A note on Howe's oscillator semigroup

Joachim Hilgert

Annales de l'institut Fourier (1989)

  • Volume: 39, Issue: 3, page 663-688
  • ISSN: 0373-0956

Abstract

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Analytic extensions of the metaplectic representation by integral operators of Gaussian type have been calculated in the L 2 ( ℝ n ) and the Bargmann-Fock realisations by Howe [How2] and Brunet-Kramer [Brunet-Kramer, Reports on Math. Phys., 17 (1980), 205-215]], respectively. In this paper we show that the resulting semigroups of operators are isomorphic and calculate the intertwining operator.

How to cite

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Hilgert, Joachim. "A note on Howe's oscillator semigroup." Annales de l'institut Fourier 39.3 (1989): 663-688. <http://eudml.org/doc/74846>.

@article{Hilgert1989,
abstract = {Analytic extensions of the metaplectic representation by integral operators of Gaussian type have been calculated in the $L^2(\{\Bbb R\}^n)$ and the Bargmann-Fock realisations by Howe [How2] and Brunet-Kramer [Brunet-Kramer, Reports on Math. Phys., 17 (1980), 205-215]], respectively. In this paper we show that the resulting semigroups of operators are isomorphic and calculate the intertwining operator.},
author = {Hilgert, Joachim},
journal = {Annales de l'institut Fourier},
keywords = {analytic extensions of the metaplectic representation by integral operators of Gaussian type; Bargmann-Fock realisation; semigroups of operators; intertwining operator},
language = {eng},
number = {3},
pages = {663-688},
publisher = {Association des Annales de l'Institut Fourier},
title = {A note on Howe's oscillator semigroup},
url = {http://eudml.org/doc/74846},
volume = {39},
year = {1989},
}

TY - JOUR
AU - Hilgert, Joachim
TI - A note on Howe's oscillator semigroup
JO - Annales de l'institut Fourier
PY - 1989
PB - Association des Annales de l'Institut Fourier
VL - 39
IS - 3
SP - 663
EP - 688
AB - Analytic extensions of the metaplectic representation by integral operators of Gaussian type have been calculated in the $L^2({\Bbb R}^n)$ and the Bargmann-Fock realisations by Howe [How2] and Brunet-Kramer [Brunet-Kramer, Reports on Math. Phys., 17 (1980), 205-215]], respectively. In this paper we show that the resulting semigroups of operators are isomorphic and calculate the intertwining operator.
LA - eng
KW - analytic extensions of the metaplectic representation by integral operators of Gaussian type; Bargmann-Fock realisation; semigroups of operators; intertwining operator
UR - http://eudml.org/doc/74846
ER -

References

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  1. [Ba1] V. BARGMANN, On a Hilbert space of analytic functions and an associated integral transform, Part I, Comm. Pure. Appl. Math., 14 (1961), 187-214. Zbl0107.09102MR28 #486
  2. [Ba2] V. BARGMANN, Group representations on Hilbert spaces of analytic functions, in Analytic Methods in Mathematical Physics, Gilbert and Newton, Eds. Gordon and Breach, New York, 1968. 
  3. [Br] M. BRUNET, The metaplectic semigroup and related topics, Reports on Math. Phys., 22 (1985), 149-170. Zbl0609.22015MR88a:22009
  4. [BrK] M. BRUNET and P. KRAMER, Complex extension of the representation of the symplectic group associated with the canonical commutation relations, Reports on Math. Phys., 17 (1980), 205-215. Zbl0485.22017MR83a:81029
  5. [HilHofL] J. HILGERT, K.H. HOFMANN and J.D. LAWSON, Lie groups, convex cones and semigroups, Oxford University Press, Oxford, 1989. Zbl0701.22001MR91k:22020
  6. [How1] R. HOWE, Quantum mechanics and partial differential equations, J. Funct. Anal., 38 (1980), 188-254. Zbl0449.35002MR83b:35166
  7. [How2] R. HOWE, The oscillator semigroup, in the mathematical heritage of Hermann Weyl, Proc. Symp. Pure Math., 48, R.O. Wells, Ed. AMS Providence, 1988. Zbl0687.47034
  8. [K] P. KRAMER, Composite particles and symplectic (semi)-groups, in Group Theoretical Methods in Physics, P. Kramer and A. Rieckers Ed., LNP, 79, Springer, Berlin, 1978. 
  9. [KMS] P. KRAMER, M. MOSHINSKY and T.H. SELIGMAN, Complex extensions of canonical transformations and quantum mechanics, in Group theory and its applications III, E. Loeble Ed., Acad. Press, New York, 1975. 
  10. [LM] M. LÜSCHER and G. MACK, Global conformal invariance in quantum field theory, Comm. Math. Phys., 41 (1975), 203-234. 
  11. [OlaØ] G. 'OLAFSSON and B. ØRSTED, The holomorphic discrete series for affine symmetric spaces I, J. Funct. Anal., 81 (1988), 126-159. Zbl0678.22008MR89m:22021
  12. [Ol'1] G.I. OL'SHANSKII, Invariant cones in Lie algebras, Lie semigroups and the holomorphic discrete series, Funct. Anal. and Appl., 15 (1981), 275-285. Zbl0503.22011MR83e:32032
  13. [Ol'2] G.I. OL'SHANSKII, Convex cones in symmetric Lie algebras, Lie semigroups, and invariant causal (order) structures on pseudo-Riemannian symmetric spaces, Sov. Math. Dokl., 26 (1982), 97-101. Zbl0512.22012
  14. [Ol'3] G.I. OL'SHANSKII, Unitary representations of the infinite symmetric group : a semigroup approach in Representations of Lie groups and Lie algebras, Akad. Kiado, Budapest, 1985. Zbl0605.22005
  15. [R] L.J.M. ROTHKRANTZ, Transformatiehalfgroepen van nietcompacte hermitesche symmetrische Ruimten, Dissertation, Univ. of Amsterdam, 1980. 
  16. [S] R.J. STANTON, Analytic extension of the holomorphic discrete series, Amer. J. of Math., 108 (1986), 1411-1424. Zbl0626.43008MR88b:22013

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