In
mathematicsMathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, on a finitedimensional
inner product spaceIn mathematics, an inner product space is a vector space with an additional structure called an inner product. This additional structure associates each pair of vectors in the space with a scalar quantity known as the inner product of the vectors...
, a
selfadjoint operator is an operator that is its own adjoint, or, equivalently, one whose
matrixIn mathematics, a matrix is a rectangular array of numbers, symbols, or expressions. The individual items in a matrix are called its elements or entries. An example of a matrix with six elements isMatrices of the same size can be added or subtracted element by element...
is
Hermitian, where a Hermitian matrix is one which is equal to its own
conjugate transposeIn mathematics, the conjugate transpose, Hermitian transpose, Hermitian conjugate, or adjoint matrix of an mbyn matrix A with complex entries is the nbym matrix A* obtained from A by taking the transpose and then taking the complex conjugate of each entry...
. By the finitedimensional
spectral theoremIn mathematics, particularly linear algebra and functional analysis, the spectral theorem is any of a number of results about linear operators or about matrices. In broad terms the spectral theorem provides conditions under which an operator or a matrix can be diagonalized...
such operators have an
orthonormal basisIn mathematics, particularly linear algebra, an orthonormal basis for inner product space V with finite dimension is a basis for V whose vectors are orthonormal. For example, the standard basis for a Euclidean space Rn is an orthonormal basis, where the relevant inner product is the dot product of...
in which the operator can be represented as a
diagonal matrixIn linear algebra, a diagonal matrix is a matrix in which the entries outside the main diagonal are all zero. The diagonal entries themselves may or may not be zero...
with entries in the
real numberIn mathematics, a real number is a value that represents a quantity along a continuum, such as 5 , 4/3 , 8.6 , √2 and π...
s. In this article, we consider
generalizationA generalization of a concept is an extension of the concept to lessspecific criteria. It is a foundational element of logic and human reasoning. Generalizations posit the existence of a domain or set of elements, as well as one or more common characteristics shared by those elements. As such, it...
s of this
conceptThe word concept is used in ordinary language as well as in almost all academic disciplines. Particularly in philosophy, psychology and cognitive sciences the term is much used and much discussed. WordNet defines concept: "conception, construct ". However, the meaning of the term concept is much...
to operators on
Hilbert spaceThe mathematical concept of a Hilbert space, named after David Hilbert, generalizes the notion of Euclidean space. It extends the methods of vector algebra and calculus from the twodimensional Euclidean plane and threedimensional space to spaces with any finite or infinite number of dimensions...
s of arbitrary dimension.
Selfadjoint operators are used in
functional analysisFunctional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limitrelated structure and the linear operators acting upon these spaces and respecting these structures in a suitable sense...
and
quantum mechanicsQuantum mechanics, also known as quantum physics or quantum theory, is a branch of physics providing a mathematical description of much of the dual particlelike and wavelike behavior and interactions of energy and matter. It departs from classical mechanics primarily at the atomic and subatomic...
. In quantum mechanics their importance lies in the fact that in the
DiracPaul Adrien Maurice Dirac, OM, FRS was an English theoretical physicist who made fundamental contributions to the early development of both quantum mechanics and quantum electrodynamics...
–von Neumann formulation of quantum mechanics, physical
observableIn physics, particularly in quantum physics, a system observable is a property of the system state that can be determined by some sequence of physical operations. For example, these operations might involve submitting the system to various electromagnetic fields and eventually reading a value off...
s such as position,
momentumIn classical mechanics, linear momentum or translational momentum is the product of the mass and velocity of an object...
,
angular momentumIn physics, angular momentum, moment of momentum, or rotational momentum is a conserved vector quantity that can be used to describe the overall state of a physical system...
and
spinIn quantum mechanics and particle physics, spin is a fundamental characteristic property of elementary particles, composite particles , and atomic nuclei.It is worth noting that the intrinsic property of subatomic particles called spin and discussed in this article, is related in some small ways,...
are represented by selfadjoint operators on a Hilbert space. Of particular significance is the
HamiltonianIn quantum mechanics, the Hamiltonian H, also Ȟ or Ĥ, is the operator corresponding to the total energy of the system. Its spectrum is the set of possible outcomes when one measures the total energy of a system...
which as an observable corresponds to the total energy of a particle of mass
m in a real potential field
V. Differential operators are an important class of
unbounded operatorIn mathematics, more specifically functional analysis and operator theory, the notion of unbounded operator provides an abstract framework for dealing with differential operators, unbounded observables in quantum mechanics, and other cases....
s.
The structure of selfadjoint operators on infinitedimensional Hilbert spaces essentially resembles the
finitedimensional case, that is to say, operators are selfadjoint if and only if they are unitarily equivalent to realvalued multiplication operators. With suitable modifications, this result can be extended to possibly unbounded operators on infinitedimensional spaces. Since an everywhere defined selfadjoint operator is necessarily bounded, one needs be more attentive to the domain issue in the unbounded case. This is explained below in more detail.
Symmetric operators
A partiallydefined linear operator
A on a
Hilbert spaceThe mathematical concept of a Hilbert space, named after David Hilbert, generalizes the notion of Euclidean space. It extends the methods of vector algebra and calculus from the twodimensional Euclidean plane and threedimensional space to spaces with any finite or infinite number of dimensions...
H is called
symmetric if

for all elements
x and
y in the domain of
A. More generally, a partiallydefined linear operator
A from a
topological vector spaceIn mathematics, a topological vector space is one of the basic structures investigated in functional analysis...
E into its continuous dual space
E^{∗} is said to be
symmetric if
for all elements
x and
y in the domain of
A. This usage is fairly standard in the functional analysis literature.
A symmetric
everywhere defined operator is
selfadjointIn mathematics, an element x of a staralgebra is selfadjoint if x^*=x.A collection C of elements of a staralgebra is selfadjoint if it is closed under the involution operation...
.
By the HellingerToeplitz theorem, a symmetric
everywhere defined operator is
boundedIn mathematics, the operator norm is a means to measure the "size" of certain linear operators. Formally, it is a norm defined on the space of bounded linear operators between two given normed vector spaces. Introduction and definition :...
.
Bounded symmetric operators are also called
Hermitian.
The previous definition agrees with the one for matrices given in the introduction to this article, if we take as
H the Hilbert space
C^{n} with the standard dot product and interpret a square matrix as a linear operator on this Hilbert space. It is however much more general as there are important infinitedimensional Hilbert spaces.
The spectrum of any bounded symmetric operator is real; in particular all its eigenvalues are real, although a symmetric operator may have no eigenvalues.
A general version of the
spectral theoremIn mathematics, particularly linear algebra and functional analysis, the spectral theorem is any of a number of results about linear operators or about matrices. In broad terms the spectral theorem provides conditions under which an operator or a matrix can be diagonalized...
which also applies to bounded symmetric operators (see Read and Simon, vol. 1, chapter VII, or other books cited) is stated below. If the set of eigenvalues for a symmetric operator is non empty, and the eigenvalues are nondegenerate, then it follows from the definition that eigenvectors corresponding to distinct eigenvalues are orthogonal.
Contrary to what is sometimes claimed in introductory physics textbooks, it is possible for symmetric operators to have no eigenvalues at all (although the
spectrumIn functional analysis, the concept of the spectrum of a bounded operator is a generalisation of the concept of eigenvalues for matrices. Specifically, a complex number λ is said to be in the spectrum of a bounded linear operator T if λI − T is not invertible, where I is the...
of any selfadjoint operator is nonempty). The example below illustrates the special case when an (unbounded) symmetric operator does have a set of eigenvectors which constitute a Hilbert space basis. The operator
A below can be seen to have a
compactIn functional analysis, compact operators on Hilbert spaces are a direct extension of matrices: in the Hilbert spaces, they are precisely the closure of finiterank operators in the uniform operator topology. As such, results from matrix theory can sometimes be extended to compact operators using...
inverse, meaning that the corresponding differential equation
A f =
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.
Example. Consider the complex Hilbert space L
^{2}[0,1] and the
differential operatorIn mathematics, a differential operator is an operator defined as a function of the differentiation operator. It is helpful, as a matter of notation first, to consider differentiation as an abstract operation, accepting a function and returning another .This article considers only linear operators,...

defined on the subspace consisting of all complexvalued infinitely differentiable functions
f on [0,1] with the boundary conditions:
Then
integration by partsIn calculus, and more generally in mathematical analysis, integration by parts is a rule that transforms the integral of products of functions into other integrals...
shows that
A is symmetric. Its eigenfunctions are the sinusoids
with the real eigenvalues
n^{2}π
^{2}; the wellknown orthogonality of the sine functions follows as a consequence of the property of being symmetric.
We consider generalizations of this operator below.
Selfadjoint operators
Given a densely defined linear operator
A on
H, its adjoint
A* is defined as follows:
 The domain of A* consists of vectors x in H such that

 (which is a densely defined linear map) is a continuous linear functional. By continuity and density of the domain of A, it extends to a unique continuous linear functional on all of H.
 By the Riesz representation theorem
There are several wellknown theorems in functional analysis known as the Riesz representation theorem. They are named in honour of Frigyes Riesz. The Hilbert space representation theorem :...
for linear functionals, if x is in the domain of A*, there is a unique vector z in H such that

 This vector z is defined to be A* x. It can be shown that the dependence of z on x is linear.
Notice that it is the denseness of the domain of the operator, along with the uniqueness part of Riesz representation, that ensures the adjoint operator is well defined.
A result of HellingerToeplitz type says that an operator having an everywhere defined bounded adjoint is bounded.
The condition for a linear operator on a Hilbert space to be
selfadjoint is stronger than to be
symmetric.
For any densely defined operator
A on Hilbert space one can define its adjoint operator
A*.
For a symmetric operator
A, the domain of the operator
A* contains the domain of the operator
A, and the restriction of the operator
A* on the domain of
A coincides with the operator
A, i.e.
, in other words
A* is extension of
A. For a selfadjoint operator
A the domain of
A* is the same as the domain of
A, and
A=
A*. See also
Extensions of symmetric operatorsIn functional analysis, one is interested in extensions of symmetric operators acting on a Hilbert space. Of particular importance is the existence, and sometimes explicit constructions, of selfadjoint extensions. This problem arises, for example, when one needs to specify domains of...
and
unbounded operatorIn mathematics, more specifically functional analysis and operator theory, the notion of unbounded operator provides an abstract framework for dealing with differential operators, unbounded observables in quantum mechanics, and other cases....
.
Geometric interpretation
There is a useful geometrical way of looking at the adjoint of an operator
A on
H as follows: we consider the graph G(
A) of
A defined by
Theorem. Let J be the
symplectic mapping

given by
Then the graph of
A* is the
orthogonal complement of JG(
A):
A densely defined operator
A is symmetric
if and only ifIn logic and related fields such as mathematics and philosophy, if and only if is a biconditional logical connective between statements....
where the subset notation
is understood to mean
An operator
A is
selfadjoint if and only if
; that is, if and only if
Example. Consider the complex Hilbert space L
^{2}(
R), and the operator which multiplies a given function by
x:
The domain of
A is the space of all L
^{2} functions for which the righthandside is squareintegrable.
A is a symmetric operator without any eigenvalues and eigenfunctions. In fact it turns out that the operator is selfadjoint, as follows from the theory outlined below.
As we will see later, selfadjoint operators have very important spectral properties; they are in fact multiplication operators on general measure spaces.
Spectral theorem
Partially defined operators
A,
B on Hilbert spaces
H,
K are
unitarily equivalent if and only if there is a
unitary transformationIn mathematics, a unitary transformation may be informally defined as a transformation that respects the inner product: the inner product of two vectors before the transformation is equal to their inner product after the transformation....
U:
H →
K such that
 U maps dom A bijectively onto dom B,
A
multiplication operatorIn operator theory, a multiplication operator is a linear operator T defined on some vector space of functions and whose value at a function φ is given by multiplication by a fixed function f...
is defined as follows: Let
be a countably additive measure space and
f a realvalued measurable function on
X. An operator
T of the form
whose domain is the space of ψ for which the righthand side above is in
L^{2} is called a multiplication operator.
Theorem. Any multiplication operator is a (densely defined) selfadjoint operator. Any selfadjoint operator is unitarily equivalent to a multiplication operator.
This version of the spectral theorem for selfadjoint operators can be proved by reduction to the spectral theorem for unitary operators. This reduction uses the
Cayley transform for selfadjoint 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 is intuitively the 'nonnegligible' range of the function. One way of thinking of the essential range of a function is the set on which the range of the function is most 'concentrated'...
of f.
Borel functional calculus
Given the representation of
T as a multiplication operator, it is easy to characterize the
Borel functional calculusIn functional analysis, a branch of mathematics, the Borel functional calculus is a functional calculus , which has particularly broad scope. Thus for instance if T is an operator, applying the squaring function s → s2 to T yields the operator T2...
: If
h is a bounded realvalued Borel function on
R, then
h(
T) is the operator of multiplication by the composition
. In order for this to be welldefined, we must show that it is the unique operation on bounded realvalued Borel functions satisfying a number of conditions.
Resolution of the identity
It has been customary to introduce the following notation
where
denotes the function which is identically 1 on the interval
. 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:
The definition of the operator integral above can be reduced to that 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 calculusIn functional analysis, a branch of mathematics, the Borel functional calculus is a functional calculus , which has particularly broad scope. Thus for instance if T is an operator, applying the squaring function s → s2 to T yields the operator T2...
using Dirac notation as follows:
If
H is Hermitian (the name for selfadjoint in the physics literature) and
f is a Borel function,
with
where the integral runs over the whole spectrum of
H. The notation suggests that
H is diagonalized by the eigenvalues Ψ
_{E}. Such a notation is purely
formalIn mathematical logic, a formal calculation is sometimes defined as a calculation which is systematic, but without a rigorous justification. This means that we are manipulating the symbols in an expression using a generic substitution, without proving that the necessary conditions hold...
. 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 rank1 projections
.
In the Dirac notation, (projective) measurements are described via eigenvalues and eigenstates, 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
, if the system is prepared in
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 spaceIn mathematics, a rigged Hilbert space is a construction designed to link the distribution and squareintegrable aspects of functional analysis. Such spaces were introduced to study spectral theory in the broad sense...
.
If
f=1, the theorem is referred to as resolution of unity:
In the case
is the sum of an Hermitian
H and a skewHermitian (see
skewHermitian matrixIn linear algebra, a square matrix with complex entries is said to be skewHermitian or antihermitian if its conjugate transpose is equal to its negative. That is, the matrix A is skewHermitian if it satisfies the relationA^\dagger = A,\;...
) operator
, one defines the
biorthogonal basis set
and write the spectral theorem as:
(See Feshbach–Fano partitioning method for the context where such operators appear in
scattering theoryIn mathematics and physics, scattering theory is a framework for studying and understanding the scattering of waves and particles. Prosaically, wave scattering corresponds to the collision and scattering of a wave with some material object, for instance sunlight scattered by rain drops to form a...
).
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 selfadjoint extensions? One answer is provided by the
Cayley transformIn mathematics, the Cayley transform, named after Arthur Cayley, has a cluster of related meanings. As originally described by , the Cayley transform is a mapping between skewsymmetric matrices and special orthogonal matrices. In complex analysis, the Cayley transform is a conformal mapping in...
of a selfadjoint operator and the deficiency indices. (We should note here that it is often of technical convenience to deal with
closed operatorIn mathematics, specifically in functional analysis, closed linear operators are an important class of linear operators on Banach spaces. They are more general than bounded operators, and therefore not necessarily continuous, but they still retain nice enough properties that one can define the...
s. In the symmetric case, the closedness requirement poses no obstacles, since it is known that all symmetric operators are closable.)
Theorem. Suppose
A is a symmetric operator. Then there is a
unique partially defined linear operator

such that

Here,
ran and
dom denote the
rangeIn mathematics, the range of a function refers to either the codomain or the image of the function, depending upon usage. This ambiguity is illustrated by the function f that maps real numbers to real numbers with f = x^2. Some books say that range of this function is its codomain, the set of all...
and the
domainIn mathematics, the domain of definition or simply the domain of a function is the set of "input" or argument values for which the function is defined...
, respectively. W(
A) is
isometricIn mathematics, an isometry is a distancepreserving map between metric spaces. Geometric figures which can be related by an isometry are called congruent.Isometries are often used in constructions where one space is embedded in another space...
on its domain. Moreover, the range of 1 − W(
A) is
denseIn topology and related areas of mathematics, a subset A of a topological space X is called dense if any point x in X belongs to A or is a limit point of A...
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)

such that

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 isometryIn functional analysis a partial isometry is a linear map W between Hilbert spaces H and K such that the restriction of W to the orthogonal complement of its kernel is an isometry...
to any symmetric denselydefined operator. Note that the mappings W and S are : 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.
Theorem. A necessary and sufficient condition for
A to be selfadjoint is that its Cayley transform W(
A) be unitary.
This immediately gives us a necessary and sufficient condition for
A to have a selfadjoint extension, as follows:
Theorem. A necessary and sufficient condition for
A to have a selfadjoint extension is that W(
A) have a unitary extension.
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 a partial isometry is a linear map W between Hilbert spaces H and K such that the restriction of W to the orthogonal complement of its kernel is an isometry...
.
Given a partial isometry
V, the
deficiency indices of
V are defined as the dimension of the
orthogonal complements of the domain and range:
Theorem. A partial isometry
V has a unitary extension if and only if the deficiency indices are identical. Moreover,
V has a
unique unitary extension if and only if the both deficiency indices are zero.
We see that there is a bijection between symmetric extensions of an operator and isometric extensions of its Cayley transform. An operator which has a unique selfadjoint extension is said to be
essentially selfadjoint. Such operators have a welldefined
Borel functional calculusIn functional analysis, a branch of mathematics, the Borel functional calculus is a functional calculus , which has particularly broad scope. Thus for instance if T is an operator, applying the squaring function s → s2 to T yields the operator T2...
. Symmetric operators which are not essentially selfadjoint may still have a
canonicalCanonical is an adjective derived from canon. Canon comes from the greek word κανών kanon, "rule" or "measuring stick" , and is used in various meanings....
selfadjoint extension. Such is the case for
nonnegative symmetric operators (or more generally, operators which are bounded below). These operators always have a canonically defined
Friedrichs extensionIn functional analysis, the Friedrichs extension is a canonical selfadjoint extension of a nonnegative densely defined symmetric operator. It is named after the mathematician Kurt Friedrichs...
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 operator), so the issue of essential adjointness for these operators is less critical.
Selfadjoint extensions in quantum mechanics
In quantum mechanics, observables correspond to selfadjoint operators. By Stone's theorem, selfadjoint operators are precisely the infinitesimal generators of unitary groups of
time evolutionTime evolution is the change of state brought about by the passage of time, applicable to systems with internal state . In this formulation, time is not required to be a continuous parameter, but may be discrete or even finite. In classical physics, time evolution of a collection of rigid bodies...
operators. However, many physical problems are formulated as a timeevolution equation involving differential operators for which the Hamiltonian is only symmetric. In such cases, either the Hamiltonian is essentially selfadjoint, in which case the physical problem has unique solutions or one attempts to find selfadjoint extensions of the Hamiltonian corresponding to different types of boundary conditions or conditions at infinity.
Example. The onedimensional Schrödinger operator with the potential
, defined initially on smooth compactly supported functions, is essentially selfadjoint (that is, has a selfadjoint closure) for
but not for
See Berezin and Schubin, page 55.
Example. There is no selfadjoint momentum operator
for a particle moving on a halfline. Nevertheless, the Hamiltonian
of a "free" particle on a halfline has several selfadjoint 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; any symmetric extension of
A is a restriction of
A*; Indeed if
B is symmetric
Theorem. Suppose
A is a densely defined symmetric operator. Let
Then
and
where the decomposition is orthogonal relative to the graph inner product of dom(
A*):
These are referred to as von Neumann's formulas in the Akhiezer and Glazman reference.
Examples
We first consider the differential operator

defined on the space of complexvalued C
^{∞} functions on [0,1] vanishing near 0 and 1.
D is a symmetric operator as can be shown by
integration by partsIn calculus, and more generally in mathematical analysis, integration by parts is a rule that transforms the integral of products of functions into other integrals...
. The spaces
N_{+},
N_{−} are given respectively by the
distributionIn mathematical analysis, distributions are objects that generalize functions. Distributions make it possible to differentiate functions whose derivatives do not exist in the classical sense. In particular, any locally integrable function has a distributional derivative...
al solutions to the equation
which are in
L^{2} [0,1]. One can show that each one of these solution spaces is 1dimensional, generated by the functions
x →
e^{ix} and
x →
e^{−ix} respectively. This shows that
D is not essentially selfadjoint, but does have selfadjoint extensions. These selfadjoint extensions are parametrized by the space of unitary mappings

which in this case happens to be the unit circle
T.
This simple example illustrates a general fact about selfadjoint extensions of symmetric differential operators
P on an open set
M. They are determined by the unitary maps between the eigenvalue spaces

where
P_{dist} is the distributional extension of
P.
We next give the example of differential operators with constant coefficients. Let
be a polynomial on
R^{n} with
real coefficients, where α ranges over a (finite) set of multiindices. Thus

and

We also use the notation
Then the operator
P(D) defined on the space of infinitely differentiable functions of compact support on
R^{n} by
is essentially selfadjoint on
L^{2}(
R^{n}).
Theorem. Let
P a polynomial function on
R^{n} with real coefficients,
F the Fourier transform considered as a unitary map
L^{2}(
R^{n}) →
L^{2}(
R^{n}). Then
F*
P(D)
F is essentially selfadjoint and its unique selfadjoint extension is the operator of multiplication by the function
P.
More generally, consider linear differential operators acting on infinitely differentiable complexvalued functions of compact support. If
M is an open subset of
R^{n}
where
a_{α} are (not necessarily constant) infinitely differentiable functions.
P is a linear operator
Corresponding to
P there is another differential operator, the
formal adjoint of
P
Theorem. The operator theoretic adjoint
P* of
P is a restriction of the distributional extension of the formal adjoint. Specifically:
Spectral multiplicity theory
The multiplication representation of a selfadjoint 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 selfadjoint 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
HahnHans Hahn was an Austrian mathematician who made contributions to functional analysis, topology, set theory, the calculus of variations, real analysis, and order theory.Biography:...
HellingerErnst David Hellinger was a German mathematician.Early years:Ernst Hellinger was born on September 30, 1883 in Striegau, Silesia, Germany to Emil and Julie Hellinger. He grew up in Breslau, attended school and graduated from the Gymnasium there in 1902...
theory of spectral multiplicity.
We first define
uniform multiplicity:
Definition. A selfadjoint 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
where
H_{n} is a Hilbert space of dimension
n. The domain of M
_{f} consists of vectorvalued functions ψ on
R such that
Nonnegative countably additive measures μ, ν are
mutually singular if and only if they are supported on disjoint Borel sets.
Theorem. Let
A be a selfadjoint operator on a
separable Hilbert space
H. Then there is an ω sequence of countably additive finite measures on
R (some of which may be identically 0)
such that the measures are pairwise singular and
A is unitarily equivalent to the operator of multiplication by the function
f(λ) = λ on
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.
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 1949 by John von Neumann in one of the papers...
s of Hilbert spaces:
Theorem. Any selfadjoint operator on a separable Hilbert space is unitarily equivalent to multiplication by the function λ → λ on
The measure equivalence class of μ (or equivalently its sets of measure 0) is uniquely determined and the measurable family
{
H_{x}}
_{x} is determined almost everywhere with respect to μ.
Example: structure of the Laplacian
The Laplacian on
R^{n} is the operator
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 nonnegative; (see
elliptic operatorIn the theory of partial differential equations, elliptic operators are differential operators that generalize the Laplace operator. They are defined by the condition that the coefficients of the highestorder derivatives be positive, which implies the key property that the principal symbol is...
).
Theorem. If
n=1, then Δ has uniform multiplicity
mult=2, otherwise Δ has uniform multiplicity
mult=ω. Moreover, the measure μ
_{mult} is Borel measure on
[0, ∞).
Pure point spectrum
A selfadjoint operator
A on
H has pure point spectrum if and only if
H has an orthonormal basis {
e_{i}}
_{i ∈ I} consisting of eigenvectors for
A.
Example. The Hamiltonian for the harmonic oscillator has a quadratic potential
V, that is
This Hamiltonian has pure point spectrum; this is typical for bound state
HamiltoniansIn quantum mechanics, the Hamiltonian H, also Ȟ or Ĥ, is the operator corresponding to the total energy of the system. Its spectrum is the set of possible outcomes when one measures the total energy of a system...
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, compact operators on Hilbert spaces are a direct extension of matrices: in the Hilbert spaces, they are precisely the closure of finiterank operators in the uniform operator topology. As such, results from matrix theory can sometimes be extended to compact operators using...
 Theoretical and experimental justification for the Schrödinger equation
The theoretical and experimental justification for the Schrödinger equation motivates the discovery of the Schrödinger equation, the equation that describes the dynamics of nonrelativistic particles...
 Unbounded operator
In mathematics, more specifically functional analysis and operator theory, the notion of unbounded operator provides an abstract framework for dealing with differential operators, unbounded observables in quantum mechanics, and other cases....