Eigenfunction Expansions and Transformation Theory

Details Eigenfunction Expansions and Transformation Theory Eigenfunction Expansions and Transformation Theory

2006-07-21

arxiv.org/abs/math/0607548v1

Mathematics

Functional Analysis

21 pages

Scientific paper

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 $\Phi^\times$ in a convenient Gelfand triplet $\Phi\subseteq{\mathcal\H}\subseteq\Phi^\times$. 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.

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