- Spectral theory
mathematics, spectral theory is an inclusive term for theories extending the eigenvectorand eigenvaluetheory of a single square matrix. The name was introduced by David Hilbertin his original formulation of Hilbert spacetheory, which was cast in terms of quadratic forms in infinitely many variables. The original spectral theoremwas therefore conceived as a version of the theorem on principal axesof an ellipsoid, in an infinite-dimensional setting. The later discovery in quantum mechanicsthat spectral theory could explain features of atomic spectrawas 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 spaceand the spectral theory of a single normal operatoron 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 algebras, which can be given abstractly. This development leads to the Gelfand representation, which covers the commutative case, and further into non-commutative harmonic analysis.
The difference can be seen in making the connection with
Fourier analysis. The Fourier transformon the real lineis in one sense the spectral theory of differentiation "qua" differential operator. But for that to cover the phenomena one has already to deal with generalized eigenfunctions (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 operators on Banach spaces have many spectral properties similar to that of matrices.
Aspects of spectral theory include:
Integral equations, Fredholm theory, compact operators
Sturm-Liouville theory, hydrogen atom
Spectral theorem, self-adjoint operator, Decomposition of spectrum (functional analysis), functional calculus
Isospectraltheory, Lax pairs.
Spectrum of an operator
Atiyah-Singer index theorem
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