A constructive presentation of rigged Hilbert spaces

[摘要] We construct a rigged Hilbert space for the square integrable functions on the line L2(R) adding to the generators of the Weyl-Heisenberg algebra a new discrete operator, related to the degree of the Hermite polynomials. All together, continuous and discrete operators, constitute the generators of the projective algebra io(2). L2(R) and the vector space of the line R are shown to be isomorphic representations of such an algebra and, as both these representations are irreducible, all operators defined on the rigged Hilbert spaces L2(R) or R are shown to belong to the universal enveloping algebra of io(2). The procedure can be extended to orthogonal and pseudo-orthogonal spaces of arbitrary dimension by tensorialization. Circumventing all formal problems the paper proposes a kind of toy model, well defined from a mathematical point of view, of rigged Hilbert spaces where, in contrast with the Hilbert spaces, operators with different cardinality are allowed.

[发布机构] Dipartimento di Fisica, Università di Firenze, Sesto Fiorentino, Firenze; I50019, Italy^1;Depatamento de Fisica Téorica, Universidad de Valladolid, Valladolid; E-47005, Spain^2

[效力级别] 力学 [学科分类] 力学，机械学

[关键词] Arbitrary dimension;Discrete operators;Heisenberg algebras;Hermite polynomials;Isomorphic representation;Rigged Hilbert space;Square integrable;Universal enveloping algebras