Spectral theory
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2008-04-13T11:57:02Z
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In [[mathematics]], '''spectral theory''' is an inclusive term for theories extending the [[eigenvector]] and [[eigenvalue]] theory of a single [[square matrix]]. The name was introduced by [[David Hilbert]] in his original formulation of [[Hilbert space]] theory, which was cast in terms of [[quadratic form]]s in infinitely many variables. The original [[spectral theorem]] was therefore conceived as a version of the theorem on [[principal axes]] of an [[ellipsoid]], in an infinite-dimensional setting. The later discovery in [[quantum mechanics]] that spectral theory could explain features of [[atomic spectra]] was therefore fortuitous.
There have been three main ways to formulate spectral theory, all of which retain their usefulness. After Hilbert's initial formulation, the later development of abstract [[Hilbert space]] and the spectral theory of a single [[normal operator]] on it did very much go in parallel with the requirements of [[physics]]; particularly at the hands of [[von Neumann]]. The further theory built on this to include [[Banach algebra]]s, which can be given abstractly. This development leads to the [[Gelfand representation]], which covers the [[commutative Banach algebra|commutative case]], and further into [[non-commutative harmonic analysis]].
The difference can be seen in making the connection with [[Fourier analysis]]. The [[Fourier transform]] on the [[real line]] is in one sense the spectral theory of [[derivative|differentiation]] ''qua'' [[differential operator]]. But for that to cover the phenomena one has already to deal with [[generalized eigenfunction]]s (for example, by means of a [[rigged Hilbert space]]). On the other hand it is simple to construct a [[group algebra]], the spectrum of which captures the Fourier transform's basic properties, and this is carried out by means of [[Pontryagin duality]].
One can also study the spectral properties of operators on [[Banach spaces]]. For example, [[compact operator]]s on Banach spaces have many spectral properties similar to that of [[Matrix (mathematics)|matrices]].
Aspects of spectral theory include:
* [[Integral equation]]s, [[Fredholm theory]], [[compact operator]]s
* [[Sturm-Liouville theory]], [[hydrogen atom]]
* [[Spectral theorem]], [[self-adjoint operator]], [[Decomposition of spectrum (functional analysis)]], [[functional calculus]]
* [[Isospectral]] theory, [[Lax pair]]s.
* [[Spectrum of an operator]]
* [[Atiyah-Singer index theorem]]
[[Category:Linear algebra]]
[[Category:Spectral theory|*]]
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