Spectral theory
[3] The name spectral theory 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 later discovery in quantum mechanics that spectral theory could explain features of atomic spectra was therefore fortuitous.Hilbert himself was surprised by the unexpected application of this theory, noting that "I developed my theory of infinitely many variables from purely mathematical interests, and even called it 'spectral analysis' without any presentiment that it would later find application to the actual spectrum of physics."[4] There have been three main ways to formulate spectral theory, each of which find use in different domains.After Hilbert's initial formulation, the later development of abstract Hilbert spaces and the spectral theory of single normal operators on them were well suited to the requirements of physics, exemplified by the work of von Neumann.[5] The further theory built on this to address Banach algebras in general.This development leads to the Gelfand representation, which covers the commutative case, and further into non-commutative harmonic analysis.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.The background in the physics of vibrations has been explained in this way:[6] Spectral theory is connected with the investigation of localized vibrations of a variety of different objects, from atoms and molecules in chemistry to obstacles in acoustic waveguides.This is a very complicated problem since every object has not only a fundamental tone but also a complicated series of overtones, which vary radically from one body to another.Such physical ideas have nothing to do with the mathematical theory on a technical level, but there are examples of indirect involvement (see for example Mark Kac's question Can you hear the shape of a drum?).Hilbert's adoption of the term "spectrum" has been attributed to an 1897 paper of Wilhelm Wirtinger on Hill differential equation (by Jean Dieudonné), and it was taken up by his students during the first decade of the twentieth century, among them Erhard Schmidt and Hermann Weyl.[7][8] It was almost twenty years later, when quantum mechanics was formulated in terms of the Schrödinger equation, that the connection was made to atomic spectra; a connection with the mathematical physics of vibration had been suspected before, as remarked by Henri Poincaré, but rejected for simple quantitative reasons, absent an explanation of the Balmer series.Consider a bounded linear transformation T defined everywhere over a general Banach space.The spectrum of T is the set of all complex numbers ζ such that Rζ fails to exist or is unbounded.The function Rζ for all ζ in ρ(T) (that is, wherever Rζ exists as a bounded operator) is called the resolvent of T. The spectrum of T is therefore the complement of the resolvent set of T in the complex plane.[14] With suitable restrictions, much can be said about the structure of the spectra of transformations in a Hilbert space.[15] In functional analysis and linear algebra the spectral theorem establishes conditions under which an operator can be expressed in simple form as a sum of simpler operators.This topic is easiest to describe by introducing the bra–ket notation of Dirac for operators.[16][17] As an example, a very particular linear operator L might be written as a dyadic product:[18][19] in terms of the "bra" ⟨b1| and the "ket" |k1⟩.Answering such questions is the realm of spectral theory and requires considerable background in functional analysis and matrix algebra.Using the bra–ket notation of the above section, the identity operator may be written as: where it is supposed as above that, one obtains: which is the generalized Fourier expansion of ψ in terms of the basis functions { ei }.In particular, the basis might consist of the eigenfunctions of some linear operator L: with the { λi } the eigenvalues of L from the spectrum of L. Then the resolution of the identity above provides the dyad expansion of L: Using spectral theory, the resolvent operator R: can be evaluated in terms of the eigenfunctions and eigenvalues of L, and the Green's function corresponding to L can be found.The Green's function of the previous section is: and satisfies: Using this Green's function property: Then, multiplying both sides of this equation by h(z) and integrating: which suggests the solution is: That is, the function ψ(x) satisfying the operator equation is found if we can find the spectrum of O, and construct G, for example by using: There are many other ways to find G, of course.It must be kept in mind that the above mathematics is purely formal, and a rigorous treatment involves some pretty sophisticated mathematics, including a good background knowledge of functional analysis, Hilbert spaces, distributions and so forth.Optimization problems may be the most useful examples about the combinatorial significance of the eigenvalues and eigenvectors in symmetric matrices, especially for the Rayleigh quotient with respect to a matrix M. Theorem Let M be a symmetric matrix and let x be the non-zero vector that maximizes the Rayleigh quotient with respect to M. Then, x is an eigenvector of M with eigenvalue equal to the Rayleigh quotient.Namely, evaluate the Rayleigh quotient with respect to x: where we used Parseval's identity in the last line.