- Spectral theory
In

mathematics ,**spectral theory**is an inclusive term for theories extending theeigenvector andeigenvalue theory of a singlesquare matrix . The name was introduced byDavid Hilbert in his original formulation ofHilbert space theory, which was cast in terms ofquadratic form s in infinitely many variables. The originalspectral theorem was therefore conceived as a version of the theorem onprincipal axes of anellipsoid , in an infinite-dimensional setting. The later discovery inquantum mechanics that spectral theory could explain features ofatomic 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 singlenormal operator on it did very much go in parallel with the requirements ofphysics ; particularly at the hands ofvon Neumann . The further theory built on this to includeBanach algebra s, which can be given abstractly. This development leads to theGelfand representation , which covers the commutative case, and further intonon-commutative harmonic analysis .The difference can be seen in making the connection with

Fourier analysis . TheFourier transform on thereal line is in one sense the spectral theory of differentiation "qua"differential operator . But for that to cover the phenomena one has already to deal withgeneralized eigenfunction s (for example, by means of arigged Hilbert space ). On the other hand it is simple to construct agroup algebra , the spectrum of which captures the Fourier transform's basic properties, and this is carried out by means ofPontryagin 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 matrices.Aspects of spectral theory include:

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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

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