Carfi put a series of papers on the arXiv claiming to do just this (and he confirmed to me in brief private correspondence that this is indeed his intent). In reverse time order the papers are:

http://arxiv.org/abs/1104.4660

http://arxiv.org/abs/1104.4651

http://arxiv.org/abs/1104.3908

http://arxiv.org/abs/1104.3380

http://arxiv.org/abs/1104.3324

http://arxiv.org/abs/1104.3647

Unfortunately, my abilities in Functional Analysis, etc, are inadequate to form a reliable opinion about these papers. They have not been published in peer-reviewed journals (afaict), but David Carfi seems to have published elsewhere in financial and economics mathematics.

So... I'm hoping that the FA experts here can spare a little time to look through Carfi's papers and figure out whether he does indeed achieve what I described above, i.e., establish a sensible spectral expansion forallelements of the distribution space ##\Omega'##, and not merely the well-behaved elements from ##\Omega##.

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# Papers by D. Carfi: extended spectral theory of distributions.

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