Eigenfunction Expansions and Transformation Theory
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Eigenfunction Expansions and Transformation Theory Manuel Gadella · Fernando Gómez-Cubillo
Received: 20 June 2007 / Accepted: 13 October 2008 / Published online: 22 October 2008 © Springer Science+Business Media B.V. 2008
Abstract Generalized eigenfunctions may be regarded as vectors of a basis in a particular direct integral of Hilbert spaces or as elements of the antidual space × in a convenient Gelfand triplet ⊆ H ⊆ × . This work presents a fit treatment for computational purposes of transformations formulas relating different generalized bases of eigenfunctions in both frameworks direct integrals and Gelfand triplets. Transformation formulas look like usual in Physics literature, as limits of integral functionals but with well defined kernels. Several approaches are feasible. Vitali approach is used. Keywords Eigenfunction expansion · Spectral measure · Direct integral of Hilbert spaces · Gelfand triplet · Rigged Hilbert space · Dirac transformation theory · Vitali system Mathematics Subject Classification (2000) 47A70 · 47N50 1 Introduction Eigenfunction expansions appear in the most varied domains as for example in the basis of Dirac formulation of Quantum Mechanics [23], where each complete set of commuting observables (csco) A1 , A2 , . . . , An is supposed to have a generalized basis of kets |λ1 , λ2 , . . . , λn satisfying the following properties: 1. The kets |λ1 , λ2 , . . . , λn are generalized eigenvectors of the observables A1 , A2 , . . . , An , i.e., Aj |λ1 , λ2 , . . . , λn = λj |λ1 , λ2 , . . . , λn
(j = 1, 2, . . . , n),
M. Gadella () Dpto. de Física Teórica, Facultad de Ciencias, Universidad de Valladolid, Prado de la Magdalena, s.n., 47005 Valladolid, Spain e-mail: [email protected] F. Gómez-Cubillo Dpto. de Análisis Matemático, Facultad de Ciencias, Universidad de Valladolid, Prado de la Magdalena, s.n., 47005 Valladolid, Spain e-mail: [email protected]
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M. Gadella, F. Gómez-Cubillo
where λj is one of the possible outcomes of a measurement of the observable Aj , j = 1, 2, . . . , n. Let us call j the set of these outcomes. 2. For each pure state ϕ, one has the following integral decomposition: λ1 , λ2 , . . . , λn |ϕ|λ1 , λ2 , . . . , λn dλ1 dλ2 · · · dλn ,
ϕ=
where = 1 × 2 × · · · × n is the Cartesian product of the j and for (almost) all (λ1 , λ2 , . . . , λn ) ∈ , λ1 , λ2 , . . . , λn |ϕ is a complex number that gives the coordinates of ϕ in the generalized basis |λ1 , λ2 , . . . , λn . Following von Neumann [50], observables in Quantum Mechanics are represented by selfadjoint operators in a Hilbert space H. Therefore, a csco A1 , . . . , An is given by a set of n self adjoint operators, also called Ai , whose respective Hilbert space spectra are the sets i . The classical version of the spectral theorem associates to the family of selfadjoint operators Ai a family of commuting Borel spectral measure spaces (i , Bi , H, Pi ), where i = 1 . . . , n; see Sect. 2.1. Consider as a metric space with the product topology and the corresponding Borel
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