In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, spectral theory is an inclusive term for theories extending the
eigenvector
In linear algebra, an eigenvector () or characteristic vector of a linear transformation is a nonzero vector that changes at most by a scalar factor when that linear transformation is applied to it. The corresponding eigenvalue, often denoted b ...
and
eigenvalue
In linear algebra, an eigenvector () or characteristic vector of a linear transformation is a nonzero vector that changes at most by a scalar factor when that linear transformation is applied to it. The corresponding eigenvalue, often denoted b ...
theory of a single
square matrix
In mathematics, a square matrix is a matrix with the same number of rows and columns. An ''n''-by-''n'' matrix is known as a square matrix of order Any two square matrices of the same order can be added and multiplied.
Square matrices are often ...
to a much broader theory of the structure of
operators in a variety of
mathematical space
In mathematics, a space is a set (sometimes called a universe) with some added structure.
While modern mathematics uses many types of spaces, such as Euclidean spaces, linear spaces, topological spaces, Hilbert spaces, or probability spaces, ...
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 matrices.
...
and the solutions of
systems of linear equations
In mathematics, a system of linear equations (or linear system) is a collection of one or more linear equations involving the same variables.
For example,
:\begin
3x+2y-z=1\\
2x-2y+4z=-2\\
-x+\fracy-z=0
\end
is a system of three equations in th ...
and their generalizations.
The theory is connected to that of
analytic functions
In mathematics, an analytic function is a function that is locally given by a convergent power series. There exist both real analytic functions and complex analytic functions. Functions of each type are infinitely differentiable, but complex a ...
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, one of the most influential mathematicians of the 19th and early 20th centuries. Hilbert discovered and developed a broad range of fundamental ideas in many a ...
in his original formulation of
Hilbert space
In mathematics, Hilbert spaces (named after David Hilbert) allow generalizing the methods of linear algebra and calculus from (finite-dimensional) Euclidean vector spaces to spaces that may be infinite-dimensional. Hilbert spaces arise natural ...
theory, which was cast in terms of
quadratic form
In mathematics, a quadratic form is a polynomial with terms all of degree two ("form" is another name for a homogeneous polynomial). For example,
:4x^2 + 2xy - 3y^2
is a quadratic form in the variables and . The coefficients usually belong to a ...
s in infinitely many variables. The original
spectral theorem
In mathematics, particularly linear algebra and functional analysis, a spectral theorem is a result about when a linear operator or matrix (mathematics), matrix can be Diagonalizable matrix, diagonalized (that is, represented as a diagonal matrix i ...
was therefore conceived as a version of the theorem on
principal axes of an
ellipsoid
An ellipsoid is a surface that may be obtained from a sphere by deforming it by means of directional scalings, or more generally, of an affine transformation.
An ellipsoid is a quadric surface; that is, a surface that may be defined as the ...
, in an infinite-dimensional setting. The later discovery in
quantum mechanics
Quantum mechanics is a fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum physics including quantum chemistry, ...
that spectral theory could explain features of
atomic spectra
Spectroscopy is the field of study that measures and interprets the electromagnetic spectra that result from the interaction between electromagnetic radiation and matter as a function of the wavelength or frequency of the radiation. Matter wa ...
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, Hilbert spaces (named after David Hilbert) allow generalizing the methods of linear algebra and calculus from (finite-dimensional) Euclidean vector spaces to spaces that may be infinite-dimensional. Hilbert spaces arise natural ...
s and the spectral theory of single
normal operator
In mathematics, especially functional analysis, a normal operator on a complex Hilbert space ''H'' is a continuous linear operator ''N'' : ''H'' → ''H'' that commutes with its hermitian adjoint ''N*'', that is: ''NN*'' = ''N*N''.
Normal operat ...
s on them were well suited to the requirements of
physics
Physics is the natural science that studies matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge which r ...
, 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 ...
.
[
] The further theory built on this to address
Banach algebra
In mathematics, especially functional analysis, a Banach algebra, named after Stefan Banach, is an associative algebra A over the real or complex numbers (or over a non-Archimedean complete normed field) that at the same time is also a Banach spa ...
s in general. This development leads to the
Gelfand representation In mathematics, the Gelfand representation in functional analysis (named after I. M. Gelfand) is either of two things:
* a way of representing commutative Banach algebras as algebras of continuous functions;
* the fact that for commutative C*-alge ...
, 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 dualit ...
.
The difference can be seen in making the connection with
Fourier analysis
In mathematics, Fourier analysis () is the study of the way general functions may be represented or approximated by sums of simpler trigonometric functions. Fourier analysis grew from the study of Fourier series, and is named after Josep ...
. The
Fourier transform
A Fourier transform (FT) is a mathematical transform that decomposes functions into frequency components, which are represented by the output of the transform as a function of frequency. Most commonly functions of time or space are transformed, ...
on the
real line
In elementary mathematics, a number line is a picture of a graduated straight line (geometry), line that serves as visual representation of the real numbers. Every point of a number line is assumed to correspond to a real number, and every real ...
is in one sense the spectral theory of
differentiation ''qua''
differential operator
In mathematics, a differential operator is an operator defined as a function of the differentiation operator. It is helpful, as a matter of notation first, to consider differentiation as an abstract operation that accepts a function and return ...
. But for that to cover the phenomena one has already to deal with
generalized eigenfunction
In mathematics, a rigged Hilbert space (Gelfand triple, nested Hilbert space, equipped Hilbert space) is a construction designed to link the distribution and square-integrable aspects of functional analysis. Such spaces were introduced to study s ...
s (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 and square-integrable aspects of functional analysis. Such spaces were introduced to study s ...
). 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
In mathematics, Pontryagin duality is a duality between locally compact abelian groups that allows generalizing Fourier transform to all such groups, which include the circle group (the multiplicative group of complex numbers of modulus one), ...
.
One can also study the spectral properties of operators on
Banach spaces
In mathematics, more specifically in functional analysis, a Banach space (pronounced ) is a complete normed vector space. Thus, a Banach space is a vector space with a metric that allows the computation of vector length and distance between vector ...
. For example,
compact operator
In functional analysis, a branch of mathematics, a compact operator is a linear operator T: X \to Y, where X,Y are normed vector spaces, with the property that T maps bounded subsets of X to relatively compact subsets of Y (subsets with compact c ...
s on Banach spaces have many spectral properties similar to that of
matrices
Matrix most commonly refers to:
* ''The Matrix'' (franchise), an American media franchise
** ''The Matrix'', a 1999 science-fiction action film
** "The Matrix", a fictional setting, a virtual reality environment, within ''The Matrix'' (franchis ...
.
Physical background
The background in the physics of
vibration
Vibration is a mechanical phenomenon whereby oscillations occur about an equilibrium point. The word comes from Latin ''vibrationem'' ("shaking, brandishing"). The oscillations may be periodic function, periodic, such as the motion of a pendulum ...
s has been explained in this way:
E. Brian Davies
Edward Brian Davies FRS (born 13 June 1944) is a former professor of Mathematics, King's College London (1981–2010), and is the author of the popular science book ''Science in the Looking Glass: What do Scientists Really Know''. In 2010, he ...
, 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
Mark Kac ( ; Polish: ''Marek Kac''; August 3, 1914 – October 26, 1984) was a Polish American mathematician. His main interest was probability theory. His question, " Can one hear the shape of a drum?" set off research into spectral theory, the ...
'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
Wilhelm Wirtinger (19 July 1865 – 16 January 1945) was an Austrian mathematician, working in complex analysis, geometry, algebra, number theory, Lie groups and knot theory.
Biography
He was born at Ybbs on the Danube and studied at the Unive ...
on
Hill differential equation
In mathematics, the Hill equation or Hill differential equation is the second-order linear ordinary differential equation
: \frac + f(t) y = 0,
where f(t) is a periodic function by minimal period \pi . By these we mean that for all t
:f(t+\ ...
(by
Jean Dieudonné
Jean Alexandre Eugène Dieudonné (; 1 July 1906 – 29 November 1992) was a French mathematician, notable for research in abstract algebra, algebraic geometry, and functional analysis, for close involvement with the Nicolas Bourbaki pseudonymo ...
), and it was taken up by his students during the first decade of the twentieth century, among them
Erhard Schmidt
Erhard Schmidt (13 January 1876 – 6 December 1959) was a Baltic German mathematician whose work significantly influenced the direction of mathematics in the twentieth century. Schmidt was born in Tartu (german: link=no, Dorpat), in the Govern ...
and
Hermann Weyl
Hermann Klaus Hugo Weyl, (; 9 November 1885 – 8 December 1955) was a German mathematician, theoretical physicist and philosopher. Although much of his working life was spent in Zürich, Switzerland, and then Princeton, New Jersey, he is assoc ...
. The conceptual basis for
Hilbert space
In mathematics, Hilbert spaces (named after David Hilbert) allow generalizing the methods of linear algebra and calculus from (finite-dimensional) Euclidean vector spaces to spaces that may be infinite-dimensional. Hilbert spaces arise natural ...
was developed from Hilbert's ideas by
Erhard Schmidt
Erhard Schmidt (13 January 1876 – 6 December 1959) was a Baltic German mathematician whose work significantly influenced the direction of mathematics in the twentieth century. Schmidt was born in Tartu (german: link=no, Dorpat), in the Govern ...
and
Frigyes Riesz
Frigyes Riesz ( hu, Riesz Frigyes, , sometimes spelled as Frederic; 22 January 1880 – 28 February 1956) was a HungarianEberhard Zeidler: Nonlinear Functional Analysis and Its Applications: Linear monotone operators. Springer, 199/ref> mathema ...
.
[
] It was almost twenty years later, when
quantum mechanics
Quantum mechanics is a fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum physics including quantum chemistry, ...
was formulated in terms of the
Schrödinger equation
The Schrödinger equation is a linear partial differential equation that governs the wave function of a quantum-mechanical system. It is a key result in quantum mechanics, and its discovery was a significant landmark in the development of the ...
, that the connection was made to
atomic spectra
Spectroscopy is the field of study that measures and interprets the electromagnetic spectra that result from the interaction between electromagnetic radiation and matter as a function of the wavelength or frequency of the radiation. Matter wa ...
; a connection with the mathematical physics of vibration had been suspected before, as remarked by
Henri Poincaré
Jules Henri Poincaré ( S: stress final syllable ; 29 April 1854 – 17 July 1912) was a French mathematician, theoretical physicist, engineer, and philosopher of science. He is often described as a polymath, and in mathematics as "The ...
, but rejected for simple quantitative reasons, absent an explanation of the
Balmer series
The Balmer series, or Balmer lines in atomic physics, is one of a set of six named series describing the spectral line emissions of the hydrogen atom. The Balmer series is calculated using the Balmer formula, an empirical equation discovered b ...
. 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 transformation ''T'' defined everywhere over a general
Banach space
In mathematics, more specifically in functional analysis, a Banach space (pronounced ) is a complete normed vector space. Thus, a Banach space is a vector space with a metric that allows the computation of vector length and distance between vector ...
. We form the transformation:
Here ''I'' is the
identity operator
Identity may refer to:
* Identity document
* Identity (philosophy)
* Identity (social science)
* Identity (mathematics)
Arts and entertainment Film and television
* ''Identity'' (1987 film), an Iranian film
* ''Identity'' (2003 film), a ...
and ζ is a
complex number
In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the form ...
. The ''inverse'' of an operator ''T'', that is ''T''
−1, is defined by:
If the inverse exists, ''T'' is called ''regular''. If it does not exist, ''T'' is called ''singular''.
With these definitions, the ''
resolvent set In linear algebra and operator theory, the resolvent set of a linear operator is a set of complex numbers for which the operator is in some sense "well-behaved". The resolvent set plays an important role in the resolvent formalism.
Definitions
...
'' of ''T'' is the set of all complex numbers ζ such that ''R
ζ'' exists and is
bounded. This set often is denoted as ''ρ''(''T''). The ''spectrum'' of ''T'' is the set of all complex numbers ζ such that ''R
ζ''
fails to exist or is unbounded. Often the spectrum of ''T'' is denoted by ''σ''(''T''). The function ''R
ζ'' for all ζ in ''ρ''(''T'') (that is, wherever ''R
ζ'' 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.
Every
eigenvalue
In linear algebra, an eigenvector () or characteristic vector of a linear transformation is a nonzero vector that changes at most by a scalar factor when that linear transformation is applied to it. The corresponding eigenvalue, often denoted b ...
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
In mathematics, a topological vector space (also called a linear topological space and commonly abbreviated TVS or t.v.s.) is one of the basic structures investigated in functional analysis.
A topological vector space is a vector space that is als ...
include Banach spaces, but can be more general.
On the other hand, Banach spaces include
Hilbert space
In mathematics, Hilbert spaces (named after David Hilbert) allow generalizing the methods of linear algebra and calculus from (finite-dimensional) Euclidean vector spaces to spaces that may be infinite-dimensional. Hilbert spaces arise natural ...
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
spectra of transformations in a Hilbert space. In particular, for
self-adjoint operator
In mathematics, a self-adjoint operator on an infinite-dimensional complex vector space ''V'' with inner product \langle\cdot,\cdot\rangle (equivalently, a Hermitian operator in the finite-dimensional case) is a linear map ''A'' (from ''V'' to itse ...
s, the spectrum lies on the
real line
In elementary mathematics, a number line is a picture of a graduated straight line (geometry), line that serves as visual representation of the real numbers. Every point of a number line is assumed to correspond to a real number, and every real ...
and (in general) is a
spectral combination of a point spectrum of discrete
eigenvalues
In linear algebra, an eigenvector () or characteristic vector of a linear transformation is a nonzero vector that changes at most by a scalar factor when that linear transformation is applied to it. The corresponding eigenvalue, often denoted b ...
and a
continuous spectrum.
Spectral theory briefly
In
functional analysis
Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure (e.g. Inner product space#Definition, inner product, Norm (mathematics)#Defini ...
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 matrices.
...
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
In quantum mechanics, bra–ket notation, or Dirac notation, is used ubiquitously to denote quantum states. The notation uses angle brackets, and , and a vertical bar , to construct "bras" and "kets".
A ket is of the form , v \rangle. Mathema ...
of
Dirac
Distributed Research using Advanced Computing (DiRAC) is an integrated supercomputing facility used for research in particle physics, astronomy and cosmology in the United Kingdom. DiRAC makes use of multi-core processors and provides a variety o ...
for operators.
As an example, a very particular linear operator ''L'' might be written as a
dyadic product In mathematics, specifically multilinear algebra, a dyadic or dyadic tensor is a second order tensor, written in a notation that fits in with vector algebra.
There are numerous ways to multiply two Euclidean vectors. The dot product takes in two v ...
:
:
in terms of the "bra" ⟨
1, and the "ket" ,
1⟩. A function is described by a ''ket'' as , ⟩. The function defined on the coordinates
is denoted as
:
and the magnitude of ''f'' by
:
where the notation (*) denotes a
complex conjugate
In mathematics, the complex conjugate of a complex number is the number with an equal real part and an imaginary part equal in magnitude but opposite in sign. That is, (if a and b are real, then) the complex conjugate of a + bi is equal to a - ...
. This
inner product
In mathematics, an inner product space (or, rarely, a Hausdorff space, Hausdorff pre-Hilbert space) is a real vector space or a complex vector space with an operation (mathematics), operation called an inner product. The inner product of two ve ...
choice defines a very specific
inner product space
In mathematics, an inner product space (or, rarely, a Hausdorff pre-Hilbert space) is a real vector space or a complex vector space with an operation called an inner product. The inner product of two vectors in the space is a scalar, often den ...
, restricting the generality of the arguments that follow.
[
The effect of ''L'' upon a function ''f'' is then described as:
:
expressing the result that the effect of ''L'' on ''f'' is to produce a new function multiplied by the inner product represented by .
A more general linear operator ''L'' might be expressed as:
:
where the are scalars and the are a ]basis
Basis may refer to:
Finance and accounting
* Adjusted basis, the net cost of an asset after adjusting for various tax-related items
*Basis point, 0.01%, often used in the context of interest rates
* Basis trading, a trading strategy consisting ...
and the a reciprocal basis for the space. The relation between the basis and the reciprocal basis is described, in part, by:
:
If such a formalism applies, the are eigenvalues
In linear algebra, an eigenvector () or characteristic vector of a linear transformation is a nonzero vector that changes at most by a scalar factor when that linear transformation is applied to it. The corresponding eigenvalue, often denoted b ...
of ''L'' and the functions are eigenfunctions
In mathematics, an eigenfunction of a linear operator ''D'' defined on some function space is any non-zero function f in that space that, when acted upon by ''D'', is only multiplied by some scaling factor called an eigenvalue. As an equation, th ...
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 In mathematics, a Schauder basis or countable basis is similar to the usual ( Hamel) basis of a vector space; the difference is that Hamel bases use linear combinations that are finite sums, while for Schauder bases they may be infinite sums. This ...
) 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
Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure (e.g. Inner product space#Definition, inner product, Norm (mathematics)#Defini ...
and matrix algebra
In abstract algebra, a matrix ring is a set of matrices with entries in a ring ''R'' that form a ring under matrix addition and matrix multiplication . The set of all matrices with entries in ''R'' is a matrix ring denoted M''n''(''R'')Lang, ''U ...
.
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:
:
where it is supposed as above that are a basis
Basis may refer to:
Finance and accounting
* Adjusted basis, the net cost of an asset after adjusting for various tax-related items
*Basis point, 0.01%, often used in the context of interest rates
* Basis trading, a trading strategy consisting ...
and the a reciprocal basis for the space satisfying the relation:
:
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:
:
valid for every positive integer ''k''.
Applying the resolution of the identity to any function in the space , one obtains:
:
which is the generalized ]Fourier expansion
A Fourier series () is a summation of harmonically related sinusoidal functions, also known as components or harmonics. The result of the summation is a periodic function whose functional form is determined by the choices of cycle length (or '' ...
of ψ in terms of the basis functions .[
See for example,
]
Here .
Given some operator equation of the form:
:
with ''h'' in the space, this equation can be solved in the above basis through the formal manipulations:
:
:
which converts the operator equation to a matrix equation
In mathematics, a matrix (plural matrices) is a rectangular array or table of numbers, symbols, or expressions, arranged in rows and columns, which is used to represent a mathematical object or a property of such an object.
For example,
\begin ...
determining the unknown coefficients ''cj'' in terms of the generalized Fourier coefficients of ''h'' and the matrix elements 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'':
:
with the the eigenvalues of ''L'' from the spectrum of ''L''. Then the resolution of the identity above provides the dyad expansion of ''L'':
:
Resolvent operator
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.
Applying ''R'' to some arbitrary function in the space, say ,
:
This function has poles
Poles,, ; singular masculine: ''Polak'', singular feminine: ''Polka'' or Polish people, are a West Slavic nation and ethnic group, who share a common history, culture, the Polish language and are identified with the country of Poland in Ce ...
in the complex ''λ''-plane at each eigenvalue of ''L''. Thus, using the calculus of residues:
:
where the line integral
In mathematics, a line integral is an integral where the function to be integrated is evaluated along a curve. The terms ''path integral'', ''curve integral'', and ''curvilinear integral'' are also used; ''contour integral'' is used as well, alt ...
is over a contour ''C'' that includes all the eigenvalues of ''L''.
Suppose our functions are defined over some coordinates , that is:
:
Introducing the notation
:
where ''δ(x − y)'' = ''δ(x1 − y1, x2 − y2, x3 − y3, ...)'' is the Dirac delta function
In mathematics, the Dirac delta distribution ( distribution), also known as the unit impulse, is a generalized function or distribution over the real numbers, whose value is zero everywhere except at zero, and whose integral over the entire ...
,
we can write
:
Then:
:
The function ''G(x, y; λ)'' defined by:
:
is called the ''Green's function
In mathematics, a Green's function is the impulse response of an inhomogeneous linear differential operator defined on a domain with specified initial conditions or boundary conditions.
This means that if \operatorname is the linear differential ...
'' for operator ''L'', and satisfies:
:
Operator equations
Consider the operator equation:
:
in terms of coordinates:
:
A particular case is ''λ'' = 0.
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.[
For example, see and ] See the articles on Green's functions
In mathematics, a Green's function is the impulse response of an inhomogeneous linear differential operator defined on a domain with specified initial conditions or boundary conditions.
This means that if \operatorname is the linear differential ...
and on 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
Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure (e.g. Inner product space#Definition, inner product, Norm (mathematics)#Defini ...
, Hilbert spaces
In mathematics, Hilbert spaces (named after David Hilbert) allow generalizing the methods of linear algebra and calculus from (finite-dimensional) Euclidean vector spaces to spaces that may be infinite-dimensional. Hilbert spaces arise naturally ...
, distributions and so forth. Consult these articles and the references for more detail.
Spectral theorem and Rayleigh quotient
Optimization problem
In mathematics, computer science and economics, an optimization problem is the problem of finding the ''best'' solution from all feasible solutions.
Optimization problems can be divided into two categories, depending on whether the variables ...
s may be the most useful examples about the combinatorial significance of the eigenvalues and eigenvectors in symmetric matrices, especially for the Rayleigh quotient
In mathematics, the Rayleigh quotient () for a given complex Hermitian matrix ''M'' and nonzero vector ''x'' is defined as:
R(M,x) = .
For real matrices and vectors, the condition of being Hermitian reduces to that of being symmetric, and the co ...
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
In mathematics, the Rayleigh quotient () for a given complex Hermitian matrix ''M'' and nonzero vector ''x'' is defined as:
R(M,x) = .
For real matrices and vectors, the condition of being Hermitian reduces to that of being symmetric, and the co ...
with respect to M. Then, x is an eigenvector of M with eigenvalue equal to the Rayleigh quotient
In mathematics, the Rayleigh quotient () for a given complex Hermitian matrix ''M'' and nonzero vector ''x'' is defined as:
R(M,x) = .
For real matrices and vectors, the condition of being Hermitian reduces to that of being symmetric, and the co ...
. Moreover, this eigenvalue is the largest eigenvalue of M. ''
Proof Assume the spectral theorem. Let the eigenvalues of M be . Since the form an orthonormal basis
In mathematics, particularly linear algebra, an orthonormal basis for an inner product space ''V'' with finite dimension is a basis for V whose vectors are orthonormal, that is, they are all unit vectors and orthogonal to each other. For example, ...
, any vector x can be expressed in this basis
Basis may refer to:
Finance and accounting
* Adjusted basis, the net cost of an asset after adjusting for various tax-related items
*Basis point, 0.01%, often used in the context of interest rates
* Basis trading, a trading strategy consisting ...
as
:
The way to prove this formula is pretty easy. Namely,
:
evaluate the Rayleigh quotient
In mathematics, the Rayleigh quotient () for a given complex Hermitian matrix ''M'' and nonzero vector ''x'' is defined as:
R(M,x) = .
For real matrices and vectors, the condition of being Hermitian reduces to that of being symmetric, and the co ...
with respect to ''x'':
:
where we used Parseval's identity In mathematical analysis, Parseval's identity, named after Marc-Antoine Parseval, is a fundamental result on the summability of the Fourier series of a function. Geometrically, it is a generalized Pythagorean theorem for inner-product spaces (which ...
in the last line. Finally we obtain that
:
so the Rayleigh quotient
In mathematics, the Rayleigh quotient () for a given complex Hermitian matrix ''M'' and nonzero vector ''x'' is defined as:
R(M,x) = .
For real matrices and vectors, the condition of being Hermitian reduces to that of being symmetric, and the co ...
is always less than .[Spielman, Daniel A. "Lecture Notes on Spectral Graph Theory" Yale University (2012) http://cs.yale.edu/homes/spielman/561/ .]
See also
* Functions of operators, Operator theory
In mathematics, operator theory is the study of linear operators on function spaces, beginning with differential operators and integral operators. The operators may be presented abstractly by their characteristics, such as bounded linear operat ...
* Lax pair In mathematics, in the theory of integrable systems, a Lax pair is a pair of time-dependent matrices or operators that satisfy a corresponding differential equation, called the ''Lax equation''. Lax pairs were introduced by Peter Lax to discuss sol ...
s
* Least-squares spectral analysis
Least-squares spectral analysis (LSSA) is a method of estimating a frequency spectrum, based on a least squares fit of sinusoids to data samples, similar to Fourier analysis. Fourier analysis, the most used spectral method in science, generally ...
* Riesz projector In mathematics, or more specifically in spectral theory, the Riesz projector is the projector onto the eigenspace corresponding to a particular eigenvalue of an operator (or, more generally, a projector onto an invariant subspace corresponding to an ...
* Self-adjoint operator
In mathematics, a self-adjoint operator on an infinite-dimensional complex vector space ''V'' with inner product \langle\cdot,\cdot\rangle (equivalently, a Hermitian operator in the finite-dimensional case) is a linear map ''A'' (from ''V'' to itse ...
* Spectrum (functional analysis)
In mathematics, particularly in functional analysis, the spectrum of a bounded linear operator (or, more generally, an unbounded linear operator) is a generalisation of the set of eigenvalues of a matrix. Specifically, a complex number \lambda is ...
, Resolvent formalism In mathematics, the resolvent formalism is a technique for applying concepts from complex analysis to the study of the spectrum of operators on Banach spaces and more general spaces. Formal justification for the manipulations can be found in the fr ...
, Decomposition of spectrum (functional analysis)
* Spectral radius
In mathematics, the spectral radius of a square matrix is the maximum of the absolute values of its eigenvalues. More generally, the spectral radius of a bounded linear operator is the supremum of the absolute values of the elements of its spectru ...
, Spectrum of an operator
In mathematics, particularly in functional analysis, the spectrum of a bounded linear operator (or, more generally, an unbounded linear operator) is a generalisation of the set of eigenvalues of a matrix. Specifically, a complex number \lambda is ...
, Spectral theorem
In mathematics, particularly linear algebra and functional analysis, a spectral theorem is a result about when a linear operator or matrix (mathematics), matrix can be Diagonalizable matrix, diagonalized (that is, represented as a diagonal matrix i ...
* Spectral theory of compact operators
In functional analysis, compact operators are linear operators on Banach spaces that map bounded sets to relatively compact sets. In the case of a Hilbert space ''H'', the compact operators are the closure of the finite rank operators in the unifo ...
* Spectral theory of normal C*-algebras
In functional analysis, every C*-algebra is isomorphic to a subalgebra of the C*-algebra \mathcal(H) of bounded linear operators on some Hilbert space H. This article describes the spectral theory of closed normal subalgebras of \mathcal(H). A ...
* Sturm–Liouville theory
In mathematics and its applications, a Sturm–Liouville problem is a second-order linear ordinary differential equation of the form:
\frac\!\!\left ,p(x)\frac\right+ q(x)y = -\lambda\, w(x)y,
for given functions p(x), q(x) and w(x), together wit ...
, Integral equation
In mathematics, integral equations are equations in which an unknown Function (mathematics), function appears under an integral sign. In mathematical notation, integral equations may thus be expressed as being of the form: f(x_1,x_2,x_3,...,x_n ; ...
s, Fredholm theory In mathematics, Fredholm theory is a theory of integral equations. In the narrowest sense, Fredholm theory concerns itself with the solution of the Fredholm integral equation. In a broader sense, the abstract structure of Fredholm's theory is given ...
* Compact operator
In functional analysis, a branch of mathematics, a compact operator is a linear operator T: X \to Y, where X,Y are normed vector spaces, with the property that T maps bounded subsets of X to relatively compact subsets of Y (subsets with compact c ...
s, Isospectral
In mathematics, two linear operators are called isospectral or cospectral if they have the same spectrum. Roughly speaking, they are supposed to have the same sets of eigenvalues, when those are counted with multiplicity.
The theory of isospectr ...
operators, Completeness
* Spectral geometry Spectral geometry is a field in mathematics which concerns relationships between geometric structures of manifolds and spectra of canonically defined differential operators. The case of the Laplace–Beltrami operator on a closed Riemannian m ...
* Spectral graph theory
In mathematics, spectral graph theory is the study of the properties of a graph in relationship to the characteristic polynomial, eigenvalues, and eigenvectors of matrices associated with the graph, such as its adjacency matrix or Laplacian matrix ...
* List of functional analysis topics
Notes
References
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External links
Evans M. Harrell II
A Short History of Operator Theory
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{{DEFAULTSORT:Spectral Theory
Linear algebra
de:Spektrum (Operatortheorie)