Eigenfunction Expansions and Transformation Theory

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 Φ × in a convenient Gelfand triplet Φ ⊆ H ⊆ Φ × . 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. Here Vitali and martingale approaches are showed.

Lie Algebra Representations and Rigged Hilbert Spaces: The So(2) Case

“…They also proved, together with Maurin [14], the nuclear spectral theorem [10,15]. The RHS formulation of Quantum Mechanics was introduced by Bohm and Roberts around 1965 [5,8].…”

“…Moreover, it is self-adjoint on H, so that it can be extended to a weakly continuous operator on Φ × as the last identity in (15) shows. Therefore on Φ × J ≡ −iD φ .…”

“…On the other hand, the Rigged Hilbert space (RHS) is a suitable framework for a description of quantum states, when the use of both discrete bases, i.e., complete orthonormal sets, and generalized continuous bases like those used in the Dirac formalism are necessary [4]- [15]. As mentioned above, this is a typical situation arisen when we deal with special functions, which hold discrete labels and depend on continuous variables.…”

Applications of rigged Hilbert spaces in quantum mechanics and signal processing

“…The answer is as follows: as a consequence of a theorem by Gelfand and Maurin, which serves for a rigorous justification of the Dirac formulation of quantum mechanics. Although not in its full generality or with its full consequences, this theorem could be given as in here: 8,14,18,25 Theorem (Gelfand-Maurin).-Let A be a self-adjoint operator, with domain D A , on a separable infinite dimensional Hilbert space H . Then, there exists a subspace Φ ⊂ H dense on H having the following properties:…”

“…The extension of Q and R to the dual Φ × is defined via a duality formula like in (18). This extension is automatically continuous on Φ × for any topology compatible with the dual pair (Φ, Φ × ).…”