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. Mathematics is the study of quantity, structure, space and change. ... In linear algebra, the eigenvectors (from the German eigen meaning inherent, characteristic) of a linear operator are non-zero vectors which, when operated on by the operator, result in a scalar multiple of themselves. ... In mathematics, a number is called an eigenvalue of a matrix if there exists a nonzero vector such that the matrix times the vector is equal to the same vector multiplied by the eigenvalue. ... For the square matrix section, see square matrix. ... David Hilbert David Hilbert (January 23, 1862 – February 14, 1943) was a German mathematician born in Wehlau, near Königsberg, Prussia (now Znamensk, near Kaliningrad, Russia) who is recognized as one of the most influential mathematicians of the 19th and early 20th centuries. ... In mathematics, a Hilbert space is an inner product space that is complete with respect to the norm defined by the inner product. ... In mathematics, a quadratic form is a homogeneous polynomial of degree two in a number of variables. ... In mathematics, particularly linear algebra and functional analysis, the spectral theorem is a collection of results about linear operators or about matrices. ... In mathematics, particularly linear algebra and functional analysis, the spectral theorem is a collection of results about linear operators or about matrices. ... Definition In mathematics, an ellipsoid is a type of quadric that is a higher dimensional analogue of an ellipse. ... Fig. ... Spectroscopy is the study of spectra, that is, the dependence of a physical quantity to frequency. ...

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 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. In mathematics, a Hilbert space is an inner product space that is complete with respect to the norm defined by the inner product. ... In functional analysis, a normal operator on a Hilbert space H is a continuous linear operator N : H → H that commutes with its hermitian adjoint N*: N N* = N* N. The main importance of this concept is that the spectral theorem applies to normal operators. ... The willingness to question previously held truths and search for new answers resulted in a period of major scientific advancements, now known as the Scientific Revolution. ... A separate article covers Saint John Neumann, the American priest. ... In 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. ... In mathematics, the Gelfand representation in functional analysis allows a complete characterisation of commutative C*-algebras as algebras of continuous complex-valued functions. ...

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 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. Harmonic analysis is the branch of mathematics which studies the representation of functions or signals as the superposition of basic waves. ... The Fourier transform, named after Jean Baptiste Joseph Fourier, is an integral transform that re-expresses a function in terms of sinusoidal basis functions, i. ... In mathematics, the real line is simply the set of real numbers. ... In mathematics, the derivative of a function is one of the two central concepts of calculus. ... In mathematics, a differential operator is a linear operator defined as a function of the differentiation operator. ... In mathematics, a rigged Hilbert space is a construction designed to link the distribution (test function) and square-integrable aspects of functional analysis. ... In the theory of group representations, the group algebra is any of various constructions to assign to a group (either a locally compact topological group, or a group without a topology, i. ... In mathematics, in particular in harmonic analysis and the theory of topological groups, Pontryagin duality explains the general properties of the Fourier transform. ...

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 pairs.
• Spectrum of an operator