Proof of Gelfand-Maurin Nuclear Spectral Theorem?

In summary, the Nuclear Spectral Theorem, which is necessary for the use of Rigged Hilbert Spaces in modern Quantum Mechanics and the Dirac bra-ket formalism, is typically referred to in the old multi-volume series by Gelfand and Vilenkin. However, it is not easily accessible locally and has a high price on Amazon. Other textbooks and free online sources may contain alternative proofs. The proof in Gelfand's Generalized Function vol 4 is not complete, as pointed out by the translator of the English version. A paper by G. G. Gould may have resolved this issue, but it is not easy to understand. The book by Gadella and Gomez also provides updated versions of the spectral theorem, but
  • #1
strangerep
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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 bra-ket formalism respectable).

Most textbooks and papers refer to the old multi-volume 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...
 
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  • #2
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
zhupihou said:
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.

Thanks for your comment! (That was indeed an unexpected and interesting first
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
I couldn't find the <incompletenes/inaccurate> statement/footnote either.
 
  • #5
Hi.

The trouble is on page 122 of vol 4 (I mean Gelfand-Vilenkin "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) 745-754) 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 user-friendly.

Maybe you can ask some experts and update this?
 
  • #6
zhupihou said:
The trouble is on page 122 of vol 4 (I mean Gelfand-Vilenkin "Generalized Functions").
At the bottom of that page, the translator expressed some concern
"... it is not clear why..."

Oh, thanks. I see it now.

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,

It sounds like you know more about this than I do. :-)

and it seems no one I know cares about rigged Hilbert space!).
I know what you mean. This is an unfortunate situation, since
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 quant-ph/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, 2225-2254

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.


In fact, after a search online, there is a paper of G. G. Gould (J. London Math. Soc.
43 (1968) 745-754) that claimed to have resolved this issue; but that paper is not
so easy to read.

Thanks. I'll take a look at it when I get a chance.

On the other hand, apart from this issue the Gelfand book is user-friendly.

Yes, it's certainly better than Maurin's text which seems to contain many typos
and/or errors. (Sometimes I'm not sure which is which.)

Maybe you can ask some experts and update this?

I don't know many experts on this directly, but I'll try.

BTW, what is your interest in RHS? Physics or maths?
 

1. What is the Gelfand-Maurin Nuclear Spectral Theorem?

The Gelfand-Maurin Nuclear Spectral Theorem is a mathematical theorem that provides a way to decompose a linear operator on a Hilbert space into a sequence of simpler operators. It is a fundamental result in functional analysis and is used in various fields of physics and engineering.

2. Who discovered the Gelfand-Maurin Nuclear Spectral Theorem?

The Gelfand-Maurin Nuclear Spectral Theorem was discovered independently by two mathematicians, Israel Gelfand and Pierre Maurin, in the mid-20th century. Gelfand published his version of the theorem in 1943, while Maurin published his in 1953.

3. What is the significance of the Gelfand-Maurin Nuclear Spectral Theorem?

The Gelfand-Maurin Nuclear Spectral Theorem is significant because it allows for a deeper understanding of linear operators on Hilbert spaces, which have numerous applications in mathematics and physics. It also provides a powerful tool for solving differential equations and studying the properties of linear differential operators.

4. How is the Gelfand-Maurin Nuclear Spectral Theorem used in practice?

The Gelfand-Maurin Nuclear Spectral Theorem is used in various fields such as quantum mechanics, signal processing, and control theory. It allows for the decomposition of complex systems into simpler components, making it easier to analyze and solve problems. It is also used in the construction and analysis of numerical methods for solving differential equations.

5. Are there any limitations to the Gelfand-Maurin Nuclear Spectral Theorem?

While the Gelfand-Maurin Nuclear Spectral Theorem is a powerful tool, it does have limitations. It only applies to compact operators and does not hold for non-compact operators. Additionally, the theorem assumes the operator is defined on a Hilbert space, which may not always be the case in practical applications.

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