Stone's theorem on one-parameter unitary groups

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 {displaystyle {mathcal {H}}} and one-parameter families {displaystyle (U_{t})_{tin mathbb {R} }} of unitary operators that are strongly continuous, i.e., {displaystyle forall t_{0}in mathbb {R} ,psi in {mathcal {H}}:qquad lim _{tto t_{0}}U_{t}(psi )=U_{t_{0}}(psi ),} and are homomorphisms, i.e., {displaystyle forall s,tin mathbb {R} :qquad U_{t+s}=U_{t}U_{s}.} Such one-parameter families are ordinarily referred to as strongly continuous one-parameter unitary groups.

The theorem was proved by Marshall Stone (1930, 1932), and John von Neumann (1932) showed that the requirement that {displaystyle (U_{t})_{tin mathbb {R} }} be strongly continuous can be relaxed to say that it is merely weakly measurable, at least when the Hilbert space is separable.

This is an impressive result, as it allows to define the derivative of the mapping {displaystyle tmapsto U_{t},} which is only supposed to be continuous. It is also related to the theory of Lie groups and Lie algebras.

Contents 1 Formal statement 2 Example 3 Applications 4 Using Fourier transform 5 Generalizations 6 References 7 Bibliography Formal statement The statement of the theorem is as follows.[1] Theorem. Let {displaystyle (U_{t})_{tin mathbb {R} }} be a strongly continuous one-parameter unitary group. Then there exists a unique (possibly unbounded) operator {displaystyle A:{mathcal {D}}_{A}to {mathcal {H}}} , that is self-adjoint on {displaystyle {mathcal {D}}_{A}} and such that {displaystyle forall tin mathbb {R} :qquad U_{t}=e^{itA}.} The domain of {displaystyle A} is defined by {displaystyle {mathcal {D}}_{A}=left{psi in {mathcal {H}}left|lim _{varepsilon to 0}{frac {-i}{varepsilon }}left(U_{varepsilon }(psi )-psi right){text{ exists}}right.right}.} Conversely, let {displaystyle A:{mathcal {D}}_{A}to {mathcal {H}}} be a (possibly unbounded) self-adjoint operator on {displaystyle {mathcal {D}}_{A}subseteq {mathcal {H}}.} Then the one-parameter family {displaystyle (U_{t})_{tin mathbb {R} }} of unitary operators defined by {displaystyle forall tin mathbb {R} :qquad U_{t}:=e^{itA}} is a strongly continuous one-parameter group.

In both parts of the theorem, the expression {displaystyle e^{itA}} is defined by means of the spectral theorem for unbounded self-adjoint operators.

The operator {displaystyle A} is called the infinitesimal generator of {displaystyle (U_{t})_{tin mathbb {R} }.} Furthermore, {displaystyle A} will be a bounded operator if and only if the operator-valued mapping {displaystyle tmapsto U_{t}} is norm-continuous.

The infinitesimal generator {displaystyle A} of a strongly continuous unitary group {displaystyle (U_{t})_{tin mathbb {R} }} may be computed as {displaystyle Apsi =-ilim _{varepsilon to 0}{frac {U_{varepsilon }psi -psi }{varepsilon }},} with the domain of {displaystyle A} consisting of those vectors {displaystyle psi } for which the limit exists in the norm topology. That is to say, {displaystyle A} is equal to {displaystyle -i} times the derivative of {displaystyle U_{t}} with respect to {displaystyle t} at {displaystyle t=0} . Part of the statement of the theorem is that this derivative exists—i.e., that {displaystyle A} is a densely defined self-adjoint operator. The result is not obvious even in the finite-dimensional case, since {displaystyle U_{t}} is only assumed (ahead of time) to be continuous, and not differentiable.

Example The family of translation operators {displaystyle left[T_{t}(psi )right](x)=psi (x+t)} is a one-parameter unitary group of unitary operators; the infinitesimal generator of this family is an extension of the differential operator {displaystyle -i{frac {d}{dx}}} defined on the space of continuously differentiable complex-valued functions with compact support on {displaystyle mathbb {R} .} Thus {displaystyle T_{t}=e^{t{frac {d}{dx}}}.} In other words, motion on the line is generated by the momentum operator.

Applications Stone's theorem has numerous applications in quantum mechanics. For instance, given an isolated quantum mechanical system, with Hilbert space of states H, time evolution is a strongly continuous one-parameter unitary group on {displaystyle {mathcal {H}}} . The infinitesimal generator of this group is the system Hamiltonian.

Further information: Stone–von Neumann theorem and Heisenberg group Using Fourier transform Stone's Theorem can be recast using the language of the Fourier transform. The real line {displaystyle mathbb {R} } is a locally compact abelian group. Non-degenerate *-representations of the group C*-algebra {displaystyle C^{*}(mathbb {R} )} are in one-to-one correspondence with strongly continuous unitary representations of {displaystyle mathbb {R} ,} i.e., strongly continuous one-parameter unitary groups. On the other hand, the Fourier transform is a *-isomorphism from {displaystyle C^{*}(mathbb {R} )} to {displaystyle C_{0}(mathbb {R} ),} the {displaystyle C^{*}} -algebra of continuous complex-valued functions on the real line that vanish at infinity. Hence, there is a one-to-one correspondence between strongly continuous one-parameter unitary groups and *-representations of {displaystyle C_{0}(mathbb {R} ).} As every *-representation of {displaystyle C_{0}(mathbb {R} )} corresponds uniquely to a self-adjoint operator, Stone's Theorem holds.

Therefore, the procedure for obtaining the infinitesimal generator of a strongly continuous one-parameter unitary group is as follows: Let {displaystyle (U_{t})_{tin mathbb {R} }} be a strongly continuous unitary representation of {displaystyle mathbb {R} } on a Hilbert space {displaystyle {mathcal {H}}} . Integrate this unitary representation to yield a non-degenerate *-representation {displaystyle rho } of {displaystyle C^{*}(mathbb {R} )} on {displaystyle {mathcal {H}}} by first defining {displaystyle forall fin C_{c}(mathbb {R} ):qquad rho (f):=int _{mathbb {R} }f(t)~U_{t}dt,} and then extending {displaystyle rho } to all of {displaystyle C^{*}(mathbb {R} )} by continuity. Use the Fourier transform to obtain a non-degenerate *-representation {displaystyle tau } of {displaystyle C_{0}(mathbb {R} )} on {displaystyle {mathcal {H}}} . By the Riesz-Markov Theorem, {displaystyle tau } gives rise to a projection-valued measure on {displaystyle mathbb {R} } that is the resolution of the identity of a unique self-adjoint operator {displaystyle A} , which may be unbounded. Then {displaystyle A} is the infinitesimal generator of {displaystyle (U_{t})_{tin mathbb {R} }.} The precise definition of {displaystyle C^{*}(mathbb {R} )} is as follows. Consider the *-algebra {displaystyle C_{c}(mathbb {R} ),} the continuous complex-valued functions on {displaystyle mathbb {R} } with compact support, where the multiplication is given by convolution. The completion of this *-algebra with respect to the {displaystyle L^{1}} -norm is a Banach *-algebra, denoted by {displaystyle (L^{1}(mathbb {R} ),star ).} Then {displaystyle C^{*}(mathbb {R} )} is defined to be the enveloping {displaystyle C^{*}} -algebra of {displaystyle (L^{1}(mathbb {R} ),star )} , i.e., its completion with respect to the largest possible {displaystyle C^{*}} -norm. It is a non-trivial fact that, via the Fourier transform, {displaystyle C^{*}(mathbb {R} )} is isomorphic to {displaystyle C_{0}(mathbb {R} ).} A result in this direction is the Riemann-Lebesgue Lemma, which says that the Fourier transform maps {displaystyle L^{1}(mathbb {R} )} to {displaystyle C_{0}(mathbb {R} ).} Generalizations The Stone–von Neumann theorem generalizes Stone's theorem to a pair of self-adjoint operators, {displaystyle (P,Q)} , satisfying the canonical commutation relation, and shows that these are all unitarily equivalent to the position operator and momentum operator on {displaystyle L^{2}(mathbb {R} ).} The Hille–Yosida theorem generalizes Stone's theorem to strongly continuous one-parameter semigroups of contractions on Banach spaces.

References ^ Hall 2013 Theorem 10.15 Bibliography Hall, B.C. (2013), Quantum Theory for Mathematicians, Graduate Texts in Mathematics, vol. 267, Springer, ISBN 978-1461471158 von Neumann, John (1932), "Über einen Satz von Herrn M. H. Stone", Annals of Mathematics, Second Series (in German), Annals of Mathematics, 33 (3): 567–573, doi:10.2307/1968535, ISSN 0003-486X, JSTOR 1968535 Stone, M. H. (1930), "Linear Transformations in Hilbert Space. III. Operational Methods and Group Theory", Proceedings of the National Academy of Sciences of the United States of America, National Academy of Sciences, 16 (2): 172–175, doi:10.1073/pnas.16.2.172, ISSN 0027-8424, JSTOR 85485, PMC 1075964, PMID 16587545 Stone, M. H. (1932), "On one-parameter unitary groups in Hilbert Space", Annals of Mathematics, 33 (3): 643–648, doi:10.2307/1968538, JSTOR 1968538 K. Yosida, Functional Analysis, Springer-Verlag, (1968) hide vte Functional analysis (topics – glossary) Spaces BanachBesovFréchetHilbertHölderNuclearOrliczSchwartzSobolevtopological vector Properties barrelledcompletedual (algebraic/topological)locally convexreflexiveseparable Theorems Hahn–BanachRiesz representationclosed graphuniform boundedness principleKakutani fixed-pointKrein–Milmanmin–maxGelfand–NaimarkBanach–Alaoglu Operators adjointboundedcompactHilbert–Schmidtnormalnucleartrace classtransposeunboundedunitary Algebras Banach algebraC*-algebraspectrum of a C*-algebraoperator algebragroup algebra of a locally compact groupvon Neumann algebra Open problems invariant subspace problemMahler's conjecture Applications Hardy spacespectral theory of ordinary differential equationsheat kernelindex theoremcalculus of variationsfunctional calculusintegral operatorJones polynomialtopological quantum field theorynoncommutative geometryRiemann hypothesisdistribution (or generalized functions) Advanced topics approximation propertybalanced setChoquet theoryweak topologyBanach–Mazur distanceTomita–Takesaki theory Categories: Theorems in functional analysis

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