In
mathematics, a self-adjoint operator on an infinite-dimensional
complex vector space
In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called ''vectors'', may be added together and multiplied ("scaled") by numbers called ''scalars''. Scalars are often real numbers, but can ...
''V'' with
inner product
In mathematics, an inner product space (or, rarely, a Hausdorff pre-Hilbert space) is a real vector space or a complex vector space with an operation called an inner product. The inner product of two vectors in the space is a scalar, often ...
(equivalently, a Hermitian operator in the finite-dimensional case) is a
linear map
In mathematics, and more specifically in linear algebra, a linear map (also called a linear mapping, linear transformation, vector space homomorphism, or in some contexts linear function) is a mapping V \to W between two vector spaces that pr ...
''A'' (from ''V'' to itself) that is its own
adjoint
In mathematics, the term ''adjoint'' applies in several situations. Several of these share a similar formalism: if ''A'' is adjoint to ''B'', then there is typically some formula of the type
:(''Ax'', ''y'') = (''x'', ''By'').
Specifically, adjoin ...
. If ''V'' is finite-dimensional with a given
orthonormal basis
In mathematics, particularly linear algebra, an orthonormal basis for an inner product space ''V'' with finite dimension is a basis for V whose vectors are orthonormal, that is, they are all unit vectors and orthogonal to each other. For examp ...
, this is equivalent to the condition that the
matrix
Matrix most commonly refers to:
* ''The Matrix'' (franchise), an American media franchise
** ''The Matrix'', a 1999 science-fiction action film
** "The Matrix", a fictional setting, a virtual reality environment, within ''The Matrix'' (franchis ...
of ''A'' is a
Hermitian matrix
In mathematics, a Hermitian matrix (or self-adjoint matrix) is a complex square matrix that is equal to its own conjugate transpose—that is, the element in the -th row and -th column is equal to the complex conjugate of the element in the -t ...
, i.e., equal to its
conjugate transpose
In mathematics, the conjugate transpose, also known as the Hermitian transpose, of an m \times n complex matrix \boldsymbol is an n \times m matrix obtained by transposing \boldsymbol and applying complex conjugate on each entry (the complex c ...
''A''. By the finite-dimensional
spectral theorem
In mathematics, particularly linear algebra and functional analysis, a spectral theorem is a result about when a linear operator or matrix can be diagonalized (that is, represented as a diagonal matrix in some basis). This is extremely useful ...
, ''V'' has an
orthonormal basis
In mathematics, particularly linear algebra, an orthonormal basis for an inner product space ''V'' with finite dimension is a basis for V whose vectors are orthonormal, that is, they are all unit vectors and orthogonal to each other. For examp ...
such that the matrix of ''A'' relative to this basis is a
diagonal matrix
In linear algebra, a diagonal matrix is a matrix in which the entries outside the main diagonal are all zero; the term usually refers to square matrices. Elements of the main diagonal can either be zero or nonzero. An example of a 2×2 diagonal m ...
with entries in the
real number
In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every ...
s. In this article, we consider
generalization
A generalization is a form of abstraction whereby common properties of specific instances are formulated as general concepts or claims. Generalizations posit the existence of a domain or set of elements, as well as one or more common characte ...
s of this
concept
Concepts are defined as abstract ideas. They are understood to be the fundamental building blocks of the concept behind principles, thoughts and beliefs.
They play an important role in all aspects of cognition. As such, concepts are studied by ...
to operators on
Hilbert spaces of arbitrary dimension.
Self-adjoint operators are used in
functional analysis
Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure (e.g. inner product, norm, topology, etc.) and the linear functions defined o ...
and
quantum mechanics
Quantum mechanics is a fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum physics including quantum chemistr ...
. In quantum mechanics their importance lies in the
Dirac–von Neumann formulation of quantum mechanics, in which physical
observable
In physics, an observable is a physical quantity that can be measured. Examples include position and momentum. In systems governed by classical mechanics, it is a real-valued "function" on the set of all possible system states. In quantum phy ...
s such as
position
Position often refers to:
* Position (geometry), the spatial location (rather than orientation) of an entity
* Position, a job or occupation
Position may also refer to:
Games and recreation
* Position (poker), location relative to the dealer
* ...
,
momentum,
angular momentum
In physics, angular momentum (rarely, moment of momentum or rotational momentum) is the rotational analog of linear momentum. It is an important physical quantity because it is a conserved quantity—the total angular momentum of a closed syst ...
and
spin are represented by self-adjoint operators on a Hilbert space. Of particular significance is the
Hamiltonian
Hamiltonian may refer to:
* Hamiltonian mechanics, a function that represents the total energy of a system
* Hamiltonian (quantum mechanics), an operator corresponding to the total energy of that system
** Dyall Hamiltonian, a modified Hamiltonian ...
operator
defined by
:
which as an observable corresponds to the total
energy
In physics, energy (from Ancient Greek: ἐνέργεια, ''enérgeia'', “activity”) is the quantitative property that is transferred to a body or to a physical system, recognizable in the performance of work and in the form of hea ...
of a particle of mass ''m'' in a real
potential field ''V''.
Differential operators are an important class of
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.
The ter ...
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 theory, a multiplication operator is an operator defined on some vector space of functions and whose value at a function is given by multiplication by a fixed function . That is,
T_f\varphi(x) = f(x) \varphi (x) \quad
for all in th ...
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
be an
unbounded (i.e. not necessarily bounded) operator with a dense domain
This condition holds automatically when
is
finite-dimensional since
for every linear operator on a finite-dimensional space.
Let the inner product
be
conjugate-linear
In mathematics, a function f : V \to W between two complex vector spaces is said to be antilinear or conjugate-linear if
\begin
f(x + y) &= f(x) + f(y) && \qquad \text \\
f(s x) &= \overline f(x) && \qquad \text \\
\end
hold for all vectors x, y ...
on the ''second'' argument. This applies to complex Hilbert spaces only. By definition, the
adjoint operator
In mathematics, specifically in operator theory, each linear operator A on a Euclidean vector space defines a Hermitian adjoint (or adjoint) operator A^* on that space according to the rule
:\langle Ax,y \rangle = \langle x,A^*y \rangle,
where ...
acts on the subspace
consisting of the elements
for which there is a
such that
for every
Setting
defines the linear operator
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 discre ...
of an (arbitrary) operator
is the set
An operator
is said to extend
if
This is written as
The densely defined operator
is called symmetric if
:
for all
As shown below,
is symmetric if and only if
The unbounded densely defined operator
is called self-adjoint if
Explicitly,
and
Every self-adjoint operator is symmetric. Conversely, a symmetric operator
for which
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
is called the resolvent set (or regular set) if for every
the (not-necessarily-bounded) operator
has a ''bounded everywhere-defined'' inverse. The complement
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. The word was first used scientifically in optics to describe the rainbow of colors ...
. In finite dimensions,
consists exclusively of
eigenvalue
In linear algebra, an eigenvector () or characteristic vector of a linear transformation is a nonzero vector that changes at most by a scalar factor when that linear transformation is applied to it. The corresponding eigenvalue, often denoted ...
s.
Bounded self-adjoint operators
A bounded operator ''A'' is self-adjoint if
:
for all
and
in ''H''. If ''A'' is symmetric and
, then, by
Hellinger–Toeplitz theorem In 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. By definition, an operator ' ...
, ''A'' is necessarily bounded.
Every bounded linear operator ''T'' : ''H'' → ''H'' on a Hilbert space ''H'' can be written in the form
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
be a bounded self-adjoint linear operator defined on
.
*
is real for all
.
*
if
* If the image of ''A'', denoted by
, is dense in ''H'' then
is invertible.
* The eigenvalues of ''A'' are real and eigenvectors belonging to different eigenvalues are orthogonal.
* If
is an eigenvalue of ''A'' then
; in particular,
.
** In general, there may not exist any eigenvalue
such that
, but if in addition ''A'' is compact then there necessarily exists an eigenvalue
, equal to either
or
, such that
,
* If a sequence of bounded self-adjoint linear operators is convergent then the limit is self-adjoint.
* There exists a number
, equal to either
or
, and a sequence
such that
and
for all ''i''.
Symmetric operators
''NOTE: symmetric operators are defined above.''
''A'' is symmetric ⇔ ''A''⊆''A''
An unbounded, densely defined operator
is symmetric if and only if
Indeed, the if-part follows directly from the definition of the adjoint operator. For the only-if-part, assuming that
is symmetric, the inclusion
follows from
the
Cauchy–Bunyakovsky–Schwarz inequality: for every
:
The equality
holds due to the equality
:
for every
the density of
and non-degeneracy of the inner product.
The
Hellinger–Toeplitz theorem In 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. By definition, an operator ' ...
says that an everywhere-defined symmetric operator is bounded and self-adjoint.
''A'' is symmetric ⇔ ∀''x'' ⟨''Ax'', ''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
is anti-linear on the ''first'' argument and linear on the second. (In the reverse scenario, we work with
instead). The symmetry of
follows from the
polarization identity
:
which holds for every
, , (''A''−''λ'')''x'', , ≥ ''d''(''λ'')⋅, , ''x'', ,
This property is used in the proof that the spectrum of a self-adjoint operator is real.
Define
and
The values
are properly defined since
and
due to symmetry. Then, for every
and every
:
where
Indeed, let
By
Cauchy-Schwarz inequality,
:
If
then
and
is called ''bounded below''.
A simple example
As noted above, the
spectral theorem
In mathematics, particularly linear algebra and functional analysis, a spectral theorem is a result about when a linear operator or matrix can be diagonalized (that is, represented as a diagonal matrix in some basis). This is extremely useful ...
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 or treaty; in more specific cases it may refer to:
* Interstate compact
* Blood compact, an ancient ritual of the Philippines
* Compact government, a type of colonial rule utilized in British ...
inverse, meaning that the corresponding differential equation ''Af'' = ''g'' is solved by some integral (and 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 ''L''
2 ,1and the
differential operator
:
with
consisting of all complex-valued infinitely
differentiable
In mathematics, a differentiable function of one real variable is a function whose derivative exists at each point in its domain. In other words, the graph of a differentiable function has a non-vertical tangent line at each interior point in its ...
functions ''f'' on
, 1
The comma is a punctuation mark that appears in several variants in different languages. It has the same shape as an apostrophe or single closing quotation mark () in many typefaces, but it differs from them in being placed on the baseline o ...
satisfying the boundary conditions
:
Then
integration by parts
In calculus, and more generally in mathematical analysis, integration by parts or partial integration is a process that finds the integral of a product of functions in terms of the integral of the product of their derivative and antiderivative. ...
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
ensure that the boundary terms in the integration by parts vanish.
The eigenfunctions of ''A'' are the sinusoids
:
with the real eigenvalues ''n''
2π
2; 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
be an unbounded symmetric operator.
is self-adjoint if and only if
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'':
:
The domain of ''A'' is the space of all ''L''
2 functions
for which
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
is self-adjoint. Symmetric operators for which
is strictly greater than
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
(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:
:
We must now specify a domain for ''A'', which amounts to choosing boundary conditions. If we choose
:
then ''A'' is not symmetric (because the boundary terms in the integration by parts do not vanish).
If we choose
:
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
of ''A'' is
:
whereas the domain of the adjoint
of ''A'' is
:
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
. If we compute
for
using integration by parts, then since
vanishes at both ends of the interval, no boundary conditions on
are needed to cancel out the boundary terms in the integration by parts. Thus, any sufficiently smooth function
is in the domain of
, with
.
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
is the same as the domain of the adjoint of ''A''. Thus, in this case, the domain of the adjoint of
is bigger than the domain of
itself, showing that
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:
:
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
for
are eigenvectors, with eigenvalues
, 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
. 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
.
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
:
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
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
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
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
:
It is then possible to show that
is not a symmetric operator, which certainly implies that
is not essentially self-adjoint. Indeed,
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
: There are functions
in the domain of
for which neither
nor
is separately in
, but the combination of them occurring in
is in
. This allows for
to be nonsymmetric, even though both
and
are symmetric operators. This sort of cancellation does not occur if we replace the repelling potential
with the confining potential
.
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
, for example, physicists would say that the eigenvectors are the functions
, which are clearly not in the Hilbert space
. (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
is replaced by a Dirac delta function
.
Although these statements may seem disconcerting to mathematicians, they can be made rigorous by use of the Fourier transform, which allows a general
function to be expressed as a "superposition" (i.e., integral) of the functions
, even though these functions are not in
. The Fourier transform "diagonalizes" the momentum operator; that is, it converts it into the operator of multiplication by
, where
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 that preserves the inner product: the inner product of two vectors before the transformation is equal to their inner product after the transformation.
Formal definition
More precisely, ...
''U'' : ''H'' → ''K'' such that
* ''U'' maps dom ''A''
bijective
In mathematics, a bijection, also known as a bijective function, one-to-one correspondence, or invertible function, is a function between the elements of two sets, where each element of one set is paired with exactly one element of the other ...
ly onto dom ''B'',
*
A
multiplication operator
In operator theory, a multiplication operator is an operator defined on some vector space of functions and whose value at a function is given by multiplication by a fixed function . That is,
T_f\varphi(x) = f(x) \varphi (x) \quad
for all in th ...
is defined as follows: Let (''X'', Σ, μ) be a countably additive
measure space and ''f'' a real-valued measurable function on ''X''. An operator ''T'' of the form
:
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 transform In mathematics, the Cayley transform, named after Arthur Cayley, is any of a cluster of related things. As originally described by , the Cayley transform is a mapping between skew-symmetric matrices and special orthogonal matrices. The transform ...
'' 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 range In mathematics, particularly measure theory, the essential range, or the set of essential values, of a function is intuitively the 'non-negligible' range of the function: It does not change between two functions that are equal almost everywhere. O ...
of f.
Functional calculus
One important application of the spectral theorem is to define a "
functional calculus
In mathematics, a functional calculus is a theory allowing one to apply mathematical functions to mathematical operators. It is now a branch (more accurately, several related areas) of the field of functional analysis, connected with spectral the ...
." That is to say, if
is a function on the real line and
is a self-adjoint operator, we wish to define the operator
. If
has a true orthonormal basis of eigenvectors
with eigenvalues
, then
is the operator with eigenvectors
and eigenvalues
. The goal of functional calculus is to extend this idea to the case where
has continuous spectrum.
Of particular importance in quantum physics is the case in which
is the Hamiltonian operator
and
is an exponential. In this case, the functional calculus should allow us to define the operator
:
which is the operator defining the time-evolution in quantum mechanics.
Given the representation of ''T'' as the operator of multiplication by
—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
.
Resolution of the identity
It has been customary to introduce the following notation
:
where
is the
characteristic function In mathematics, the term "characteristic function" can refer to any of several distinct concepts:
* The indicator function of a subset, that is the function
::\mathbf_A\colon X \to \,
:which for a given subset ''A'' of ''X'', has value 1 at points ...
(indicator function)of the interval
. The family of projection operators E
''T''(λ) is called resolution of the identity for ''T''. Moreover, the following
Stieltjes integral
Thomas Joannes Stieltjes (, 29 December 1856 – 31 December 1894) was a Dutch mathematician. He was a pioneer in the field of moment problems and contributed to the study of continued fractions. The Thomas Stieltjes Institute for Mathematics at ...
representation for ''T'' can be proved:
:
The definition of the operator integral above can be reduced to that of a scalar valued Stieltjes integral using the
weak operator topology
In functional analysis, the weak operator topology, often abbreviated WOT, is the weakest topology on the set of bounded operators on a Hilbert space H, such that the functional sending an operator T to the complex number \langle Tx, y\rangle is c ...
. 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, a branch of mathematics, the Borel functional calculus is a ''functional calculus'' (that is, an assignment of operators from commutative algebras to functions defined on their spectra), which has particularly broad scope. ...
using
Dirac notation
Distributed Research using Advanced Computing (DiRAC) is an integrated supercomputing facility used for research in particle physics, astronomy and cosmology in the United Kingdom. DiRAC makes use of multi-core processors and provides a variety o ...
as follows:
If ''H'' is self-adjoint and ''f'' is a
Borel function
In mathematics and in particular measure theory, a measurable function is a function between the underlying sets of two measurable spaces that preserves the structure of the spaces: the preimage of any measurable set is measurable. This is in d ...
,
:
with
:
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. 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
. In the Dirac notation, (projective) measurements are described via
eigenvalues
In linear algebra, an eigenvector () or characteristic vector of a linear transformation is a nonzero vector that changes at most by a scalar factor when that linear transformation is applied to it. The corresponding eigenvalue, often denoted b ...
and
eigenstates
In quantum physics, a quantum state is a mathematical entity that provides a probability distribution for the outcomes of each possible measurement on a system. Knowledge of the quantum state together with the rules for the system's evolution i ...
, 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
In mathematics, the spectral theory of ordinary differential equations is the part of spectral theory concerned with the determination of the spectrum and eigenfunction expansion associated with a linear ordinary differential equation. In his disse ...
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 space
In mathematics, a rigged Hilbert space (Gelfand triple, nested Hilbert space, equipped Hilbert space) is a construction designed to link the distribution and square-integrable aspects of functional analysis. Such spaces were introduced to study s ...
.
If , the theorem is referred to as resolution of unity:
:
In the case
is the sum of an Hermitian ''H'' and a skew-Hermitian (see
skew-Hermitian matrix
__NOTOC__
In linear algebra, a square matrix with complex entries is said to be skew-Hermitian or anti-Hermitian if its conjugate transpose is the negative of the original matrix. That is, the matrix A is skew-Hermitian if it satisfies the relation ...
) operator
, one defines the
biorthogonal basis set
:
and write the spectral theorem as:
:
(See
Feshbach–Fano partitioning
In quantum mechanics, and in particular in scattering theory, the Feshbach–Fano method, named after Herman Feshbach and Ugo Fano, separates (partitions) the resonant and the background components of the wave function and therefore of the associat ...
method for the context where such operators appear in
scattering theory
In mathematics and physics, scattering theory is a framework for studying and understanding the scattering of waves and particles. Wave scattering corresponds to the collision and scattering of a wave with some material object, for instance sunli ...
).
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
and
have trivial kernels. That is to say, ''A'' ''fails to be'' self-adjoint if and only if
has an eigenvector with eigenvalue
or
.
Another way of looking at the issue is provided by the
Cayley transform In mathematics, the Cayley transform, named after Arthur Cayley, is any of a cluster of related things. As originally described by , the Cayley transform is a mapping between skew-symmetric matrices and special orthogonal matrices. The transform ...
of a self-adjoint operator and the deficiency indices. (It is often of technical convenience to deal with
closed 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.
The ter ...
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 is a visual representation of something. It can be two-dimensional, three-dimensional, or somehow otherwise feed into the visual system to convey information. An image can be an artifact, such as a photograph or other two-dimensiona ...
(in other words, range) and the
domain
Domain may refer to:
Mathematics
*Domain of a function, the set of input values for which the (total) function is defined
**Domain of definition of a partial function
**Natural domain of a partial function
**Domain of holomorphy of a function
* Do ...
, respectively. W(''A'') is
isometric on its domain. Moreover, the range of 1 − W(''A'') is
dense
Density (volumetric mass density or specific mass) is the substance's mass per unit of volume. The symbol most often used for density is ''ρ'' (the lower case Greek letter rho), although the Latin letter ''D'' can also be used. Mathematically ...
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 isometry to any symmetric densely defined operator. Note that the mappings W and S are
monotone
Monotone refers to a sound, for example music or speech, that has a single unvaried tone. See: monophony.
Monotone or monotonicity may also refer to:
In economics
*Monotone preferences, a property of a consumer's preference ordering.
*Monotonic ...
: 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 isometry In functional analysis a partial isometry is a linear map between Hilbert spaces such that it is an isometry on the orthogonal complement of its kernel.
The orthogonal complement of its kernel is called the initial subspace and its range is called ...
.
Given a partial isometry ''V'', the deficiency indices of ''V'' are defined as the dimension of the
orthogonal complement In the mathematical fields of linear algebra and functional analysis, the orthogonal complement of a subspace ''W'' of a vector space ''V'' equipped with a bilinear form ''B'' is the set ''W''⊥ of all vectors in ''V'' that are orthogonal to every ...
s of the domain and range:
:
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
The adjective canonical is applied in many contexts to mean "according to the canon" the standard, rule or primary source that is accepted as authoritative for the body of knowledge or literature in that context. In mathematics, "canonical example ...
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 In functional analysis, the Friedrichs extension is a canonical form, canonical self-adjoint operator, self-adjoint extension of a non-negative densely defined self-adjoint operator#Symmetric operators, symmetric operator. It is named after the math ...
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, the Laplace operator or Laplacian is a differential operator given by the divergence of the gradient of a scalar function on Euclidean space. It is usually denoted by the symbols \nabla\cdot\nabla, \nabla^2 (where \nabla is the ...
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, Stone's theorem on one-parameter unitary groups is a basic theorem of functional analysis that establishes a one-to-one correspondence between self-adjoint operators on a Hilbert space \mathcal and one-parameter families
:(U_)_
o ...
, 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, applicable to systems with internal state (also called ''stateful systems''). In this formulation, ''time'' is not required to be a continuous parameter, but may be disc ...
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
, 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
has a counterpart in the classical dynamics of a particle with potential
: 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
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