A mathematical interpretation of Dirac's formalism; Part B: Generalized eigenfunctions in trajectory spaces
type: Item
instance of: scholarly article
title: A mathematical interpretation of Dirac's formalism; Part B: Generalized eigenfunctions in trajectory spaces
zbMATH Open document ID: 0614.46034
DOI: 10.1016/0034-4877(85)90049-7
author: Stephanus J. L. Van Eijndhoven
author: J. de Graaf
published in: Reports on Mathematical Physics
publication date: 1985-01-01T00:00:00Z
review text: Starting with a Hilbert spce \(L_ 2({\mathbb{R}},\mu)\) the authors introduce the dense subspace \({\mathcal R}(L_ 2({\mathbb{R}},\mu))\) where \({\mathcal R}\) is a positive self-adjoint Hilbert-Schmidt operator on \(L_ 2({\mathbb{R}},\mu)\). For the space \({\mathcal R}(L_ 2({\mathbb{R}},\mu))\) a measure-theoretical Sobolev lemma is proved. The results for the spaces of type \({\mathcal R}(L_ 2({\mathbb{R}},\mu))\) are applied to nuclear analyticity spaces \(S_{X,{\mathcal A}}=\cup_{t>0}e^{-t{\mathcal A}}(X)\) where \(e^{-t{\mathcal A}}\) is a Hilbert-Schmidt operator on the Hilbert space X for each \(t>0\). The so-called generalized eigenvalue problem for a general self-adjoint operator \({\mathcal P}\) in X is solved completely. Meanwhile a detailed account of Dirac's formalism has appeared in our book: A mathematical introduction to Dirac's formalism. North-Holland Math. Libr. 36 (1986). [For part A see the review above.]
Mathematics Subject Classification ID: 46F10
Mathematics Subject Classification ID: 47A70
Mathematics Subject Classification ID: 81P05
zbMATH DE Number: 3994292
zbMATH Keywords: positive self-adjoint Hilbert-Schmidt operator
zbMATH Keywords: measure-theoretical Sobolev lemma
zbMATH Keywords: nuclear analyticity spaces
zbMATH Keywords: generalized eigenvalue problem
zbMATH Keywords: Dirac's formalism
MaRDI profile type: Publication
full work available at URL: 0034-4877(85)90049-7
OpenAlex ID: W2082051114
cites work: Q3247887
cites work: Q4741200
cites work: A Characterization of the Spaces $\mathfrak{S}_{{1/{k + 1}}}^{{k /{k + 1}}} $ by Means of Holomorphic Semigroups
cites work: Q5594071
cites work: Q3909512
cites work: Q4135799
Recommended article: A mathematical interpretation of Dirac's formalism; Part A: Dirac bases in trajectory spaces
Recommended article: A mathematical interpretation of Dirac's formalism. Part C: Free field operators
Recommended article: Q3972205
Recommended article: Q3759371
Recommended article: Dirac theory in spacetime algebra. I: The generalized bivector Dirac equation
Recommended article: Q5842855
Recommended article: A Generalization of the Dirac Equations
Recommended article: Q4655983
Recommended article: Generalizations of the Dirac equation in covariant and Hamiltonian form
Recommended article: The form of the spectral functions associated with Dirac equations
description: scientific article; zbMATH DE number 3994292