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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 forms 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 algebraIn functional analysis, a Banach algebra named after Stefan Banach, is an associative algebra A over the real or complex numbers which at the same time is also a Banach space. The algebra multiplication and the Banach space norm are required to be relateds, which can be given abstractly. This development leads to the Gelfand representationFunctional analysis In mathematics, the Gelfand representation in functional analysis allows a complete characterisation of commutative C -algebras as algebras of continuous complex-valued functions. The Gelfand representation theorem is one avenue in the, 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 transformThe Fourier transform named for Jean Baptiste Joseph Fourier, is an integral transform that re-expresses a function in terms of sinusoidal basis functions, i. as a sum or integral of sinusoidal functions multiplied by some coefficients ("amplitudes"). on the real lineIn mathematics, the real line is simply the set of real numbers. However, this term is usually used when R is to be treated as a space of some sort, such as a topological space or a vector space. The real line has been studied at least since the days of t is in one sense the spectral theory of differentiationCalculus In mathematics, the derivative of a function is one of the two central concepts of calculus. The inverse of a derivative is called the antiderivative, or indefinite integral. The derivative of a function at a certain point is a measure of the rat qua differential operatorMultivariate calculus Differential operators In mathematics, a differential operator is a linear operator defined as a function of the differentiation operator. It is helpful, as a matter of notation first, to consider differentiation as an abstract opera. 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.

Aspects of spectral theory include:

• Integral equations, Fredholm theory , compact operators
• Sturm-Liouville theory, hydrogen atom
• Spectral theorem, self-adjoint operator, continuous spectrum, functional calculus
• Isospectral theory, Lax pair s.
• Spectrum of an operator