Comptes Rendus
Analyse harmonique, Théorie des représentations
Completeness of coherent state subsystems for nilpotent Lie groups
Jordy Timo van Velthoven1
1 Delft University of Technology, Mekelweg 4, Building 36, 2628 CD Delft, The Netherlands
Comptes Rendus. Mathématique, Volume 360 (2022), pp. 799-808.

Let G be a nilpotent Lie group and let π be a coherent state representation of G. The interplay between the cyclicity of the restriction π| Γ to a lattice Γ≤G and the completeness of subsystems of coherent states based on a homogeneous G-space is considered. In particular, it is shown that necessary density conditions for Perelomov’s completeness problem can be obtained via density conditions for the cyclicity of π| Γ .

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DOI : 10.5802/crmath.342
Classification : 22E27, 42C30, 42C40, 81R30
Jordy Timo van Velthoven 1

1 Delft University of Technology, Mekelweg 4, Building 36, 2628 CD Delft, The Netherlands
Licence : CC-BY 4.0
Droits d'auteur : Les auteurs conservent leurs droits 
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Jordy Timo van Velthoven. Completeness of coherent state subsystems for nilpotent Lie groups. Comptes Rendus. Mathématique, Volume 360 (2022), pp. 799-808. doi : 10.5802/crmath.342. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.342/

[1] Syed Twareque Ali; Jean-Pierre Antoine; Jean-Pierre Gazeau Coherent states, wavelets, and their generalizations, Theoretical and Mathematical Physics (Cham), Springer, 2014, xviii+577 pages | Zbl

[2] Valentine Bargmann; Paolo Butera; Luciano Girardello; John R. Klauder On the completeness of the coherent states, Rep. Math. Phys., Volume 2 (1971) no. 4, pp. 221-228 | DOI | MR

[3] Bachir Bekka Square integrable representations, von Neumann algebras and an application to Gabor analysis, J. Fourier Anal. Appl., Volume 10 (2004) no. 4, pp. 325-349 | DOI | MR | Zbl

[4] Bachir Bekka; Jean Ludwig Complemented *-primitive ideals in L 1 -algebras of exponential Lie groups and of motion groups, Math. Z., Volume 204 (1990) no. 4, pp. 515-526 | DOI | MR | Zbl

[5] Leonardo Biliotti On the moment map on symplectic manifolds, Bull. Belg. Math. Soc. Simon Stevin, Volume 16 (2009) no. 1, pp. 107-116 | MR | Zbl

[6] Lawrence J. Corwin; Frederick P. Greenleaf Representations of nilpotent Lie groups and their applications. Part 1: Basic theory and examples, Cambridge Studies in Advanced Mathematics, 18, Cambridge University Press, 1990, viii+269 pages | Zbl

[7] Karlheinz Gröchenig Multivariate Gabor frames and sampling of entire functions of several variables, Appl. Comput. Harmon. Anal., Volume 31 (2011) no. 2, pp. 218-227 | DOI | MR | Zbl

[8] Victor Guillemin; Shlomo Sternberg Symplectic techniques in physics, Cambridge University Press, 1984, xi+468 pages | Zbl

[9] Vaughan Jones Bergman space zero sets, modular forms, von Neumann algebras and ordered groups (2020) (https://arxiv.org/abs/2006.16419)

[10] D. Kelly-Lyth Uniform lattice point estimates for co-finite Fuchsian groups, Proc. Lond. Math. Soc., Volume 78 (1999) no. 1, pp. 29-51 | DOI | MR | Zbl

[11] Anthony W. Knapp Lie groups beyond an introduction, Progress in Mathematics, 140, Birkhäuser, 2002, xviii+812 pages | MR | Zbl

[12] Bertram Kostant; Shlomo Sternberg Symplectic projective orbits, New directions in applied mathematics (Papers presented April 25/26, 1980, on the occasion of the case centennial celebration), Springer, 1982, pp. 81-84 | Zbl

[13] Wojciech Lisiecki Kaehler coherent state orbits for representations of semisimple Lie groups, Ann. Inst. Henri Poincaré, Phys. Théor., Volume 53 (1990) no. 2, pp. 245-258 | Numdam | MR | Zbl

[14] Wojciech Lisiecki A classification of coherent state representations of unimodular Lie groups, Bull. Am. Math. Soc., Volume 25 (1991) no. 1, pp. 37-43 | DOI | MR | Zbl

[15] Wojciech Lisiecki Coherent state representations. A survey, Rep. Math. Phys., Volume 35 (1995) no. 2-3, pp. 327-358 | DOI | MR | Zbl

[16] Henri Moscovici Coherent state representations of nilpotent Lie groups, Commun. Math. Phys., Volume 54 (1977), pp. 63-68 | DOI | MR | Zbl

[17] Henri Moscovici; Andrei Verona Coherent states and square integrable representations, Ann. Inst. Henri Poincaré, Nouv. Sér., Sect. A, Volume 29 (1978), pp. 139-156 | Numdam | MR | Zbl

[18] Karl-Hermann Neeb Coherent states, holomorphic extensions, and highest weight representations, Pac. J. Math., Volume 174 (1996) no. 2, pp. 497-542 | DOI | MR | Zbl

[19] Karl-Hermann Neeb Square integrable highest weight representations, Glasg. Math. J., Volume 39 (1997) no. 3, pp. 295-321 | DOI | MR | Zbl

[20] Karl-Hermann Neeb Holomorphy and convexity in Lie theory, de Gruyter Expositions in Mathematics, 28, Walter de Gruyter, 1999, xxi+778 pages | Zbl

[21] Yuriĭ A. Neretin Perelomov problem and inversion of the Segal-Bargmann transform, Funct. Anal. Appl., Volume 40 (2006) no. 4, pp. 330-333 | DOI | Zbl

[22] Anatol Odzijewicz Coherent states and geometric quantization, Commun. Math. Phys., Volume 150 (1992) no. 2, pp. 385-413 | DOI | MR | Zbl

[23] Askold M. Perelomov Remark on the completeness of the coherent state system, Teor. Mat. Fiz., Volume 6 (1971) no. 2, pp. 213-224 | MR

[24] Askold M. Perelomov Coherent states for arbitrary Lie group, Commun. Math. Phys., Volume 26 (1972), pp. 222-236 | DOI | MR | Zbl

[25] Askold M. Perelomov Coherent states for the Lobačevskiĭ plane, Funkts. Anal. Prilozh., Volume 7 (1973) no. 3, pp. 57-66 | MR

[26] Askold M. Perelomov Generalized coherent states and their applications, Texts and Monographs in Physics, Springer, 1986 | DOI | Zbl

[27] Götz E. Pfander; Peter Rashkov Remarks on multivariate Gaussian Gabor frames, Monatsh. Math., Volume 172 (2013) no. 2, pp. 179-187 | DOI | MR | Zbl

[28] Jayakumar Ramanathan; Tim Steger Incompleteness of sparse coherent states, Appl. Comput. Harmon. Anal., Volume 2 (1995) no. 2, pp. 148-153 | DOI | MR | Zbl

[29] John H. Rawnsley Coherent states and Kähler manifolds, Q. J. Math., Oxf. II. Ser., Volume 28 (1977), pp. 403-415 | DOI | Zbl

[30] Marc A. Rieffel von Neumann algebras associated with pairs of lattices in Lie groups, Math. Ann., Volume 257 (1981) no. 4, pp. 403-418 | DOI | MR | Zbl

[31] José Luis Romero; Jordy Timo van Velthoven The density theorem for discrete series representations restricted to lattices, Expo. Math., Volume 40 (2022) no. 2, pp. 265-301 | DOI | MR | Zbl

[32] Hugo Rossi; Michele Vergne Representations of certain solvable Lie groups on Hilbert spaces of holomorphic functions and the application to the holomorphic discrete series of a semisimple Lie group, J. Funct. Anal., Volume 13 (1973), pp. 324-389 | DOI | MR | Zbl

[33] Norman J. Wildberger Convexity and unitary representations of nilpotent Lie groups, Invent. Math., Volume 98 (1989) no. 2, pp. 281-292 | DOI | MR | Zbl

[34] Nicholas M. J. Woodhouse Geometric quantization, Oxford Math. Monogr., Clarendon Press, 1992, xi+307 pages | Zbl

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