spectral theory
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mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, spectral theory is an inclusive term for theories extending the eigenvector and eigenvalue theory of a single square matrix to a much broader theory of the structure of operators in a variety of mathematical spaces. 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 (mathemat ...
and the solutions of systems of linear equations and their generalizations. The theory is connected to that of analytic functions 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 in his original formulation of
Hilbert space In mathematics, a Hilbert space is a real number, real or complex number, complex inner product space that is also a complete metric space with respect to the metric induced by the inner product. It generalizes the notion of Euclidean space. The ...
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 t ...
s in infinitely many variables. The original
spectral theorem In linear algebra and functional analysis, a spectral theorem is a result about when a linear operator or matrix can be diagonalized (that is, represented as a diagonal matrix in some basis). This is extremely useful because computations involvin ...
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 Quantum mechanics is the fundamental physical Scientific theory, theory that describes the behavior of matter and of light; its unusual characteristics typically occur at and below the scale of atoms. Reprinted, Addison-Wesley, 1989, It is ...
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." 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, a Hilbert space is a real number, real or complex number, complex inner product space that is also a complete metric space with respect to the metric induced by the inner product. It generalizes the notion of Euclidean space. The ...
s and the spectral theory of single normal operators on them were well suited to the requirements of
physics Physics is the scientific study of matter, its Elementary particle, 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 whi ...
, exemplified by the work of von Neumann. 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. The difference can be seen in making the connection with Fourier analysis. The
Fourier transform In mathematics, the Fourier transform (FT) is an integral transform that takes a function as input then outputs another function that describes the extent to which various frequencies are present in the original function. The output of the tr ...
on the
real line A number line is a graphical representation of a straight line that serves as spatial representation of numbers, usually graduated like a ruler with a particular origin (geometry), origin point representing the number zero and evenly spaced mark ...
is in one sense the spectral theory of differentiation as a 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. 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.


Physical background

The background in the physics of
vibration Vibration () is a mechanical phenomenon whereby oscillations occur about an equilibrium point. Vibration may be deterministic if the oscillations can be characterised precisely (e.g. the periodic motion of a pendulum), or random if the os ...
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 Hermann Klaus Hugo Weyl (; ; 9 November 1885 – 8 December 1955) was a German mathematician, theoretical physicist, logician and philosopher. Although much of his working life was spent in Zürich, Switzerland, and then Princeton, New Jersey, ...
. The conceptual basis for
Hilbert space In mathematics, a Hilbert space is a real number, real or complex number, complex inner product space that is also a complete metric space with respect to the metric induced by the inner product. It generalizes the notion of Euclidean space. The ...
was developed from Hilbert's ideas by Erhard Schmidt and Frigyes Riesz. It was almost twenty years later, when
quantum mechanics Quantum mechanics is the fundamental physical Scientific theory, theory that describes the behavior of matter and of light; its unusual characteristics typically occur at and below the scale of atoms. Reprinted, Addison-Wesley, 1989, It is ...
was formulated in terms of the
Schrödinger equation The Schrödinger equation is a partial differential equation that governs the wave function of a non-relativistic quantum-mechanical system. Its discovery was a significant landmark in the development of quantum mechanics. It is named after E ...
, 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 Jules Henri Poincaré (, ; ; 29 April 185417 July 1912) was a French mathematician, Theoretical physics, theoretical physicist, engineer, and philosophy of science, philosopher of science. He is often described as a polymath, and in mathemati ...
, 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 transformation ''T'' defined everywhere over a general
Banach space In mathematics, more specifically in functional analysis, a Banach space (, ) 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 vectors and ...
. We form the transformation: R_ = \left( \zeta I - T \right)^. Here ''I'' is the identity operator 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 for ...
. 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. 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 of ''T''. The ''spectrum'' of ''T'' is therefore the complement of the ''resolvent set'' of ''T'' in the complex plane. Every 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, a Hilbert space is a real number, real or complex number, complex inner product space that is also a complete metric space with respect to the metric induced by the inner product. It generalizes the notion of Euclidean space. The ...
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 operators, the spectrum lies on the
real line A number line is a graphical representation of a straight line that serves as spatial representation of numbers, usually graduated like a ruler with a particular origin (geometry), origin point representing the number zero and evenly spaced mark ...
and (in general) is a spectral combination of a point spectrum of discrete eigenvalues 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 (for example, Inner product space#Definition, inner product, Norm (mathematics ...
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 (mathemat ...
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 Bra–ket notation, also called Dirac notation, is a notation for linear algebra and linear operators on complex vector spaces together with their dual space both in the finite-dimensional and infinite-dimensional case. It is specifically de ...
of 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 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 numbers, then the complex conjugate of a + bi is a - ...
. This
inner product 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, ofte ...
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 and the \ a 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 In mathematics, an eigenfunction of a linear map, linear operator ''D'' defined on some function space is any non-zero function (mathematics), function f in that space that, when acted upon by ''D'', is only multiplied by some scaling factor calle ...
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 (for example, Inner product space#Definition, inner product, Norm (mathematics ...
and 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 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 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 In mathematics, a matrix (: matrices) is a rectangle, rectangular array or table of numbers, symbol (formal), symbols, or expression (mathematics), expressions, with elements or entries arranged in rows and columns, which is used to represent ...
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
poles Pole or poles may refer to: People *Poles (people), another term for Polish people, from the country of Poland * Pole (surname), including a list of people with the name * Pole (musician) (Stefan Betke, born 1967), German electronic music artist ...
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 In mathematics, a line integral is an integral where the function (mathematics), function to be integrated is evaluated along a curve. The terms ''path integral'', ''curve integral'', and ''curvilinear integral'' are also used; ''contour integr ...
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 In mathematical analysis, the Dirac delta function (or distribution), also known as the unit impulse, is a generalized function on the real numbers, whose value is zero everywhere except at zero, and whose integral over the entire real line ...
, 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 functions 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 (for example, Inner product space#Definition, inner product, Norm (mathematics ...
, Hilbert spaces, distributions and so forth. Consult these articles and the references for more detail.


Spectral theorem and Rayleigh quotient

Optimization problem In mathematics, engineering, computer science and economics Economics () is a behavioral science that studies the Production (economics), production, distribution (economics), distribution, and Consumption (economics), consumption of goo ...
s 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 \leq \lambda_2 \leq \cdots \leq \lambda_n. Since the \ form an
orthonormal basis In mathematics, particularly linear algebra, an orthonormal basis for an inner product space V with finite Dimension (linear algebra), dimension is a Basis (linear algebra), basis for V whose vectors are orthonormal, that is, they are all unit vec ...
, any vector x can be expressed in this 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 = & \sum_ v_i^ x v_j^ v_i \\ pt= & (v_j^T x ) v_j^T v_j \\ pt= & v_j^T x \end evaluate the Rayleigh quotient with respect to ''x'': : \begin x^T M x = & \left(\sum_i (v_i^T x) v_i\right)^T M \left(\sum_j (v_j^T x) v_j\right) \\ pt= & \left(\sum_i (v_i^T x) v_i^T\right) \left(\sum_j (v_j^T x) v_j\lambda_j \right) \\ pt= & \sum_ (v_i^T x) v_i^T(v_j^T x) v_j\lambda_j \\ pt= & \sum_j (v_j^T x)(v_j^T x)\lambda_j \\ pt= & \sum_ (v_j^T x)^2\lambda_j\le\lambda_n \sum_j (v_j^T x)^2 \\ pt= & \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

* Functions of operators, Operator theory * Lax pairs *
Least-squares spectral analysis Least-squares spectral analysis (LSSA) is a method of estimating a Spectral density estimation#Overview, frequency spectrum based on a least-squares fit of Sine wave, sinusoids to data samples, similar to Fourier analysis. Fourier analysis, the ...
* Riesz projector * Self-adjoint operator * Spectrum (functional analysis), Resolvent formalism, Decomposition of spectrum (functional analysis) *
Spectral radius ''Spectral'' is a 2016 Hungarian-American military science fiction action film co-written and directed by Nic Mathieu. Written with Ian Fried (screenwriter), Ian Fried & George Nolfi, the film stars James Badge Dale as DARPA research scientist Ma ...
, Spectrum of an operator,
Spectral theorem In linear algebra and functional analysis, a spectral theorem is a result about when a linear operator or matrix can be diagonalized (that is, represented as a diagonal matrix in some basis). This is extremely useful because computations involvin ...
* Spectral theory of compact operators * Spectral theory of normal C*-algebras * Sturm–Liouville theory, Integral equations, Fredholm theory * Compact operators, Isospectral operators, Completeness * 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 Linear algebra de:Spektrum (Operatortheorie)