Editing Spectral theory (section)

==Physical background==
The background in the physics of [[vibration]]s has been explained in this way:<ref name=Davies>[[E. Brian Davies]], quoted on the King's College London analysis group website {{Cite web |url=http://www.kcl.ac.uk/schools/pse/maths/research/analysis/research.html |title=Research at the analysis group}}</ref>

{{Blockquote|Spectral theory is connected with the investigation of localized vibrations of a variety of different objects, from [[atom]]s and [[molecule]]s in [[chemistry]] to obstacles in [[Waveguide (acoustics)|acoustic waveguide]]s. These vibrations have [[frequency|frequencies]], and the issue is to decide when such localized vibrations occur, and how to go about computing the frequencies. This is a very complicated problem since every object has not only a [[fundamental tone]] but also a complicated series of [[overtone]]s, which vary radically from one body to another.}}

Such physical ideas have nothing to do with the mathematical theory on a technical level, but there are examples of indirect involvement (see for example [[Mark Kac]]'s question ''[[Can you hear the shape of a drum?]]''). Hilbert's adoption of the term "spectrum" has been attributed to an 1897 paper of [[Wilhelm Wirtinger]] on [[Hill differential equation]] (by [[Jean Dieudonné]]), and it was taken up by his students during the first decade of the twentieth century, among them [[Erhard Schmidt]] and [[Hermann Weyl]]. The conceptual basis for [[Hilbert space]] was developed from Hilbert's ideas by [[Erhard Schmidt]] and [[Frigyes Riesz]].<ref name=Young>{{Cite book |title=An introduction to Hilbert space |author=Nicholas Young |url=https://books.google.com/books?id=_igwFHKwcyYC&pg=PA3 |page=3 |isbn=0-521-33717-8 |publisher=Cambridge University Press |year=1988}}</ref><ref name=Dorier>{{Cite book |title=On the teaching of linear algebra; ''Vol. 23 of'' Mathematics education library |author=Jean-Luc Dorier |url=https://books.google.com/books?id=gqZUGMKtNuoC&q=%22thinking+geometrically+in+Hilbert%27s+%22&pg=PA50 |isbn=0-7923-6539-9 |publisher=Springer |year=2000 }}

</ref>  It was almost twenty years later, when [[quantum mechanics]] was formulated in terms of the [[Schrödinger equation]], that the connection was made to [[atomic spectra]]; a connection with the mathematical physics of vibration had been suspected before, as remarked by [[Henri Poincaré]], but rejected for simple quantitative reasons, absent an explanation of the [[Balmer series]].<ref>Cf. [http://www.dm.unito.it/personalpages/capietto/Spectra.pdf Spectra in mathematics and in physics] {{webarchive|url=https://web.archive.org/web/20110727024805/http://www.dm.unito.it/personalpages/capietto/Spectra.pdf |date=2011-07-27 }} by [[Jean Mawhin]], p.4 and pp. 10-11.</ref> The later discovery in quantum mechanics that spectral theory could explain features of atomic spectra was therefore fortuitous, rather than being an object of Hilbert's spectral theory.