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Infunctional analysis, a branch ofmathematics, theBorel functional calculus is afunctional calculus (that is, an assignment ofoperators fromcommutative algebras to functions defined on theirspectra), which has particularly broad scope.[1][2] Thus for instance ifT is an operator, applying the squaring functions →s2 toT yields the operatorT2. Using the functional calculus for larger classes of functions, we can for example define rigorously the "square root" of the (negative)Laplacian operator−Δ or the exponential
The 'scope' here means the kind offunction of an operator which is allowed. The Borel functional calculus is more general than thecontinuous functional calculus, and its focus is different than theholomorphic functional calculus.
More precisely, the Borel functional calculus allows for applying an arbitraryBorel function to aself-adjoint operator, in a way that generalizes applying apolynomial function.
IfT is a self-adjoint operator on a finite-dimensionalinner product spaceH, thenH has anorthonormal basis{e1, ...,eℓ} consisting ofeigenvectors ofT, that is
Thus, for any positive integern,
If only polynomials inT are considered, then one gets theholomorphic functional calculus. The relation also holds for more general functions ofT. Given aBorel functionh, one can define an operatorh(T) by specifying its behavior on the basis:
Generally, any self-adjoint operatorT isunitarily equivalent to a multiplication operator; this means that for many purposes,T can be considered as an operatoracting onL2 of somemeasure space. The domain ofT consists of those functions whose above expression is inL2. In such a case, one can define analogously
For many technical purposes, the previous formulation is good enough. However, it is desirable to formulate the functional calculus in a way that does not depend on the particular representation ofT as a multiplication operator. That's what we do in the next section.
Formally, the bounded Borel functional calculus of a self adjoint operatorT onHilbert spaceH is a mapping defined on the space of bounded complex-valued Borel functionsf on the real line,such that the following conditions hold
Theorem— Any self-adjoint operatorT has a unique Borel functional calculus.
This defines the functional calculus forbounded functions applied to possiblyunbounded self-adjoint operators. Using the bounded functional calculus, one can prove part of theStone's theorem on one-parameter unitary groups:
Theorem— IfA is a self-adjoint operator, thenis a 1-parameter strongly continuous unitary group whoseinfinitesimal generator isiA.
As an application, we consider theSchrödinger equation, or equivalently, thedynamics of a quantum mechanical system. Innon-relativisticquantum mechanics, theHamiltonian operatorH models the totalenergyobservable of a quantum mechanical systemS. The unitary group generated byiH corresponds to the time evolution ofS.
We can also use the Borel functional calculus to abstractly solve some linearinitial value problems such as the heat equation, or Maxwell's equations.
The existence of a mapping with the properties of a functional calculus requires proof. For the case of a bounded self-adjoint operatorT, the existence of a Borel functional calculus can be shown in an elementary way as follows:
First pass from polynomial tocontinuous functional calculus by using theStone–Weierstrass theorem. The crucial fact here is that, for a bounded self adjoint operatorT and a polynomialp,
Consequently, the mappingis an isometry and a densely defined homomorphism on the ring of polynomial functions. Extending by continuity definesf(T) for a continuous functionf on the spectrum ofT. TheRiesz-Markov theorem then allows us to pass from integration on continuous functions tospectral measures, and this is the Borel functional calculus.
Alternatively, the continuous calculus can be obtained via theGelfand transform, in the context of commutative Banach algebras. Extending to measurable functions is achieved by applying Riesz-Markov, as above. In this formulation,T can be anormal operator.
Given an operatorT, the range of the continuous functional calculush →h(T) is the (abelian) C*-algebraC(T) generated byT. The Borel functional calculus has a larger range, that is the closure ofC(T) in theweak operator topology, a (still abelian)von Neumann algebra.
We can also define the functional calculus for not necessarily bounded Borel functionsh; the result is an operator which in general fails to be bounded. Using the multiplication by a functionf model of a self-adjoint operator given by the spectral theorem, this is multiplication by the composition ofh withf.
Theorem— LetT be a self-adjoint operator onH,h a real-valued Borel function onR. There is a unique operatorS such that
The operatorS of the previous theorem is denotedh(T).
More generally, a Borel functional calculus also exists for (bounded) normal operators.
Let be a self-adjoint operator. If is a Borel subset ofR, and is theindicator function ofE, then is a self-adjoint projection onH. Then mappingis aprojection-valued measure. The measure ofR with respect to is the identity operator onH. In other words, the identity operator can be expressed as the spectral integral
Stone's formula[3] expresses the spectral measure in terms of theresolvent:
Depending on the source, theresolution of the identity is defined, either as a projection-valued measure,[4] or as a one-parameter family of projection-valued measures with.[5]
In the case of a discrete measure (in particular, whenH is finite-dimensional), can be written asin the Dirac notation, where each is a normalized eigenvector ofT. The set is an orthonormal basis ofH.
In physics literature, using the above as heuristic, one passes to the case when the spectral measure is no longer discrete and write the resolution of identity asand speak of a "continuous basis", or "continuum of basis states", Mathematically, unless rigorous justifications are given, this expression is purely formal.