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In
mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and their changes (cal ...
, spectral theory is an inclusive term for theories extending the
eigenvector In linear algebra Linear algebra is the branch of mathematics concerning linear equations such as: :a_1x_1+\cdots +a_nx_n=b, linear maps such as: :(x_1, \ldots, x_n) \mapsto a_1x_1+\cdots +a_nx_n, and their representations in vector spaces an ...

eigenvector
and
eigenvalue In linear algebra, an eigenvector () or characteristic vector of a Linear map, linear transformation is a nonzero Vector space, vector that changes at most by a Scalar (mathematics), scalar factor when that linear transformation is applied to i ...

eigenvalue
theory of a single
square matrix In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...
to a much broader theory of the structure of operators in a variety of
mathematical space In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities an ...
s. It is a result of studies of
linear algebra Linear algebra is the branch of mathematics concerning linear equations such as: :a_1x_1+\cdots +a_nx_n=b, linear maps such as: :(x_1, \ldots, x_n) \mapsto a_1x_1+\cdots +a_nx_n, and their representations in vector spaces and through matrix (mat ...
and the solutions of
systems of linear equations In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no gen ...
and their generalizations. The theory is connected to that of
analytic functions In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...
because the spectral properties of an operator are related to analytic functions of the spectral parameter.


Mathematical background

The name ''spectral theory'' was introduced by
David Hilbert David Hilbert (; ; 23 January 1862 – 14 February 1943) was a German mathematician This is a List of German mathematician A mathematician is someone who uses an extensive knowledge of mathematics Mathematics (from Ancient Greek, G ...
in his original formulation of
Hilbert space In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
theory, which was cast in terms of
quadratic form In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no gene ...
s in infinitely many variables. The original spectral theorem was therefore conceived as a version of the theorem on
principal axesPrincipal axis may refer to: * One of the two principal axes of a Hyperbola#Nomenclature and features, hyperbola * Principal axis (mechanics) * Aircraft principal axes * Principal axis theorem * Principal axis (crystallography) * Optical axis * The ...
of an
ellipsoid An ellipsoid is a surface that may be obtained from a sphere by deforming it by means of directional Scaling (geometry), scalings, or more generally, of an affine transformation. An ellipsoid is a quadric surface;  that is, a Surface (mathemat ...

ellipsoid
, in an infinite-dimensional setting. The later discovery in
quantum mechanics Quantum mechanics is a fundamental theory A theory is a reason, rational type of abstraction, abstract thinking about a phenomenon, or the results of such thinking. The process of contemplative and rational thinking is often associated with ...
that spectral theory could explain features of
atomic spectra Spectroscopy is the study of the interaction Interaction is a kind of action that occurs as two or more objects have an effect upon one another. The idea of a two-way effect is essential in the concept of interaction, as opposed to a one-way ...
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." 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 space In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
s and the spectral theory of single
normal operator In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...
s on them were well suited to the requirements of
physics Physics is the natural science that studies matter, its Elementary particle, fundamental constituents, its Motion (physics), motion and behavior through Spacetime, space and time, and the related entities of energy and force. "Physical scie ...

physics
, exemplified by the work of
von Neumann Von Neumann may refer to: * John von Neumann (1903–1957), a Hungarian American mathematician * Von Neumann family * Von Neumann (surname), a German surname * Von Neumann (crater), a lunar impact crater See also

* Von Neumann algebra * Von Ne ...

von Neumann
. The further theory built on this to address
Banach algebra Banach is a Polish-language surname of several possible origins."Banach"
at genezanazwisk.pl (the webpage cites the sources)
s in general. This development leads to the
Gelfand representationIn mathematics, the Gelfand representation in functional analysis (named after I. M. Gelfand) has two related meanings: * a way of representing commutative Banach algebras as algebras of continuous functions; * the fact that for commutative C*-algeb ...
, which covers the commutative case, and further into
non-commutative harmonic analysis In mathematics, noncommutative harmonic analysis is the field in which results from Fourier analysis are extended to topological groups that are not commutative. Since locally compact abelian groups have a well-understood theory, Pontryagin duality, ...
. The difference can be seen in making the connection with
Fourier analysis In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities ...
. The
Fourier transform#REDIRECT Fourier transform In mathematics, a Fourier transform (FT) is a Integral transform, mathematical transform that decomposes function (mathematics), functions depending on space or time into functions depending on spatial or temporal frequenc ...
on the
real line In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities an ...
is in one sense the spectral theory of
differentiation Differentiation may refer to: Business * Differentiation (economics), the process of making a product different from other similar products * Product differentiation, in marketing * Differentiated service, a service that varies with the identity o ...

differentiation
''qua''
differential operator 300px, A harmonic function defined on an annulus. Harmonic functions are exactly those functions which lie in the kernel of the Laplace operator, an important differential operator. In mathematics, a differential operator is an Operator (mathe ...
. But for that to cover the phenomena one has already to deal with generalized eigenfunctions (for example, by means of a
rigged Hilbert space In mathematics, a rigged Hilbert space (Gelfand triple, nested Hilbert space, equipped Hilbert space) is a construction designed to link the distribution (mathematics), distribution and square-integrable aspects of functional analysis. Such spaces ...
). 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 300px, The p-adic integer, 2-adic integers, with selected corresponding characters on Prüfer group, their Pontryagin dual group In mathematics, Pontryagin duality is a duality (mathematics), duality between locally compact abelian groups that a ...
. One can also study the spectral properties of operators on
Banach spaces In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and th ...
. For example,
compact operator In functional analysis Image:Drum vibration mode12.gif, 200px, One of the possible modes of vibration of an idealized circular drum head. These modes are eigenfunctions of a linear operator on a function space, a common construction in functional a ...
s on Banach spaces have many spectral properties similar to that of
matrices Matrix or MATRIX may refer to: Science and mathematics * Matrix (mathematics) In , a matrix (plural matrices) is a array or table of s, s, or s, arranged in rows and columns, which is used to represent a or a property of such an object. Fo ...
.


Physical background

The background in the physics of
vibration Vibration is a mechanical phenomenon whereby oscillation Oscillation is the repetitive variation, typically in time Time is the indefinite continued sequence, progress of existence and event (philosophy), events that occur in an apparentl ...

vibration
s has been explained in this way: E. Brian Davies, quoted on the King's College London analysis group website 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. The conceptual basis for
Hilbert space In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
was developed from Hilbert's ideas by Erhard Schmidt and Frigyes Riesz. It was almost twenty years later, when
quantum mechanics Quantum mechanics is a fundamental theory A theory is a reason, rational type of abstraction, abstract thinking about a phenomenon, or the results of such thinking. The process of contemplative and rational thinking is often associated with ...
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. The later discovery in quantum mechanics that spectral theory could explain features of atomic spectra was therefore fortuitous, rather than being an object of Hilbert's spectral theory.


A definition of spectrum

Consider a Bounded linear operator, bounded linear transformation ''T'' defined everywhere over a general Banach space. We form the transformation: : R_ = \left( \zeta I - T \right)^. Here ''I'' is the identity operator and ζ is a complex number. The ''inverse'' of an operator ''T'', that is ''T''−1, is defined by: :T T^ = T^ T = I. If the inverse exists, ''T'' is called ''regular''. If it does not exist, ''T'' is called ''singular''. With these definitions, the ''resolvent set'' of ''T'' is the set of all complex numbers ζ such that ''Rζ'' exists and is Bounded operator, bounded. This set often is denoted as ''ρ(T)''. The ''spectrum'' of ''T'' is the set of all complex numbers ζ such that ''Rζ'' fails to exist or is unbounded. Often the spectrum of ''T'' is denoted by ''σ(T)''. The function ''Rζ'' for all ζ in ''ρ(T)'' (that is, wherever ''Rζ'' exists as a bounded operator) is called the Resolvent formalism, resolvent of ''T''. The ''spectrum'' of ''T'' is therefore the complement of the ''resolvent set'' of ''T'' in the complex plane. Every
eigenvalue In linear algebra, an eigenvector () or characteristic vector of a Linear map, linear transformation is a nonzero Vector space, vector that changes at most by a Scalar (mathematics), scalar factor when that linear transformation is applied to i ...

eigenvalue
of ''T'' belongs to ''σ(T)'', but ''σ(T)'' may contain non-eigenvalues. This definition applies to a Banach space, but of course other types of space exist as well, for example, topological vector spaces include Banach spaces, but can be more general. On the other hand, Banach spaces include
Hilbert space In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
s, and it is these spaces that find the greatest application and the richest theoretical results. With suitable restrictions, much can be said about the structure of the Hilbert space#Spectral theory, spectra of transformations in a Hilbert space. In particular, for self-adjoint operators, the spectrum lies on the
real line In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities an ...
and (in general) is a Decomposition of spectrum (functional analysis), spectral combination of a point spectrum of discrete Eigenvalues#Computation of eigenvalues.2C and the characteristic equation, eigenvalues and a continuous spectrum.


Spectral theory briefly

In functional analysis and
linear algebra Linear algebra is the branch of mathematics concerning linear equations such as: :a_1x_1+\cdots +a_nx_n=b, linear maps such as: :(x_1, \ldots, x_n) \mapsto a_1x_1+\cdots +a_nx_n, and their representations in vector spaces and through matrix (mat ...
the spectral theorem establishes conditions under which an operator can be expressed in simple form as a sum of simpler operators. As a full rigorous presentation is not appropriate for this article, we take an approach that avoids much of the rigor and satisfaction of a formal treatment with the aim of being more comprehensible to a non-specialist. This topic is easiest to describe by introducing the bra–ket notation of Paul Dirac, Dirac for operators. As an example, a very particular linear operator ''L'' might be written as a dyadic product: : L = , k_1 \rangle \langle b_1 , , in terms of the "bra" ⟨1, and the "ket" , 1⟩. A function is described by a ''ket'' as , ⟩. The function defined on the coordinates (x_1, x_2, x_3, \dots) is denoted as : f(x)=\langle x, f\rangle and the magnitude of ''f'' by : \, f \, ^2 = \langle f, f\rangle =\int \langle f, x\rangle \langle x, f \rangle \, dx = \int f^*(x) f(x) \, dx where the notation '*' denotes a complex conjugate. This inner product choice defines a very specific inner product space, restricting the generality of the arguments that follow. The effect of ''L'' upon a function ''f'' is then described as: : L , f\rangle = , k_1 \rangle \langle b_1 , f \rangle expressing the result that the effect of ''L'' on ''f'' is to produce a new function , k_1 \rangle multiplied by the inner product represented by \langle b_1 , f \rangle . A more general linear operator ''L'' might be expressed as: : L = \lambda_1 , e_1\rangle\langle f_1, + \lambda_2 , e_2\rangle \langle f_2, + \lambda_3 , e_3\rangle\langle f_3, + \dots , where the \ are scalars and the \ are a Basis (linear algebra), basis and the \ a Dual basis, reciprocal basis for the space. The relation between the basis and the reciprocal basis is described, in part, by: : \langle f_i , e_j \rangle = \delta_ If such a formalism applies, the \ are eigenvalues of ''L'' and the functions \ are eigenfunctions of ''L''. The eigenvalues are in the ''spectrum'' of ''L''. Some natural questions are: under what circumstances does this formalism work, and for what operators ''L'' are expansions in series of other operators like this possible? Can any function ''f'' be expressed in terms of the eigenfunctions (are they a Schauder basis) and under what circumstances does a point spectrum or a continuous spectrum arise? How do the formalisms for infinite-dimensional spaces and finite-dimensional spaces differ, or do they differ? Can these ideas be extended to a broader class of spaces? Answering such questions is the realm of spectral theory and requires considerable background in functional analysis and Matrix (mathematics), matrix algebra.


Resolution of the identity

This section continues in the rough and ready manner of the above section using the bra–ket notation, and glossing over the many important details of a rigorous treatment. See discussion in Dirac's book referred to above, and A rigorous mathematical treatment may be found in various references.See, for example, the fundamental text of and , , In particular, the dimension ''n'' of the space will be finite. Using the bra–ket notation of the above section, the identity operator may be written as: :I = \sum _ ^ , e_i \rangle \langle f_i , where it is supposed as above that are a Basis (linear algebra), basis and the a reciprocal basis for the space satisfying the relation: :\langle f_i , e_j\rangle = \delta_ . This expression of the identity operation is called a ''representation'' or a ''resolution'' of the identity. This formal representation satisfies the basic property of the identity: : I^k = I\, valid for every positive integer ''k''. Applying the resolution of the identity to any function in the space '', \psi \rangle'', one obtains: :I , \psi \rangle = , \psi \rangle = \sum_^ , e_i \rangle \langle f_i , \psi \rangle = \sum_^ \ c_i , e_i \rangle which is the generalized Fourier series, Fourier expansion of ψ in terms of the basis functions . See for example, Here ''c_i = \langle f_i , \psi \rangle''. Given some operator equation of the form: :O , \psi \rangle = , h \rangle with ''h'' in the space, this equation can be solved in the above basis through the formal manipulations: : O , \psi \rangle = \sum_^ c_i \left( O , e_i \rangle \right) = \sum_^ , e_i \rangle \langle f_i , h \rangle , :\langle f_j, O, \psi \rangle = \sum_^ c_i \langle f_j, O , e_i \rangle = \sum_^ \langle f_j, e_i \rangle \langle f_i , h \rangle = \langle f_j , h \rangle, \quad \forall j which converts the operator equation to a matrix equation determining the unknown coefficients ''cj'' in terms of the generalized Fourier coefficients \langle f_j , h \rangle of ''h'' and the matrix elements O_= \langle f_j, O , e_i \rangle of the operator ''O''. The role of spectral theory arises in establishing the nature and existence of the basis and the reciprocal basis. In particular, the basis might consist of the eigenfunctions of some linear operator ''L'': :L , e_i \rangle = \lambda_i , e_i \rangle \, ; with the the eigenvalues of ''L'' from the spectrum of ''L''. Then the resolution of the identity above provides the dyad expansion of ''L'': :LI = L = \sum_^ L , e_i \rangle \langle f_i, = \sum_^ \lambda _i , e_i \rangle \langle f_i , .


Resolvent operator

Using spectral theory, the resolvent operator ''R'': :R = (\lambda I - L)^,\, can be evaluated in terms of the eigenfunctions and eigenvalues of ''L'', and the Green's function corresponding to ''L'' can be found. Applying ''R'' to some arbitrary function in the space, say \varphi, :R , \varphi \rangle = (\lambda I - L)^ , \varphi \rangle = \sum_^n \frac , e_i \rangle \langle f_i , \varphi \rangle. This function has Pole (complex analysis), poles in the complex ''λ''-plane at each eigenvalue of ''L''. Thus, using the calculus of residues: :\frac \oint_C R , \varphi \rangle d \lambda = -\sum_^n , e_i \rangle \langle f_i , \varphi \rangle = -, \varphi \rangle, where the line integral is over a contour ''C'' that includes all the eigenvalues of ''L''. Suppose our functions are defined over some coordinates , that is: :\langle x, \varphi \rangle = \varphi (x_1, x_2, ...). Introducing the notation : \langle x , y \rangle = \delta (x-y), where ''δ(x − y)'' = ''δ(x1 − y1, x2 − y2, x3 − y3, ...)'' is the Dirac delta function, we can write :\langle x, \varphi \rangle = \int \langle x , y \rangle \langle y, \varphi \rangle dy. Then: :\begin \left\langle x, \frac \oint_C \frac d \lambda\right\rangle &= \frac\oint_C d \lambda \left \langle x, \frac \right \rangle\\ &= \frac \oint_C d \lambda \int dy \left \langle x, \frac \right \rangle \langle y, \varphi \rangle \end The function ''G(x, y; λ)'' defined by: :\begin G(x, y; \lambda) &= \left \langle x, \frac \right \rangle \\ &= \sum_^n \sum_^n \langle x, e_i \rangle \left \langle f_i, \frac \right \rangle \langle f_j , y\rangle \\ &= \sum_^n \frac \\ &= \sum_^n \frac, \end is called the ''Green's function'' for operator ''L'', and satisfies: :\frac\oint_C G(x,y;\lambda) \, d \lambda = -\sum_^n \langle x, e_i \rangle \langle f_i , y\rangle = -\langle x, y\rangle = -\delta (x-y).


Operator equations

Consider the operator equation: :(O-\lambda I ) , \psi \rangle = , h \rangle; in terms of coordinates: :\int \langle x, (O-\lambda I)y \rangle \langle y, \psi \rangle \, dy = h(x). A particular case is ''λ'' = 0. The Green's function of the previous section is: :\langle y, G(\lambda) z\rangle = \left \langle y, (O-\lambda I)^ z \right \rangle = G(y, z; \lambda), and satisfies: :\int \langle x, (O - \lambda I) y \rangle \langle y, G(\lambda) z \rangle \, dy = \int \langle x, (O-\lambda I) y \rangle \left \langle y, (O-\lambda I)^ z \right \rangle \, dy = \langle x , z \rangle = \delta (x-z). Using this Green's function property: :\int \langle x, (O-\lambda I) y \rangle G(y, z; \lambda ) \, dy = \delta (x-z). Then, multiplying both sides of this equation by ''h''(''z'') and integrating: :\int dz \, h(z) \int dy \, \langle x, (O-\lambda I)y \rangle G(y, z; \lambda)=\int dy \, \langle x, (O-\lambda I) y \rangle \int dz \, h(z)G(y, z; \lambda) = h(x), which suggests the solution is: :\psi(x) = \int h(z) G(x, z; \lambda) \, dz. 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: :G(x, z; \lambda) = \sum_^n \frac. There are many other ways to find ''G'', of course. For example, see and See the articles on Green's function#Green.27s functions for solving inhomogeneous boundary value problems, Green's functions and on Fredholm theory#Inhomogeneous equations, Fredholm integral equations. 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, Distribution (mathematics), distributions and so forth. Consult these articles and the references for more detail.


Spectral theorem and Rayleigh quotient

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. Moreover, this eigenvalue is the largest eigenvalue of M. '' Proof Assume the spectral theorem. Let the eigenvalues of M be \lambda_1\le\lambda_2\le\cdots\le\lambda_n. Since the form an orthonormal basis, any vector x can be expressed in this Basis (linear algebra), basis as : x = \sum_i v_i^T x v_i The way to prove this formula is pretty easy. Namely, : \begin v_j^T \sum_i v_i^T x v_i \\[4pt] = & \sum_ v_i^ x v_j^ v_i \\[4pt] = & (v_j^T x ) v_j^T v_j \\[4pt] = & v_j^T x \end evaluate the Rayleigh quotient with respect to x: : \begin & x^T M x \\[4pt] = & \left(\sum_i (v_i^T x) v_i\right)^T M \left(\sum_j (v_j^T x) v_j\right) \\[4pt] = & \left(\sum_i (v_i^T x) v_i^T\right) \left(\sum_j (v_j^T x) v_j\lambda_j \right) \\[4pt] = & \sum_ (v_i^T x) v_i^T(v_j^T x) v_j\lambda_j \\[4pt] = & \sum_j (v_j^T x)(v_j^T x)\lambda_j \\[4pt] = & \sum_ (v_j^T x)^2\lambda_j\le\lambda_n \sum_j (v_j^T x)^2 \\[4pt] = & \lambda_n x^T x, \end where we used Parseval's identity in the last line. Finally we obtain that :\frac\le \lambda_n so the Rayleigh quotient is always less than \lambda_n.Spielman, Daniel A. "Lecture Notes on Spectral Graph Theory" Yale University (2012) http://cs.yale.edu/homes/spielman/561/ .


See also

* Spectrum (functional analysis), Resolvent formalism, Decomposition of spectrum (functional analysis) * Spectral radius, Spectrum of an operator, Spectral theorem * Spectral theory of compact operators * Spectral theory of normal C*-algebras * functional calculus, Functions of operators, Operator theory * Riesz projector * Self-adjoint operator * Sturm–Liouville theory, Integral equations, Fredholm theory * Compact operators, Isospectral operators, complete metric space, Completeness * Lax pairs * Spectral geometry * Spectral graph theory * List of functional analysis topics


Notes


References

* * * * * * * * *


External links


Evans M. Harrell II
A Short History of Operator Theory * * {{DEFAULTSORT:Spectral Theory Spectral theory, Linear algebra de:Spektrum (Operatortheorie)