Talk:Spectral theory

Quotation from Davies

This quotation seems out of place here: it is not connected to the mathematical ideas presented in the intro, and seems to require some physical (rather than mathematical) understanding not in evidence. Brews ohare (talk) 19:47, 11 December 2009 (UTC)[reply]

This problem was corrected by Charles Matthews. Brews ohare (talk) 18:52, 13 December 2009 (UTC)[reply]

Self-adjoint operators

It appears that Hilbert_space#Spectral_theory expects some discussion in this article on this subject, about which nothing appears at the moment. Brews ohare (talk) 19:41, 11 December 2009 (UTC)[reply]

Unsubstantiated statement

The statement: The later discovery in quantum mechanics that spectral theory could explain features of atomic spectra was therefore fortuitous. may not be exactly what was meant. More probably it is meant that the math and the physics followed parallel paths, and were not originally connected. Brews ohare (talk) 01:53, 13 December 2009 (UTC)[reply]

I have referenced some history of the term spectrum, to place this remark in context. It commonly happens here, unfortunately, that such heuristic and historical remarks get targeted. This is basically unfortunate for the site, because for the non-expert reader they may convey a disproportionately high amount of understanding (i.e. here that the mathematics anticipated the needs of mathematical physics, by pursuing its own ends, a broad theme in 20th century intellectual history). I have had to make such comments on too many talk pages. Of course all content must be verifiable and most content can be improved, but it is a mistake, I think, to bear down on non-technical content. Charles Matthews (talk) 08:07, 13 December 2009 (UTC)[reply]

I agree upon the value of historical remarks and general context. However, I doubt that the application of Hilbert space by John Von Neumann, first published by Springer-Verlag in 1932 was "fortuitous". Hence, my request for a more accurate wording. Brews ohare (talk) 16:45, 13 December 2009 (UTC)[reply]

Well, no, what von Neumann did was completely self-conscious in founding Dirac-style QM on Hilbert's ideas. But that was with the benefit of hindsight. Charles Matthews (talk) 18:59, 13 December 2009 (UTC)[reply]

but is also not separate from them

This claim could be interpreted to suggest that the math is not logically distinct from the physics, where a correct statement might be that historically they have developed in tandem. Of course, from a logical viewpoint there is no necessary connection between the two. Could this statement be amended to avoid this ambiguity? Brews ohare (talk) 16:57, 13 December 2009 (UTC)[reply]

Tweaked. Charles Matthews (talk) 19:03, 13 December 2009 (UTC)[reply]

Notation

The way the bras and kets are typeset makes the article extremely difficult to parse to the point that I can barely read it. I suggest using the standard $\langle x|$ and $|x\rangle$ (with the vertical bar). Sławomir Biały (talk) 14:11, 27 December 2009 (UTC)[reply]

Ha ha

I thought this introductory comment by Brews ohare was very funny, but also very unencyclopedic. Hence, at the cost of a reduction in amusement value, I had to snip it.

A mathematician's approach to this topic is an acquired taste, and a normal reaction is like that to one's first encounter with beer

Elroch (talk) 16:56, 23 January 2010 (UTC)[reply]

Moved from Hilbert space

I have moved the following from Hilbert space as it was clearly out of place there. My concern is that the text is convoluted, yet seems to be saying very little in the way of actual content. (In fact, it's sort of obvious.) I'm posting it here in case someone thinks there is anything worth keeping. Sławomir Biały (talk) 21:35, 12 January 2012 (UTC)[reply]

==;Reverse spectral theory in Hilbert space. Direct sums of operators==
The reverse spectral theory of operators acting in a Hilbert space is important due to its numerous applications in quantum physics ^[1] . The goal of such theory is to investigate the dependence of spectral properties of operators, built as direct sums of other operators.
As a special case of self-adjoint operators in Hilbert space, let us have a number (possibly infinite) of self-adjoint operators A_i, each acting in a Hilbert space H_i, the direct sum operator A is defined as acting on the direct sum space H as A_{H_i} = A_i. This direct sum operator is also called a self-adjoint vector-operator and the operators which comprise it are called coordinate operators. The spectral theory for self-adjoint vector-operators shows that their spectral properties are not straightforwardly inherited from the coordinate spectral properties as direct sums^[2]. The major reason for this is the behaviour of spectral multiplicity when coordinate operators are being united as direct sums. Certain constructive process should be followed to build the vector-operator's spectral properties.

References

Incoherent section

The section titled Spectral theory briefly is incoherent.

If notation like bra and ket are going to be introduced, please do readers the courtesy of defining them in mathematical terms. That would mean re-expressing them in terms of standard mathematical notation as well as, preferably, in term of standard mathematical terminology. Such a clarification must be intelligible to people who don't already know what bra and ket mean.50.205.142.50 (talk) 20:26, 13 March 2020 (UTC)[reply]

Not to discourage a brave pedagogical soul from taking up the task, but... The bra & ket notation, suitably linked, is the mainstream and standard notation of record for this sort of thing. Asking to have it translated/vulgarized to something else would in no sense do a service to the thoughtful reader...Cuzkatzimhut (talk) 01:07, 14 March 2020 (UTC)[reply]

This article lacks an explanation in ‘layman terms’.

I don’t know is that good or bad, but I can understand most words on this page, AND can’t figure out what is ‘Spectral Theory’ about. 2A00:1370:8135:B3F3:81B1:BF8A:5A11:E53E (talk) 20:16, 5 December 2021 (UTC)[reply]

[1] ttp://projecteuclid.org/DPubS/Repository/1.0/Disseminate?view=body&id=pdf_1&handle=euclid.rmjm/1250126808

[2] ttp://arxiv.org/abs/math/0409012

[1]

[2]