Balslev, E.; Grossmann, A.; Paul, T. A characterisation of dilation-analytic operators. (English) Zbl 0624.47022 Ann. Inst. Henri Poincaré, Phys. Théor. 45, 277-292 (1986). The main aim of this paper is to analyse the class of dilation-analytic operators in a new representation of quantum mechanics on a space of analytic functions on the upper half-plane to a Hilbert space. A characterization of dilation-analytic operators is given in terms of certain analytic continuation properties of the integral kernel. The characterization of dilation-analytic vectors, which was first obtained by D. Babbitt and E. Balslev [J. Funct. Analysis 18, 1-14 (1975; Zbl 0304.47009)], is recovered in an easy way. A dilation analyticity criterion is given for an integral operator in momentum space. Reviewer: Wu Jingbo MSC: 47B38 Linear operators on function spaces (general) 47A20 Dilations, extensions, compressions of linear operators 81R10 Infinite-dimensional groups and algebras motivated by physics, including Virasoro, Kac-Moody, \(W\)-algebras and other current algebras and their representations 45P05 Integral operators Keywords:dilation-analytic operators; new representation of quantum mechanics on a space of analytic functions on the upper half-plane to a Hilbert space; analytic continuation properties of the integral kernel; dilation- analytic vectors; integral operator in momentum space Citations:Zbl 0304.47009 PDFBibTeX XMLCite \textit{E. Balslev} et al., Ann. Inst. Henri Poincaré, Phys. Théor. 45, 277--292 (1986; Zbl 0624.47022) Full Text: Numdam EuDML References: [1] J. Aguilar and J.M. Combes , A class of analytic perturbations for one-body Schrödinger Hamiltonians , Comm. Math. Phys. , t. 22 , 1971 , p. 269 - 279 . Article | MR 345551 | Zbl 0219.47011 · Zbl 0219.47011 · doi:10.1007/BF01877510 [2] E. Balslev and J.M. Combes , Spectral properties of many-body Schrödinger operators with dilation-analytic interactions , Comm. Math. Phys. , t. 22 , 1971 , p. 280 - 299 . Article | MR 345552 | Zbl 0219.47005 · Zbl 0219.47005 · doi:10.1007/BF01877511 [3] D. Babbitt and E. Balslev , A characterisation of dilation-analytic potentials and vectors , J. Funct. Analysis , t. 18 , 1975 , p. 1 - 14 . MR 384008 | Zbl 0304.47009 · Zbl 0304.47009 · doi:10.1016/0022-1236(75)90026-9 [4] A. Dionisi Vici , A characterisation of dilation analytic integral kernels , Lett. Math. Phys. , t. 3 , 1979 , p. 533 - 541 . MR 555337 | Zbl 0434.47039 · Zbl 0434.47039 · doi:10.1007/BF00401935 [5] T. Paul , Functions analytic on the half-plane as quantum mechanical states , J. Math. Phys. , t. 25 , 1984 , p. 3252 - 3263 . MR 761848 [6] T. Paul , Affine coherent states for the radial Schrödinger equation 1. Radial harmonic oscillator and hydrogen atom . Preprint CPT 84/P. 1710 , Marseille . Submitted to Ann. I. H. P. [7] J. Weidmann , Linear Operators in Hilbert Spaces , Springer Verlag , 1980 . MR 566954 | Zbl 0434.47001 · Zbl 0434.47001 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.