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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 ...
, a self-adjoint operator on an infinite-dimensional complex vector space ''V'' with
inner product In mathematics, an inner product space or a Hausdorff space, Hausdorff pre-Hilbert space is a vector space with a binary operation called an inner product. This operation associates each pair of vectors in the space with a Scalar (mathematics), ...
\langle\cdot,\cdot\rangle (equivalently, a Hermitian operator in the finite-dimensional case) is a
linear map 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 ...

linear map
''A'' (from ''V'' to itself) that is its own adjoint. If ''V'' is finite-dimensional with a given
orthonormal basisIn linear algebra, two vectors in an inner product space are orthonormal if they are orthogonal (or perpendicular along a line) unit vectors. A set of vectors form an orthonormal set if all vectors in the set are mutually orthogonal and all of u ...
, this is equivalent to the condition that the
matrix Matrix or MATRIX may refer to: Science and mathematics * Matrix (mathematics), a rectangular array of numbers, symbols, or expressions * Matrix (logic), part of a formula in prenex normal form * Matrix (biology), the material in between a eukaryoti ...
of ''A'' is a
Hermitian matrix{{short description, Wikipedia list article Numerous things are named after the French mathematician Charles Hermite (1822–1901): Hermite * Cubic Hermite spline, a type of third-degree spline * Gauss–Hermite quadrature, an extension of Gaussi ...
, i.e., equal to its
conjugate transpose In mathematics, the conjugate transpose (or Hermitian transpose) of an ''m''-by-''n'' matrix (mathematics), matrix \boldsymbol with complex number, complex entries is the ''n''-by-''m'' matrix obtained from \boldsymbol by taking the transpose and ...
''A''. By the finite-dimensional spectral theorem, ''V'' has an
orthonormal basisIn linear algebra, two vectors in an inner product space are orthonormal if they are orthogonal (or perpendicular along a line) unit vectors. A set of vectors form an orthonormal set if all vectors in the set are mutually orthogonal and all of u ...
such that the matrix of ''A'' relative to this basis is a
diagonal matrix 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 and t ...
with entries in the
real number 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 g ...
s. In this article, we consider
generalization A generalization is a form of abstraction Abstraction in its main sense is a conceptual process where general rules and concept Concepts are defined as abstract ideas or general notions that occur in the mind, in speech, or in thought. They ...

generalization
s of this
concept Concepts are defined as abstract ideas A mental representation (or cognitive representation), in philosophy of mind Philosophy of mind is a branch of philosophy that studies the ontology and nature of the mind and its relationship with the bo ...

concept
to operators on
Hilbert space 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 ge ...
s of arbitrary dimension. Self-adjoint operators are used in
functional analysis 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 analysis. Functional analysis is a branch of mathemat ...
and
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 ...
. In quantum mechanics their importance lies in the Dirac–von Neumann formulation of quantum mechanics, in which physical
observable In physics Physics (from grc, φυσική (ἐπιστήμη), physikḗ (epistḗmē), knowledge of nature, from ''phýsis'' 'nature'), , is the natural science that studies matter, its Motion (physics), motion and behavior through ...
s such as position,
momentum In Newtonian mechanics, linear momentum, translational momentum, or simply momentum is the product of the mass Mass is the quantity Quantity is a property that can exist as a multitude or magnitude, which illustrate discontinui ...

momentum
,
angular momentum In , angular momentum (rarely, moment of momentum or rotational momentum) is the rotational equivalent of . It is an important quantity in physics because it is a —the total angular momentum of a closed system remains constant. In three , the ...

angular momentum
and
spin Spin or spinning may refer to: Businesses * or South Pacific Island Network * , an American scooter-sharing system * , a chain of table tennis lounges Computing * , 's tool for formal verification of distributed software systems * , a Mach-like ...
are represented by self-adjoint operators on a Hilbert space. Of particular significance is the Hamiltonian operator \hat defined by :\hat \psi = -\frac \nabla^2 \psi + V \psi, which as an observable corresponds to the total
energy In physics Physics is the that studies , its , its and behavior through , and the related entities of and . "Physical science is that department of knowledge which relates to the order of nature, or, in other words, to the regula ...
of a particle of mass ''m'' in a real potential field ''V''.
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 ...
s are an important class of
unbounded operatorIn 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 ha ...
s. The structure of self-adjoint operators on infinite-dimensional Hilbert spaces essentially resembles the finite-dimensional case. That is to say, operators are self-adjoint if and only if they are unitarily equivalent to real-valued
multiplication operator In operator theoryIn 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 ...
s. With suitable modifications, this result can be extended to possibly unbounded operators on infinite-dimensional spaces. Since an everywhere-defined self-adjoint operator is necessarily bounded, one needs be more attentive to the domain issue in the unbounded case. This is explained below in more detail.


Definitions

Let A be an unbounded (i.e. not necessarily bounded) operator with a dense domain \mathopA \subseteq H. This condition holds automatically when H is
finite-dimensional In mathematics, the dimension of a vector space ''V'' is the cardinality (i.e. the number of vectors) of a Basis (linear algebra), basis of ''V'' over its base Field (mathematics), field. p. 44, §2.36 It is sometimes called Hamel dimension (after ...
since \mathopA = H for every linear operator on a finite-dimensional space. Let the inner product \langle \cdot, \cdot\rangle be conjugate-linear on the ''second'' argument. This applies to complex Hilbert spaces only. By definition, the
adjoint operatorIn 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 ha ...
A^* acts on the subspace \mathop A^* \subseteq H consisting of the elements y for which there is a z \in H such that \langle Ax,y \rangle = \langle x,z \rangle, for every x \in \mathop A. Setting A^*y = z defines the linear operator A^*. The
graph Graph may refer to: Mathematics *Graph (discrete mathematics), a structure made of vertices and edges **Graph theory, the study of such graphs and their properties *Graph (topology), a topological space resembling a graph in the sense of discret ...

graph
of an (arbitrary) operator A is the set G(A) = \. An operator B is said to extend A if G(A) \subseteq G(B). This is written as A \subseteq B. The densely defined operator A is called symmetric if : \langle Ax , y \rangle = \lang x , Ay \rangle, for all x,y\in \mathopA. As shown below, A is symmetric if and only if A \subseteq A^*. The unbounded densely defined operator A is called self-adjoint if G(A)= G(A^*). Explicitly, \mathopA = \mathopA^* and A = A^*. Every self-adjoint operator is symmetric. Conversely, a symmetric operator A for which \mathopA = \mathopA^* is self-adjoint. In physics, the term Hermitian refers to symmetric as well as self-adjoint operators alike. The subtle difference between the two is generally overlooked. A subset \rho(A) \subseteq \C is called the resolvent set (or regular set) if for every \lambda \in \rho(A), the (not-necessarily-bounded) operator A - \lambda I has a ''bounded everywhere-defined'' inverse. The complement \sigma(A) = \C \setminus \rho(A) is called
spectrum A spectrum (plural ''spectra'' or ''spectrums'') is a condition that is not limited to a specific set of values but can vary, without gaps, across a continuum Continuum may refer to: * Continuum (measurement) Continuum theories or models expla ...
. In finite dimensions, \sigma(A) consists exclusively of
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
s.


Bounded self-adjoint operators

A bounded operator ''A'' is self-adjoint if :\langle Ax\mid y\rangle = \langle x\mid Ay\rangle for all x and y in ''H''. If ''A'' is symmetric and \mathrm(A)=H, then, by
Hellinger–Toeplitz theoremIn functional analysis, a branch of mathematics, the Hellinger–Toeplitz theorem states that an everywhere-defined symmetric operator on a Hilbert space with inner product \langle \cdot , \cdot \rangle is bounded operator, bounded. By definition, ...
, ''A'' is necessarily bounded. Every bounded linear operator ''T'' : ''H'' → ''H'' on a Hilbert space ''H'' can be written in the form T = A + i B where ''A'' : ''H'' → ''H'' and ''B'' : ''H'' → ''H'' are bounded self-adjoint operators.


Properties of bounded self-adjoint operators

Let ''H'' be a Hilbert space and let A : H \to H be a bounded self-adjoint linear operator defined on \operatorname\left( A \right) = H. * \left\langle h, A h \right\rangle is real for all h \in H. * \left\, A \right\, = \sup \left\ if \operatorname H \neq 0. * If the image of ''A'', denoted by \operatorname A, is dense in ''H'' then A : H \to \operatorname A is invertible. * The eigenvalues of ''A'' are real and eigenvectors belonging to different eigenvalues are orthogonal. * If \lambda is an eigenvalue of ''A'' then , \lambda , \leq \, A \, ; in particular, , \lambda , \leq \sup \left\. ** In general, there may not exist any eigenvalue \lambda such that , \lambda , = \sup \left\, but if in addition ''A'' is compact then there necessarily exists an eigenvalue \lambda, equal to either \, A \, or - \, A \, , such that , \lambda , = \sup \left\, * If a sequence of bounded self-adjoint linear operators is convergent then the limit is self-adjoint. * There exists a number \lambda, equal to either \, A \, or - \, A \, , and a sequence \left( x_i \right)_^ \subseteq H such that \lim_ A x_i - \lambda x_i = 0 and \, x_i \, = 1 for all ''i''.


Symmetric operators

''NOTE: symmetric operators are defined above.''


''A'' is symmetric ⇔ ''A''⊆''A''

An unbounded, densely defined operator A is symmetric if and only if A \subseteq A^*. Indeed, the if-part follows directly from the definition of the adjoint operator. For the only-if-part, assuming that A is symmetric, the inclusion \mathop(A) \subseteq \mathop(A^*) follows from the Cauchy–Bunyakovsky–Schwarz inequality: for every x,y \in \mathop(A), : , \langle Ax,y\rangle, = , \langle x,Ay\rangle, \leq \, x\, \cdot \, Ay\, . The equality A=A^*, _ holds due to the equality :\langle x,A^*y\rangle = \langle Ax,y\rangle = \langle x,Ay\rangle, for every x,y \in \mathopA \subseteq \mathopA^*, the density of \mathop A, and non-degeneracy of the inner product. The
Hellinger–Toeplitz theoremIn functional analysis, a branch of mathematics, the Hellinger–Toeplitz theorem states that an everywhere-defined symmetric operator on a Hilbert space with inner product \langle \cdot , \cdot \rangle is bounded operator, bounded. By definition, ...
says that an everywhere-defined symmetric operator is bounded and self-adjoint.


''A'' is symmetric ⇔ ''∀x ∈ R''

The only-if part follows directly from the definition (see above). To prove the if-part, assume without loss of generality that the inner product \langle \cdot, \cdot \rangle is anti-linear on the ''first'' argument and linear on the second. (In the reverse scenario, we work with \langle x,y\rangle_\text \stackrel \ \langle y, x \rangle instead). The symmetry of A follows from the
polarization identity In linear algebra, a branch of 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 (mathem ...
: \begin \langle Ax,y\rangle = & \langle A(x+y),x+y\rangle - \langle A(x-y),x-y\rangle \\ mm& - i\langle A(x+iy),x+iy\rangle + i\langle A(x-iy),x-iy\rangle \end which holds for every x,y \in \mathopA.


, , (A-λ)x, , ≥ d(λ)⋅, , x, ,

This property is used in the proof that the spectrum of a self-adjoint operator is real. Define S=\, \textstyle m=\inf_ \langle Ax,x \rangle, and \textstyle M=\sup_ \langle Ax,x \rangle. The values m,M \in \mathbb \cup \ are properly defined since S \neq \emptyset, and \langle Ax,x\rangle \in \mathbb, due to symmetry. Then, for every \lambda \in \C and every x \in \mathopA, : \Vert A - \lambda x\Vert \geq d(\lambda)\cdot \Vert x\Vert, where \textstyle d(\lambda) = \inf_ , r - \lambda, . Indeed, let x \in \mathopA \setminus \. By Cauchy-Schwarz inequality, : \Vert A - \lambda x\Vert \geq \frac =\left, \left\langle A\frac,\frac\right\rangle - \lambda\ \cdot \Vert x\Vert \geq d(\lambda)\cdot \Vert x\Vert. If \lambda \notin ,M then d(\lambda) > 0, and A - \lambda I is called ''bounded below''.


A simple example

As noted above, the spectral theorem applies only to self-adjoint operators, and not in general to symmetric operators. Nevertheless, we can at this point give a simple example of a symmetric operator that has an orthonormal basis of eigenvectors. (This operator is actually "essentially self-adjoint.") The operator ''A'' below can be seen to have a
compact Compact as used in politics may refer broadly to a pact A pact, from Latin ''pactum'' ("something agreed upon"), is a formal agreement. In international relations International relations (IR), international affairs (IA) or internationa ...
inverse, meaning that the corresponding differential equation ''Af'' = ''g'' is solved by some integral, therefore compact, operator ''G''. The compact symmetric operator ''G'' then has a countable family of eigenvectors which are complete in . The same can then be said for ''A''. Consider the complex Hilbert space L2 ,1and the
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 ...
: A = -\frac with \mathrm(A) consisting of all complex-valued infinitely
differentiable In calculus (a branch of mathematics), a differentiable function of one Real number, real variable is a function whose derivative exists at each point in its Domain of a function, domain. In other words, the Graph of a function, graph of a differen ...

differentiable
functions ''f'' on
, 1 The comma is a punctuation Punctuation (or sometimes interpunction) is the use of spacing, conventional signs (called punctuation marks), and certain typographical devices as aids to the understanding and correct reading of written text, ...
satisfying the boundary conditions :f(0) = f(1) = 0. Then
integration by parts In calculus Calculus, originally called infinitesimal calculus or "the calculus of infinitesimals", is the mathematics, mathematical study of continuous change, in the same way that geometry is the study of shape and algebra is the study of ...

integration by parts
of the inner product shows that ''A'' is symmetric. The reader is invited to perform integration by parts twice and verify that the given boundary conditions for \operatorname(A) ensure that the boundary terms in the integration by parts vanish. The eigenfunctions of ''A'' are the sinusoids : f_n(x) = \sin(n \pi x) \qquad n= 1, 2, \ldots with the real eigenvalues ''n''2π2; the well-known orthogonality of the sine functions follows as a consequence of the property of being symmetric. We consider generalizations of this operator below.


Spectrum of self-adjoint operators

Let A be an unbounded symmetric operator. A is self-adjoint if and only if \sigma(A) \subseteq \mathbb.


Essential self-adjointness

A symmetric operator ''A'' is always closable; that is, the closure of the graph of ''A'' is the graph of an operator. A symmetric operator ''A'' is said to be essentially self-adjoint if the closure of ''A'' is self-adjoint. Equivalently, ''A'' is essentially self-adjoint if it has a ''unique'' self-adjoint extension. In practical terms, having an essentially self-adjoint operator is almost as good as having a self-adjoint operator, since we merely need to take the closure to obtain self-adjoint operator.


Example: f(x) → x·f(x)

Consider the complex Hilbert space ''L''2(R), and the operator which multiplies a given function by ''x'': :A f(x) = xf(x) The domain of ''A'' is the space of all ''L''2 functions f(x) for which xf(x) is also square-integrable. Then ''A'' is self-adjoint. On the other hand, ''A'' does not have any eigenfunctions. (More precisely, ''A'' does not have any ''normalizable'' eigenvectors, that is, eigenvectors that are actually in the Hilbert space on which ''A'' is defined.) As we will see later, self-adjoint operators have very important spectral properties; they are in fact multiplication operators on general measure spaces.


Symmetric vs self-adjoint operators

As has been discussed above, although the distinction between a symmetric operator and a self-adjoint (or essentially self-adjoint) operator is a subtle one, it is important since self-adjointness is the hypothesis in the spectral theorem. Here we discuss some concrete examples of the distinction; see the section below on extensions of symmetric operators for the general theory.


A note regarding domains

Every self-adjoint operator is symmetric. Conversely, every symmetric operator for which \mathop(A^*) \subseteq \mathop(A) is self-adjoint. Symmetric operators for which \mathop(A^*) is strictly greater than \mathop(A) cannot be self-adjoint.


Boundary conditions

In the case where the Hilbert space is a space of functions on a bounded domain, these distinctions have to do with a familiar issue in quantum physics: One cannot define an operator—such as the momentum or Hamiltonian operator—on a bounded domain without specifying ''boundary conditions''. In mathematical terms, choosing the boundary conditions amounts to choosing an appropriate domain for the operator. Consider, for example, the Hilbert space L^2(
, 1 The comma is a punctuation Punctuation (or sometimes interpunction) is the use of spacing, conventional signs (called punctuation marks), and certain typographical devices as aids to the understanding and correct reading of written text, ...
(the space of square-integrable functions on the interval ,1. Let us define a "momentum" operator ''A'' on this space by the usual formula, setting Planck's constant equal to 1: : Af = -i\frac. We must now specify a domain for ''A'', which amounts to choosing boundary conditions. If we choose : \operatorname(A) = \left\, then ''A'' is not symmetric (because the boundary terms in the integration by parts do not vanish). If we choose : \operatorname(A) = \left\, then using integration by parts, one can easily verify that ''A'' is symmetric. This operator is not essentially self-adjoint, however, basically because we have specified too many boundary conditions on the domain of ''A'', which makes the domain of the adjoint too big. (This example is discussed also in the "Examples" section below.) Specifically, with the above choice of domain for ''A'', the domain of the closure A^ of ''A'' is :\operatorname\left(A^\right) = \left\, whereas the domain of the adjoint A^* of ''A'' is :\operatorname\left(A^*\right) = \left\. That is to say, the domain of the closure has the same boundary conditions as the domain of ''A'' itself, just a less stringent smoothness assumption. Meanwhile, since there are "too many" boundary conditions on ''A'', there are "too few" (actually, none at all in this case) for A^*. If we compute \langle g, Af\rangle for f \in \operatorname(A) using integration by parts, then since f vanishes at both ends of the interval, no boundary conditions on g are needed to cancel out the boundary terms in the integration by parts. Thus, any sufficiently smooth function g is in the domain of A^*, with A^*g = -i\,dg/dx. Since the domain of the closure and the domain of the adjoint do not agree, ''A'' is not essentially self-adjoint. After all, a general result says that the domain of the adjoint of A^\mathrm is the same as the domain of the adjoint of ''A''. Thus, in this case, the domain of the adjoint of A^\mathrm is bigger than the domain of A^\mathrm itself, showing that A^\mathrm is not self-adjoint, which by definition means that ''A'' is not essentially self-adjoint. The problem with the preceding example is that we imposed too many boundary conditions on the domain of ''A''. A better choice of domain would be to use periodic boundary conditions: :\operatorname(A) = \. With this domain, ''A'' is essentially self-adjoint. In this case, we can understand the implications of the domain issues for the spectral theorem. If we use the first choice of domain (with no boundary conditions), all functions f_\beta(x) = e^ for \beta \in \mathbb C are eigenvectors, with eigenvalues -i \beta, and so the spectrum is the whole complex plane. If we use the second choice of domain (with Dirichlet boundary conditions), ''A'' has no eigenvectors at all. If we use the third choice of domain (with periodic boundary conditions), we can find an orthonormal basis of eigenvectors for ''A'', the functions f_n(x) := e^. Thus, in this case finding a domain such that ''A'' is self-adjoint is a compromise: the domain has to be small enough so that ''A'' is symmetric, but large enough so that D(A^*)=D(A).


Schrödinger operators with singular potentials

A more subtle example of the distinction between symmetric and (essentially) self-adjoint operators comes from Schrödinger operators in quantum mechanics. If the potential energy is singular—particularly if the potential is unbounded below—the associated Schrödinger operator may fail to be essentially self-adjoint. In one dimension, for example, the operator :\hat := \frac - X^4 is not essentially self-adjoint on the space of smooth, rapidly decaying functions. In this case, the failure of essential self-adjointness reflects a pathology in the underlying classical system: A classical particle with a -x^4 potential escapes to infinity in finite time. This operator does not have a ''unique'' self-adjoint, but it does admit self-adjoint extensions obtained by specifying "boundary conditions at infinity". (Since \hat is a real operator, it commutes with complex conjugation. Thus, the deficiency indices are automatically equal, which is the condition for having a self-adjoint extension. See the discussion of extensions of symmetric operators below.) In this case, if we initially define \hat on the space of smooth, rapidly decaying functions, the adjoint will be "the same" operator (i.e., given by the same formula) but on the largest possible domain, namely :\operatorname\left(\hat^*\right) = \left\. It is then possible to show that \hat^* is not a symmetric operator, which certainly implies that \hat is not essentially self-adjoint. Indeed, \hat^* has eigenvectors with pure imaginary eigenvalues, which is impossible for a symmetric operator. This strange occurrence is possible because of a cancellation between the two terms in \hat^*: There are functions f in the domain of \hat^* for which neither d^2 f/dx^2 nor x^4f(x) is separately in L^2(\mathbb), but the combination of them occurring in \hat^* is in L^2(\mathbb). This allows for \hat^* to be nonsymmetric, even though both d^2/dx^2 and X^4 are symmetric operators. This sort of cancellation does not occur if we replace the repelling potential -x^4 with the confining potential x^4. Conditions for Schrödinger operators to be self-adjoint or essentially self-adjoint can be found in various textbooks, such as those by Berezin and Shubin, Hall, and Reed and Simon listed in the references.


Spectral theorem

In the physics literature, the spectral theorem is often stated by saying that a self-adjoint operator has an orthonormal basis of eigenvectors. Physicists are well aware, however, of the phenomenon of "continuous spectrum"; thus, when they speak of an "orthonormal basis" they mean either an orthonormal basis in the classic sense ''or'' some continuous analog thereof. In the case of the momentum operator P = -i\frac, for example, physicists would say that the eigenvectors are the functions f_p(x) := e^, which are clearly not in the Hilbert space L^2(\mathbb). (Physicists would say that the eigenvectors are "non-normalizable.") Physicists would then go on to say that these "eigenvectors" are orthonormal in a continuous sense, where the usual Kronecker delta \delta_ is replaced by a Dirac delta function \delta\left(p - p'\right). Although these statements may seem disconcerting to mathematicians, they can be made rigorous by use of the Fourier transform, which allows a general L^2 function to be expressed as a "superposition" (i.e., integral) of the functions e^, even though these functions are not in L^2. The Fourier transform "diagonalizes" the momentum operator; that is, it converts it into the operator of multiplication by p, where p is the variable of the Fourier transform. The spectral theorem in general can be expressed similarly as the possibility of "diagonalizing" an operator by showing it is unitarily equivalent to a multiplication operator. Other versions of the spectral theorem are similarly intended to capture the idea that a self-adjoint operator can have "eigenvectors" that are not actually in the Hilbert space in question.


Statement of the spectral theorem

Partially defined operators ''A'', ''B'' on Hilbert spaces ''H'', ''K'' are unitarily equivalent if and only if there is a
unitary transformation In mathematics, a unitary transformation is a transformation Transformation may refer to: Science and mathematics In biology and medicine * Metamorphosis, the biological process of changing physical form after birth or hatching * Malignant tr ...
''U'' : ''H'' → ''K'' such that * ''U'' maps dom ''A''
bijective 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 ...
ly onto dom ''B'', * B U \xi = U A \xi ,\qquad \forall \xi \in \operatornameA. A
multiplication operator In operator theoryIn 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 ...
is defined as follows: Let (''X'', Σ, μ) be a countably additive
measure space A measure space is a basic object of measure theory Measure is a fundamental concept of mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures ...
and ''f'' a real-valued measurable function on ''X''. An operator ''T'' of the form : \psi(x) = f(x) \psi(x) whose domain is the space of ψ for which the right-hand side above is in ''L''2 is called a multiplication operator. One version of the spectral theorem can be stated as follows. Other versions of the spectral theorem can be found in the spectral theorem article linked to above. The spectral theorem for unbounded self-adjoint operators can be proved by reduction to the spectral theorem for unitary (hence bounded) operators. This reduction uses the ''
Cayley transformCayley may refer to: __NOTOC__ People * Cayley (surname) * Cayley Illingworth (1759–1823), Anglican Archdeacon of Stow * Cayley Mercer (born 1994), Canadian women's ice hockey player Places * Cayley, Alberta, Canada, a hamlet * Mount Cayley, a vol ...
'' for self-adjoint operators which is defined in the next section. We might note that if T is multiplication by f, then the spectrum of T is just the
essential rangeIn mathematics, particularly measure theory, the essential range of a Function (mathematics), function is intuitively the 'non-negligible' range of the function: It does not change between two functions that are equal almost everywhere. One way of th ...
of f.


Functional calculus

One important application of the spectral theorem is to define a "
functional calculusIn 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 ha ...
." That is to say, if h is a function on the real line and T is a self-adjoint operator, we wish to define the operator h(T). If T has a true orthonormal basis of eigenvectors e_j with eigenvalues \lambda_j, then h(T) is the operator with eigenvectors e_j and eigenvalues h\left(\lambda_j\right). The goal of functional calculus is to extend this idea to the case where T has continuous spectrum. Of particular importance in quantum physics is the case in which T is the Hamiltonian operator \hat and h(x) := e^ is an exponential. In this case, the functional calculus should allow us to define the operator :U(t) := h\left(\hat\right) = e^\frac, which is the operator defining the time-evolution in quantum mechanics. Given the representation of ''T'' as the operator of multiplication by f—as guaranteed by the spectral theorem—it is easy to characterize the functional calculus: If ''h'' is a bounded real-valued Borel function on R, then ''h''(''T'') is the operator of multiplication by the composition h \circ f.


Resolution of the identity

It has been customary to introduce the following notation :\operatorname_T(\lambda) = \mathbf_ (T) where \mathbf_ is the
characteristic functionIn 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 ha ...
(indicator function)of the interval (-\infty, \lambda]. The family of projection operators E''T''(λ) is called resolution of the identity for ''T''. Moreover, the following Stieltjes integral representation for ''T'' can be proved: : T = \int_^ \lambda d \operatorname_T(\lambda). The definition of the operator integral above can be reduced to that of a scalar valued Stieltjes integral using the weak operator topology. In more modern treatments however, this representation is usually avoided, since most technical problems can be dealt with by the functional calculus.


Formulation in the physics literature

In physics, particularly in quantum mechanics, the spectral theorem is expressed in a way which combines the spectral theorem as stated above and the
Borel functional calculus 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 ...
using
Dirac notation Paul Adrien Maurice Dirac (; 8 August 1902 – 20 October 1984) was an English theoretical physicist who is regarded as one of the most significant physicists of the 20th century. Dirac made fundamental contributions to the early developme ...
as follows: If ''H'' is self-adjoint and ''f'' is a
Borel function 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 ...
, :f(H) = \int dE \left, \Psi_E \rangle f(E) \langle \Psi_E \ with :H \left, \Psi_E\right\rangle = E \left, \Psi_E\right\rangle where the integral runs over the whole spectrum of ''H''. The notation suggests that ''H'' is diagonalized by the eigenvectors Ψ''E''. Such a notation is purely
formal Formal, formality, informal or informality imply the complying with, or not complying with, some set theory, set of requirements (substantial form, forms, in Ancient Greek). They may refer to: Dress code and events * Formal wear, attire for forma ...
. One can see the similarity between Dirac's notation and the previous section. The resolution of the identity (sometimes called projection valued measures) formally resembles the rank-1 projections \left, \Psi_E\right\rangle \left\langle\Psi_E\. In the Dirac notation, (projective) measurements are described via
eigenvalues 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 it ...
and
eigenstates In quantum physics 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 qua ...
, both purely formal objects. As one would expect, this does not survive passage to the resolution of the identity. In the latter formulation, measurements are described using the spectral measure of , \Psi \rangle, if the system is prepared in , \Psi \rangle prior to the measurement. Alternatively, if one would like to preserve the notion of eigenstates and make it rigorous, rather than merely formal, one can replace the state space by a suitable
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 ...
. If , the theorem is referred to as resolution of unity: :I = \int dE \left, \Psi_E\right\rangle \left\langle\Psi_E\ In the case H_\text = H - i\Gamma is the sum of an Hermitian ''H'' and a skew-Hermitian (see
skew-Hermitian matrix __NOTOC__ In linear algebra, a square matrix 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 ...
) operator -i\Gamma, one defines the biorthogonal basis set :H^*_\text \left, \Psi_E^*\right\rangle = E^* \left, \Psi_E^*\right\rangle and write the spectral theorem as: :f\left(H_\text\right) = \int dE \left, \Psi_E\right\rangle f(E) \left\langle\Psi_E^*\ (See Feshbach–Fano partitioning method for the context where such operators appear in
scattering theory 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 ...
).


Extensions of symmetric operators

The following question arises in several contexts: if an operator ''A'' on the Hilbert space ''H'' is symmetric, when does it have self-adjoint extensions? An operator that has a unique self-adjoint extension is said to be essentially self-adjoint; equivalently, an operator is essentially self-adjoint if its closure (the operator whose graph is the closure of the graph of ''A'') is self-adjoint. In general, a symmetric operator could have many self-adjoint extensions or none at all. Thus, we would like a classification of its self-adjoint extensions. The first basic criterion for essential self-adjointness is the following: Equivalently, ''A'' is essentially self-adjoint if and only if the operators A^* - i and A^* + i have trivial kernels. That is to say, ''A'' ''fails to be'' self-adjoint if and only if A^* has an eigenvector with eigenvalue i or -i. Another way of looking at the issue is provided by the
Cayley transformCayley may refer to: __NOTOC__ People * Cayley (surname) * Cayley Illingworth (1759–1823), Anglican Archdeacon of Stow * Cayley Mercer (born 1994), Canadian women's ice hockey player Places * Cayley, Alberta, Canada, a hamlet * Mount Cayley, a vol ...
of a self-adjoint operator and the deficiency indices. (It is often of technical convenience to deal with
closed operatorIn 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 ha ...
s. In the symmetric case, the closedness requirement poses no obstacles, since it is known that all symmetric operators are closable.) Here, ''ran'' and ''dom'' denote the
image An image (from la, imago) is an artifact that depicts visual perception Visual perception is the ability to interpret the surrounding environment (biophysical), environment through photopic vision (daytime vision), color vision, sco ...
(in other words, range) and the
domain Domain may refer to: Mathematics *Domain of a function In mathematics, the domain of a Function (mathematics), function is the Set (mathematics), set of inputs accepted by the function. It is sometimes denoted by \operatorname(f), where is th ...
, respectively. W(''A'') is
isometric The term ''isometric'' comes from the Greek for "having equal measurement". isometric may mean: * Cubic crystal system, also called isometric crystal system * Isometre, a rhythmic technique in music. * "Isometric (Intro)", a song by Madeon from t ...
on its domain. Moreover, the range of 1 − W(''A'') is
dense The density (more precisely, the volumetric mass density; also known as specific mass), of a substance is its mass Mass is both a property Property (''latin: Res Privata'') in the Abstract and concrete, abstract is what belongs to or ...
in ''H''. Conversely, given any partially defined operator ''U'' which is isometric on its domain (which is not necessarily closed) and such that 1 − ''U'' is dense, there is a (unique) operator S(''U'') : \operatorname(U) : \operatorname(1 - U) \to \operatorname(1 + U) such that : \operatorname(U)(x - Ux) = i(x + U x) \qquad x \in \operatorname(U). The operator S(''U'') is densely defined and symmetric. The mappings W and S are inverses of each other. The mapping W is called the Cayley transform. It associates a partially defined isometry to any symmetric densely defined operator. Note that the mappings W and S are monotone: This means that if ''B'' is a symmetric operator that extends the densely defined symmetric operator ''A'', then W(''B'') extends W(''A''), and similarly for S. This immediately gives us a necessary and sufficient condition for ''A'' to have a self-adjoint extension, as follows: A partially defined isometric operator ''V'' on a Hilbert space ''H'' has a unique isometric extension to the norm closure of dom(''V''). A partially defined isometric operator with closed domain is called a
partial isometryIn 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 an ...
. Given a partial isometry ''V'', the deficiency indices of ''V'' are defined as the dimension of the
orthogonal complementIn the mathematical Mathematics (from Greek Greek may refer to: Greece Anything of, from, or related to Greece Greece ( el, Ελλάδα, , ), officially the Hellenic Republic, is a country located in Southeast Europe. Its population is ...
s of the domain and range: :\begin n_+(V) &= \dim \operatorname(V)^\perp \\ n_-(V) &= \dim \operatorname(V)^\perp \end We see that there is a bijection between symmetric extensions of an operator and isometric extensions of its Cayley transform. The symmetric extension is self-adjoint if and only if the corresponding isometric extension is unitary. A symmetric operator has a unique self-adjoint extension if and only if both its deficiency indices are zero. Such an operator is said to be essentially self-adjoint. Symmetric operators which are not essentially self-adjoint may still have a
canonical Canonical may refer to: Science and technology * Canonical form In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geo ...
self-adjoint extension. Such is the case for ''non-negative'' symmetric operators (or more generally, operators which are bounded below). These operators always have a canonically defined Friedrichs extension and for these operators we can define a canonical functional calculus. Many operators that occur in analysis are bounded below (such as the negative of the
Laplacian 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 ...
operator), so the issue of essential adjointness for these operators is less critical.


Self-adjoint extensions in quantum mechanics

In quantum mechanics, observables correspond to self-adjoint operators. By
Stone's theorem on one-parameter unitary groups 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 ...
, self-adjoint operators are precisely the infinitesimal generators of unitary groups of
time evolution Time evolution is the change of state brought about by the passage of time Time is the continued of and that occurs in an apparently succession from the , through the , into the . It is a component quantity of various s used to events ...
operators. However, many physical problems are formulated as a time-evolution equation involving differential operators for which the Hamiltonian is only symmetric. In such cases, either the Hamiltonian is essentially self-adjoint, in which case the physical problem has unique solutions or one attempts to find self-adjoint extensions of the Hamiltonian corresponding to different types of boundary conditions or conditions at infinity. Example. The one-dimensional Schrödinger operator with the potential V(x) = -(1 + , x, )^\alpha, defined initially on smooth compactly supported functions, is essentially self-adjoint (that is, has a self-adjoint closure) for but not for . See Berezin and Schubin, pages 55 and 86, or Section 9.10 in Hall. The failure of essential self-adjointness for \alpha > 2 has a counterpart in the classical dynamics of a particle with potential V(x): The classical particle escapes to infinity in finite time. Example. There is no self-adjoint momentum operator ''p'' for a particle moving on a half-line. Nevertheless, the Hamiltonian p^2 of a "free" particle on a half-line has several self-adjoint extensions corresponding to different types of boundary conditions. Physically, these boundary conditions are related to reflections of the particle at the origin (see Reed and Simon, vol.2).


Von Neumann's formulas

Suppose ''A'' is symmetric densely defined. Then any symmetric extension of ''A'' is a restriction of ''A''*. Indeed, ''A'' ⊆ ''B'' and ''B'' symmetric yields ''B'' ⊆ ''A''* by applying the definition of dom(''A''*). These are referred to as von Neumann's formulas in the Akhiezer and Glazman reference.


Examples


A symmetric operator that is not essentially self-adjoint

We first consider the Hilbert space L^2
, 1 The comma is a punctuation Punctuation (or sometimes interpunction) is the use of spacing, conventional signs (called punctuation marks), and certain typographical devices as aids to the understanding and correct reading of written text, ...
/math> and the differential operator : D: \phi \mapsto \frac \phi' defined on the space of continuously differentiable complex-valued functions on ,1 satisfying the boundary conditions :\phi(0) = \phi(1) = 0. Then ''D'' is a symmetric operator as can be shown by
integration by parts In calculus Calculus, originally called infinitesimal calculus or "the calculus of infinitesimals", is the mathematics, mathematical study of continuous change, in the same way that geometry is the study of shape and algebra is the study of ...

integration by parts
. The spaces ''N''+, ''N'' (defined below) are given respectively by the
distributionDistribution may refer to: Mathematics *Distribution (mathematics) Distributions, also known as Schwartz distributions or generalized functions, are objects that generalize the classical notion of functions in mathematical analysis. Distr ...
al solutions to the equation :\begin -i u' &= i u \\ -i u' &= -i u \end which are in ''L''2
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One can show that each one of these solution spaces is 1-dimensional, generated by the functions ''x'' → ''e''''−x'' and ''x'' → ''e''''x'' respectively. This shows that ''D'' is not essentially self-adjoint, but does have self-adjoint extensions. These self-adjoint extensions are parametrized by the space of unitary mappings ''N''+ → ''N'', which in this case happens to be the unit circle T. In this case, the failure of essential self-adjointenss is due to an "incorrect" choice of boundary conditions in the definition of the domain of D. Since D is a first-order operator, only one boundary condition is needed to ensure that D is symmetric. If we replaced the boundary conditions given above by the single boundary condition : \phi(0) = \phi(1), then ''D'' would still be symmetric and would now, in fact, be essentially self-adjoint. This change of boundary conditions gives one particular essentially self-adjoint extension of ''D''. Other essentially self-adjoint extensions come from imposing boundary conditions of the form \phi(1) = e^\phi(0). This simple example illustrates a general fact about self-adjoint extensions of symmetric differential operators ''P'' on an open set ''M''. They are determined by the unitary maps between the eigenvalue spaces : N_\pm = \left\ where ''P''dist is the distributional extension of ''P''.


Constant-coefficient operators

We next give the example of differential operators with constant coefficients. Let :P\left(\vec\right) = \sum_\alpha c_\alpha x^\alpha be a polynomial on R''n'' with ''real'' coefficients, where α ranges over a (finite) set of multi-indices. Thus : \alpha = (\alpha_1, \alpha_2, \ldots, \alpha_n) and : x^\alpha = x_1^ x_2^ \cdots x_n^. We also use the notation :D^\alpha = \frac \partial_^\partial_^ \cdots \partial_^. Then the operator ''P''(D) defined on the space of infinitely differentiable functions of compact support on R''n'' by : P(\operatorname) \phi = \sum_\alpha c_\alpha \operatorname^\alpha \phi is essentially self-adjoint on ''L''2(R''n''). More generally, consider linear differential operators acting on infinitely differentiable complex-valued functions of compact support. If ''M'' is an open subset of R''n'' :P \phi(x) = \sum_\alpha a_\alpha (x) \left ^\alpha \phi\rightx) where ''a''α are (not necessarily constant) infinitely differentiable functions. ''P'' is a linear operator : C_0^\infty(M) \to C_0^\infty(M). Corresponding to ''P'' there is another differential operator, the formal adjoint of ''P'' : P^\mathrm \phi = \sum_\alpha D^\alpha \left(\overline \phi\right)


Spectral multiplicity theory

The multiplication representation of a self-adjoint operator, though extremely useful, is not a canonical representation. This suggests that it is not easy to extract from this representation a criterion to determine when self-adjoint operators ''A'' and ''B'' are unitarily equivalent. The finest grained representation which we now discuss involves spectral multiplicity. This circle of results is called the '' Hahn
HellingerHellinger is a surname. Notable people with the surname include: *Bert Hellinger (1925–2019), German psychotherapist *Ernst Hellinger (1883–1950), German mathematician **Hellinger distance, used to quantify the similarity between two probabilit ...
theory of spectral multiplicity''.


Uniform multiplicity

We first define ''uniform multiplicity'': Definition. A self-adjoint operator ''A'' has uniform multiplicity ''n'' where ''n'' is such that 1 ≤ ''n'' ≤ ω if and only if ''A'' is unitarily equivalent to the operator M''f'' of multiplication by the function ''f''(λ) = λ on : L^2_\mu\left(\mathbf, \mathbf_n\right) = \left\ where H''n'' is a Hilbert space of dimension ''n''. The domain of M''f'' consists of vector-valued functions ψ on R such that : \int_\mathbf , \lambda, ^2\ \, \psi(\lambda)\, ^2 \, d\mu(\lambda) < \infty. Non-negative countably additive measures μ, ν are mutually singular if and only if they are supported on disjoint Borel sets. This representation is unique in the following sense: For any two such representations of the same ''A'', the corresponding measures are equivalent in the sense that they have the same sets of measure 0.


Direct integrals

The spectral multiplicity theorem can be reformulated using the language of
direct integralIn mathematics and functional analysis a direct integral is a generalization of the concept of direct sum. The theory is most developed for direct integrals of Hilbert spaces and direct integrals of von Neumann algebras. The concept was introduced in ...
s of Hilbert spaces: Unlike the multiplication-operator version of the spectral theorem, the direct-integral version is unique in the sense that the measure equivalence class of μ (or equivalently its sets of measure 0) is uniquely determined and the measurable function \lambda\mapsto\mathrm(H_) is determined almost everywhere with respect to μ. The function \lambda \mapsto \operatorname\left(H_\lambda\right) is the spectral multiplicity function of the operator. We may now state the classification result for self-adjoint operators: Two self-adjoint operators are unitarily equivalent if and only if (1) their spectra agree as sets, (2) the measures appearing in their direct-integral representations have the same sets of measure zero, and (3) their spectral multiplicity functions agree almost everywhere with respect to the measure in the direct integral. Proposition 7.24


Example: structure of the Laplacian

The Laplacian on R''n'' is the operator :\Delta = \sum_^n \partial_^2. As remarked above, the Laplacian is diagonalized by the Fourier transform. Actually it is more natural to consider the ''negative'' of the Laplacian −Δ since as an operator it is non-negative; (see
elliptic operator In the theory of partial differential equations In , a partial differential equation (PDE) is an equation which imposes relations between the various s of a . The function is often thought of as an "unknown" to be solved for, similarly to how ...
).


Pure point spectrum

A self-adjoint operator ''A'' on ''H'' has pure point spectrum if and only if ''H'' has an orthonormal basis ''i'' ∈ I consisting of eigenvectors for ''A''. Example. The Hamiltonian for the harmonic oscillator has a quadratic potential ''V'', that is :-\Delta + , x, ^2. This Hamiltonian has pure point spectrum; this is typical for bound state Hamiltonians in quantum mechanics. As was pointed out in a previous example, a sufficient condition that an unbounded symmetric operator has eigenvectors which form a Hilbert space basis is that it has a compact inverse.


See also

*
Compact operator on Hilbert space In functional analysis, the concept of a compact operator on Hilbert space is an extension of the concept of a matrix acting on a finite-dimensional vector space; in Hilbert space, compact operators are precisely the closure of finite-rank operato ...
* Theoretical and experimental justification for the Schrödinger equation *
Unbounded operatorIn 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 ha ...
*
Hermitian adjointIn 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 ha ...
*
Positive operator Positive is a property of positivity and may refer to: Mathematics and science * Converging lens or positive lens, in optics * Plus sign The plus and minus signs, and , are mathematical symbols used to represent the notions of positive an ...


Citations


References

* * * * * * * * * * * * * * {{DEFAULTSORT:Self-Adjoint Operator Operator theory Hilbert space