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Proof of GelfandMaurin Nuclear Spectral Theorem? 
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#1
Feb2209, 07:11 PM

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I want to study a detailed proof of the Nuclear Spectral Theorem
(which underpins the use of Rigged Hilbert Spaces in modern QM to make the Dirac braket formalism respectable). Most textbooks and papers refer to the old multivolume series on generalized functions by Gelfand and Vilenkin, but I cannot borrow it locally and the price from Amazon is ridiculous. Does anyone know of proofs in other textbooks, or maybe from a (free) online source? Thanks in advance for any suggestions... 


#2
Aug1610, 05:22 PM

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The proof in Gelfand's Generalized Function vol 4 is incorrect (at least
not complete), as pointed out by the translator of the English version. 


#3
Aug1710, 12:20 AM

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post in this forum, at least to me. :) I now have a copy of the (English version of) Gelfand & Vilenkin vol4, but I cannot find where the translator says this. (I looked at the translator's notes near the beginning, but I couldn't find where he says this.) If you have a copy at hand, could you possibly give me a more specific page reference to where the translator says this? Thanks again. 


#4
Aug1710, 04:46 PM

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Proof of GelfandMaurin Nuclear Spectral Theorem?
I couldn't find the <incompletenes/inaccurate> statement/footnote either.



#5
Aug1810, 06:43 PM

P: 2

Hi.
The trouble is on page 122 of vol 4 (I mean GelfandVilenkin "Generalized Functions"). At the bottom of that page, the translator expressed some concern "... it is not clear why..." As I read through the proof, this concern is serious, and I don't know how to fix it (this is not my field so I am far from being an expert, and it seems no one I know cares about rigged Hilbert space!). In fact, after a search online, there is a paper of G. G. Gould (J. London Math. Soc. 43 (1968) 745754) that claimed to have resolved this issue; but that paper is not so easy to read. On the other hand, apart from this issue the Gelfand book is userfriendly. Maybe you can ask some experts and update this? 


#6
Aug1810, 10:04 PM

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RHS theory silently underpins much of modern quantum theory. Rafael de la Madrid has, in recent years, written a number of papers trying to emphasize RHS (eg his tutorial paper quantph/0502053, and quite a few others), but these are mainly applications of RHS without giving details of the heavy proofs that underlie it. There's also this paper: M. Gadella & F. Gomez, "On the Mathematical Basis of the Dirac Formulation of Quantum Mechanics", IJTP, vol 42, No 10, Oct 2003, 22252254 Gadella & Gomez give updated version of the spectral theorem(s) near the end, but not detailed proofs, afaict. But much of this paper is over my head, and I haven't yet had time to try and chase down the further references therein. If you haven't previously seen this stuff, I'd be interested to hear your comments. and/or errors. (Sometimes I'm not sure which is which.) BTW, what is your interest in RHS? Physics or maths? 


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