Spectrum (operator theory)
The spectrum of a linear operator is a term from functional analysis , a branch of mathematics . In finite-dimensional linear algebra , one considers endomorphisms , which are represented by matrices , and their eigenvalues . The generalization into the infinite dimensional is considered in the functional analysis. The spectrum of an operator can be thought of as a set of generalized eigenvalues. These are called spectral values .
Relationship between spectral theory and eigenvalue theory
The spectral theory of linear operators from functional analysis is a generalization of the eigenvalue theory from linear algebra . In linear algebra, endomorphisms on finite-dimensional vector spaces are considered. The numbers for which the equation
Solutions , i.e. not equal to the zero vector , are called eigenvalues , where is a representation matrix of the selected endomorphism. So eigenvalues are numbers for which the inverse of the identity matrix does not exist, that is, the matrix not bijektv is. In finite-dimensional terms, this is to be equated with the fact that the endomorphisms are not injective and therefore not surjective either . However, if one considers infinitely dimensional spaces, it is necessary to distinguish whether the operator is invertible, not injective and / or not surjective. In the infinite-dimensional case, surjectivity does not automatically follow from the injectivity of an endomorphism, as is the case in the finite-dimensional case. The term spectrum in functional analysis is explained below.
is not constrained-invertible, that is, there are no inverses or they are not constrained .
The spectrum of the operator is denoted by and the elements of the spectrum are called spectral values .
The spectrum of linear operators
The above definition can be applied in different contexts. In this section the spectrum of linear operators of a vector space is considered. The spectral theory of linear operators can only be expanded to a certain extent if the set of operators to be considered is specified. For example, one could restrict oneself to restricted , compact or self-adjoint operators. In the following we assume a linear operator on a complex Banach space .
The resolvent set consists of all complex numbers , so that there is an operator limited to defined with
The operator is called the resolvent of the operator . The complement to the resolvent set is the set of complex numbers for which the resolvent does not exist or is unbounded, i.e. the spectrum of the operator , that is, it applies . In the literature there is also a definition of what leads to a different sign of the resolvent. The resolvent set is independent of this sign convention, since an operator is invertible if and only if the operator multiplied by −1 is invertible.
Division of the spectrum
The spectrum can be broken down into different parts. Once a subdivision is made into the point spectrum, the continuous spectrum and the residual spectrum. These components of the spectrum differ somewhat in the reason for the non-existence of a constrained resolvent. Another decomposition of the spectrum is that into the discrete and the essential spectrum. For the spectrum of a self-adjoint operator, there is also the third possibility of dividing it into a point and a continuous spectrum; this is described in the section on self-adjoint operators. The continuous spectrum of a self-adjoint operator is not equivalent to the continuous spectrum that is defined in the following subsection.
The point spectrum (eigenvalue spectrum, discontinuous spectrum)
If the operator is not injective , that is , then is an element of the point spectrum of . The elements of the point spectrum are called eigenvalues.
The continuous spectrum (continuous spectrum, continuity spectrum, line spectrum)
If the operator is injective, but not surjective , but has a dense image, that is, there is an inverse that is only defined on a dense subspace of the Banach space , then is an element of the continuous spectrum of .
The residual spectrum (residual spectrum)
If the operator is injective but does not have a dense image in Banach space , then is an element of the residual spectrum of . In particular, this is not surjective. The operator that is too inverse exists, but is only defined on a non-dense subspace of .
Discrete and essential spectrum
The set of all isolated spectral values with finite multiplicity is called the discrete spectrum and is also noted. The complement is called the essential spectrum of . However, there are other definitions of the essential and discrete spectrum that are not equivalent to this definition.
Approximate point spectrum
If there is a sequence in with
so one calls an approximate eigenvalue of . The set of all approximate eigenvalues is called the approximate point spectrum or approximate eigenvalue spectrum . The following applies:
If is a bounded operator, then also holds
Multiplication operator for functions
An interesting example is the multiplication operator on a function space, which maps the function to the function, i.e. with .
- If one considers the space of the limited functions with the supremum norm, then its spectrum is the interval and all spectral values belong to the point spectrum.
- If you consider it on the Hilbert space of the square-integrable functions , the spectrum is again the interval and all spectral values belong to the continuous spectrum.
- If you finally consider it in the space of continuous functions , then its spectrum is again the interval and all spectral values belong to the residual spectrum.
Multiplication operator for sequences
If there is a restricted sequence in , then is
is a continuous, linear operator on the Hilbert space of the square summable sequences and it is
the completion of the set of sequential terms. In particular, every compact subset of also occurs as a spectrum of an operator. If there is such a set, choose a dense , countable subset and consider the operator above.
Spectra of compact operators
The compact operators map limited sets of the Banach space to relatively compact sets of the same Banach space. This class of operators forms a Banach algebra, which also forms a norm-closed ideal within the algebra of all restricted operators.
The spectrum of compact operators is amazingly simple in the sense that it consists almost entirely of eigenvalues. This result goes back to Frigyes Riesz and reads precisely:
For a compact operator on an infinite-dimensional Banach space it holds that a spectral value and each one is an eigenvalue with finite multiplicity , that is, the kernel of is finite-dimensional and has no accumulation point of different from .
Spectra of self-adjoint operators
In quantum mechanics , the self-adjoint operators appear on Hilbert spaces as a mathematical formalization of the observable quantities, so-called observables. The elements of the spectrum are possible measured values. Therefore, the following statements are of fundamental importance:
The spectrum of a self adjoint operator is contained in. If it is self-adjoint and bounded, its spectrum lies in the interval and contains one of the boundary points. Is , then applies
Eigenspaces with different eigenvalues are orthogonal to one another. Self-adjoint operators on a separable Hilbert space therefore have at most a countable number of eigenvalues. The residual spectrum of a self-adjoint operator is empty.
If a self-adjoint operator, which can also be an unbounded operator , is in a Hilbert space , then this operator is assigned a spectral family of orthogonal projections . For each function is the distribution function of a measure on . The properties of these measures give rise to the definition of subspaces to which the operator can be restricted. The spectra of these constraints are then part of the spectrum of . This gives new descriptions of the parts of the spectrum already mentioned above and further subdivisions.
The point spectrum
is discontinuous subspace of respect. . It applies
- , that is, the spectrum of the restriction is the conclusion of the point spectrum of .
It is true , and it is said, has a pure point spectrum .
The steady spectrum
is continuous subspace of respect. .
- is the continuous spectrum of .
It is true , and it is said, has a purely continuous spectrum .
The singular spectrum
- is the singular spectrum of .
It is true , and it is said, has a purely singular spectrum .
The singular continuous spectrum
called singular continuous subspace of respect. .
- is the singular continuous spectrum of .
It is true , and it is said, has a purely singular continuous spectrum .
The absolutely constant spectrum
is absolutely continuous subspace of respect. . The measure to one is then absolutely constant in relation to the Lebesgue measure.
- is the absolute continuous spectrum of .
It is true , and it is said, has a purely absolutely constant spectrum .
Relations of the spectra
It is , , . This results in
The parts , , and are completed, because it is spectrum. This does not generally apply to the point spectrum.
Spectral theory for elements of a Banach algebra
If one deletes the additional requirement that the inverse is bounded, the above definition can also be applied to elements of an operator algebra . An operator algebra is usually understood to be a Banach algebra with one element and inverting elements only makes sense in this context if the inverse is in turn an element of algebra. Since such operators are not defined by their effect on any vector space (i.e. they do not actually operate at all), there is also no a priori concept of the boundedness of such operators. However, one can this always as linear operators on a vector space representing , for example, as multiplication operators on the Banach himself. Then these operators to bounded operators on a Banach space are. In particular, the set of bounded operators forms the standard example of an operator algebra. The compact operators already mentioned form an operator algebra. Hence, spectral theory for Banach algebras includes these two classes of linear operators.
In linear algebra , the n × n matrices with complex entries form an algebra with regard to the usual addition and scalar multiplication (component-wise) as well as the matrix multiplication . The matrices can therefore be viewed both as an example of actual operators in their property as linear mappings of the , and as an example of an operator algebra, although in this context it is irrelevant which operator norm is chosen for the matrices. Since all linear mappings of a finite-dimensional space are automatically restricted to themselves , this term can be disregarded in the definition here.
A matrix is invertible if there is a matrix such that ( identity matrix ) is. This is exactly the case when the determinant does not vanish: . Hence a number is a spectral value if holds. Since this is precisely the characteristic polynomial of the matrix in , it is a spectral value if and only if is an eigenvalue of the matrix. In linear algebra, the spectrum of a matrix therefore describes the set of eigenvalues.
The continuous functions on the interval with values in the complex numbers form a Banach algebra (e.g. with the supremum norm as the norm , but this is not relevant here), whereby the sum of two functions and the product of two functions is defined point by point :
In this algebra, a function is said to be invertible if there is another function such that ( one function ) is, that is, if there is a function whose values are precisely the reciprocal values of . You can now quickly see that a function can be inverted if and only if it does not have the function value and the inverse in this case has the inverse function values (reciprocal values) of the original function point by point:
- if anywhere.
A number is therefore a spectral value if the function cannot be inverted, i.e. has the function value. This is of course the case if and only if is a function value of . The spectrum of a function is therefore precisely its image .
The spectral theory of the elements of Banach algebras with one is an abstraction of the theory of bounded linear operators on a Banach space. The introductory examples are special cases of this theory, whereby in the first example the norm of the observed functions has to be specified. If you choose z. B. the Banach space of continuous functions on a compact space with the supremum norm, this example represents the most important case of an Abelian Banach algebra with one. The second example finds its place in this theory as a typical finite-dimensional example of a non-Abelian Banach algebra, where one suitable standard for the matrices must be selected. In the first case, the spectrum of an operator was the range of values and, since the functions under consideration are continuous on a compact form, a compact subset in . In the second case the spectrum is a finite set of points and is therefore also compact. This fact can also be proven in the abstract case:
- The spectrum of an element of a Banach algebra with one is always non-empty (see Gelfand-Mazur theorem ) and compact.
From this theorem it follows immediately that there is a spectral value that is greatest in terms of absolute value, because the supremum
is adopted on the compact spectrum. This value is called the spectral radius of . In the example of the algebra of continuous functions, one immediately sees that the spectral radius corresponds to the norm of the elements. However, from linear algebra we know that this is not the case for matrices in general, since e.g. B. the matrix
only possesses the eigenvalue , and therefore is , but the norm of the matrix (whichever one) is not . The spectral radius is actually smaller than the norm in general, but it applies
- Theorem: In a Banach algebra with one, the limit value exists for every element and is equal to the spectral radius of .
- In quantum mechanics , one mainly deals with the spectrum of the Hamilton operator . These are the possible energy values that can be measured on the system under consideration. The Hamilton operator is therefore (and because it determines the dynamics of the system, see Mathematical Structure of Quantum Mechanics ) a particularly important special case for the mostly unrestricted self-adjoint operators on the Hilbert space . The elements of this space represent the quantum mechanical states, while the self adjoint operators represent the observables (measurable quantities). As already mentioned above, the self-adjointness of the operator ensures that the possible measured values (spectral values) are without exception real numbers. This can be used to 1.) add up or 2.) integrate, which is related to the spectral type:
- The spectrum is divided into what physicists call discrete (mathematically more precisely: the point spectrum - 1.) - which can also be non-discrete, analogous to the rational numbers), which is the point measure (“finite differences ”, In contrast to the“ differential ”) corresponds to a discontinuous monotonic function, a so-called step function; and secondly in the so-called continuous spectrum (more precisely: in the absolutely continuous spectrum , - 2.) -, analogous to the differential of a continuous and everywhere differentiable strictly monotonically increasing smooth function .) The transition from 1.) to 2.) corresponds the transition from the summation to the integral, which can be done approximately using Riemann sums .
- In very rare cases, for example with hierarchically ordered incommensurable segments of potential energy or with certain magnetic fields, there is also a third spectral component, the so-called singular-continuous spectrum, analogous to a monotonically growing Cantor function , a function that is steady and monotonous is growing, but cannot be differentiated anywhere (e.g. the so-called Devil's Staircase ).
- In classical mechanics and statistical mechanics , observables are modeled by functions on the phase space . Analogous to quantum mechanics, it is also true in this case that the possible measured values are the spectral values of the observables, i.e. in this case simply the function values.
- In algebraic quantum theory , observables are abstractly introduced as elements of so-called C * -algebras (special Banach algebras). Without specifying a concrete representation of this algebra as a set of linear operators on a Hilbert space, the spectral calculus of these algebras then allows the possible measured values of the observables to be calculated. The states of the physical system are then not introduced as vectors in Hilbert space, but as linear functionals in algebra. The classical theories like classical (statistical) mechanics can be seen in this picture as special cases in which the C * algebra is Abelian.
- Gerald Teschl : Mathematical Methods in Quantum Mechanics; With Applications to Schrödinger Operators, American Mathematical Society, 2009 ( Free online version )
- Dirk Werner : functional analysis , Springer-Verlag, Berlin, 2007, ISBN 978-3-540-72533-6
References and footnotes
- Reinhold Meise, Dietmar Vogt: Introduction to Functional Analysis , Vieweg, Braunschweig 1992, ISBN 3-528-07262-8 , §17
- Włodzimierz Mlak : Hilbert Spaces and Operator Theory , Polish Scientific Publishers (1991), ISBN 83-01-09965-8 , chapter 3.4
- Hellmut Baumgärtel , Manfred Wollenberg: Mathematical scattering theory . Birkhäuser, Basel 1983, ISBN 3-7643-1519-9 , p. 54 .
- Harro Heuser : Functional Analysis: Theory and Application . 3rd ed., BG Teubner, Stuttgart 1992. ISBN 3-519-22206-X , pp. 520-521.
- Helmut Fischer, Helmut Kaul: Mathematics for Physicists, Volume 2: Ordinary and partial differential equations, mathematical foundations of quantum mechanics . 2nd Edition. BG Teubner, Wiesbaden 2004, ISBN 3-519-12080-1 , §21 section 5.5 and §23 section 5.2, p. 572-573, 665-666 .
- H. Triebel : Higher Analysis , Verlag Harri Deutsch, ISBN 3-87144-583-5 , §18: The spectrum of self-adjoint operators
- Wlodzimierz Mlak: Hilbert Spaces and Operator Theory , Polish Scientific Publishers (1991), ISBN 83-01-09965-8 , Theorem 4.1.5
- J. Weidmann : Linear Operators in Hilbert Spaces , Teubner-Verlag (1976), ISBN 3-519-02204-4 , Chapter 7.4: Spectra of self-adjoint operators
- It is already a significant simplification if the operator is only unlimited on one side, for example upwards, but is limited on the other. Otherwise one is led to auxiliary constructions such as the so-called Dirac lake in order to give physical meaning to certain quantities.