Analyse harmonique, Théorie des représentations
Completeness of coherent state subsystems for nilpotent Lie groups
Comptes Rendus. Mathématique, Tome 360 (2022) no. G7, 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 ΓG 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 .

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

1 Delft University of Technology, Mekelweg 4, Building 36, 2628 CD Delft, The Netherlands
@article{CRMATH_2022__360_G7_799_0,
     author = {van Velthoven, Jordy Timo},
     title = {Completeness of coherent state subsystems for nilpotent {Lie} groups},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {799--808},
     publisher = {Acad\'emie des sciences, Paris},
     volume = {360},
     number = {G7},
     year = {2022},
     doi = {10.5802/crmath.342},
     language = {en},
     url = {http://www.numdam.org/articles/10.5802/crmath.342/}
}
TY  - JOUR
AU  - van Velthoven, Jordy Timo
TI  - Completeness of coherent state subsystems for nilpotent Lie groups
JO  - Comptes Rendus. Mathématique
PY  - 2022
SP  - 799
EP  - 808
VL  - 360
IS  - G7
PB  - Académie des sciences, Paris
UR  - http://www.numdam.org/articles/10.5802/crmath.342/
DO  - 10.5802/crmath.342
LA  - en
ID  - CRMATH_2022__360_G7_799_0
ER  - 
%0 Journal Article
%A van Velthoven, Jordy Timo
%T Completeness of coherent state subsystems for nilpotent Lie groups
%J Comptes Rendus. Mathématique
%D 2022
%P 799-808
%V 360
%N G7
%I Académie des sciences, Paris
%U http://www.numdam.org/articles/10.5802/crmath.342/
%R 10.5802/crmath.342
%G en
%F CRMATH_2022__360_G7_799_0
van Velthoven, Jordy Timo. Completeness of coherent state subsystems for nilpotent Lie groups. Comptes Rendus. Mathématique, Tome 360 (2022) no. G7, pp. 799-808. doi : 10.5802/crmath.342. http://www.numdam.org/articles/10.5802/crmath.342/

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

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

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

[6] Corwin, Lawrence J.; Greenleaf, Frederick P. 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] Gröchenig, Karlheinz 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] Guillemin, Victor; Sternberg, Shlomo Symplectic techniques in physics, Cambridge University Press, 1984, xi+468 pages | Zbl

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Cité par Sources :