We deal with an operational realization of the so-called n-th true S-poly-Bargmann space. We show that it can be realized as the range of the iterated sliced creation operator, and that it is close to the spectral analysis of a sliced magnetic Laplacian for which we provide a geometrical realization à la Hodge. Some integral representations of the considered spaces are also investigated.
REFERENCES
1.
Abreu
, L. D.
, “Sampling and interpolation in Bargmann–Fock spaces of polyanalytic functions
,” Appl. Comput. Harmonic Anal.
29
(3
), 287
–302
(2010
).2.
Abreu
, L. D.
and Feichtinger
, H. G.
, “Function spaces of polyanalytic functions
,” in Harmonic and Complex Analysis and its Applications
, Trends in Mathematics
(Birkhäuser/Springer
, Cham
, 2014
), pp. 1
–38
.3.
Ghanmi
, A.
and Imlal
, L.
, “Complex creation operator and planar automorphic functions
,” Math. Phys. Anal. Geom.
26
(4
), 28
(2023
).4.
Alpay
, D.
, Colombo
, F.
, Sabadini
, I.
, and Salomon
, G.
, “The Fock space in the slice hyperholomorphic setting
,” in Hypercomplex Analysis: New Perspectives and Applications
, Trends in Mathematics
(Birkhäuser/Springer
, Cham
, 2014
), pp. 43
–59
.5.
Alpay
, D.
, Diki
, K.
, and Sabadini
, I.
, “On slice polyanalytic functions of a quaternionic variable
,” Results Math.
74
(1
), 17
(2019
).6.
Arai
, A.
, Analysis on Fock Spaces and Mathematical Theory of Quantum Fields. An Introduction to Mathematical Analysis of Quantum Fields
(World Scientific Publishing Co. Pte. Ltd.
, Hackensack, NJ
, 2018
).7.
Askour
, N.
, Intissar
, A.
, and Mouayn
, Z.
, “Explicit formulas for reproducing kernels of generalized Bargmann spaces of Cn
,” J. Math. Phys.
41
(5
), 3057
–3067
(2000
).8.
Balk
, M. B.
, Polyanalytic Functions
, Mathematical Research Vol. 63
(Akademie-Verlag
, Berlin
, 1991
).9.
Bargmann
, V.
, “On a Hilbert space of analytic functions and an associated integral transform part I
,” Commun. Pure Appl. Math.
14
, 187
–214
(1961
).10.
Benahmadi
, A.
, El Hamyani
, A.
, and Ghanmi
, A.
, “S-polyregular Bargmann spaces
,” Adv. Appl. Clifford Algebras
29
(4
), 84
(2019
).11.
Benahmadi
, A.
and Ghanmi
, A.
, “On a novel class of polyanalytic Hermite polynomials
,” Results Math.
74
(4
), 186
(2019
).12.
Benahmadi
, A.
and Ghanmi
, A.
, “Solving the heat equation for a perturbed magnetic Laplacian on the complex plane
,” J. Math. Anal. Appl.
527
(1, part 1
), 127417
(2023
).13.
Benahmadi
, A.
and Ghanmi
, A.
, “On a new characterization of the true-poly-analytic Bargmann spaces
,” Complex Anal. Oper. Theory
18
(2
), 22
(2024
).14.
Christ
, M.
and Fu
, S.
, “Compactness in the -Neumann problem, magnetic Schrödinger operators, and the Aharonov–Bohm effect
,” Adv. Math.
197
(1
), 1
–40
(2005
).15.
Colombo
, F.
, Sabadini
, I.
, and Struppa
, D. C.
, Entire Slice Regular Functions
, SpringerBriefs in Mathematics
(Springer
, Cham
, 2016
).16.
Cycon
, H. L.
, Froese
, R. G.
, Kirsch
, W.
, and Simon
, B.
, Schrödinger Operators with Application to Quantum Mechanics and Global Geometry
, Springer Study ed., Texts and Monographs in Physics (Springer-Verlag
, Berlin
, 1987
).17.
Diki
, K.
and Ghanmi
, A.
, “A quaternionic analogue of the Segal–Bargmann transform
,” Complex Anal. Oper. Theory
11
(2
), 457
–473
(2017
).18.
El Hamyani
, A.
and Ghanmi
, A.
, “On some analytic properties of slice poly-regular Hermite polynomials
,” Math. Methods Appl. Sci.
41
(17
), 7985
–8002
(2018
).19.
Folland
, G. B.
, Harmonic Analysis in Phase Space
, Annals of Mathematics Studies Vol. 122
(Princeton University Press
, Princeton, NJ
, 1989
).20.
Gentili
, G.
, Stoppato
, C.
, and Struppa
, D. C.
, Regular Functions of a Quaternionic Variable
, 2nd ed., Springer Monographs in Mathematics
(Springer
, Cham
, 2022
).21.
Gentili
, G.
and Struppa
, D. C.
, “A new theory of regular functions of a quaternionic variable
,” Adv. Math.
216
(1
), 279
–301
(2007
).22.
Ghanmi
, A.
, “Operational formulae for the complex Hermite polynomials
,” Integr. Transforms Spec. Funct.
24
(11
), 884
–895
(2013
).23.
Ghanmi
, A.
and Intissar
, A.
, “Asymptotic of complex hyperbolic geometry and L2-spectral analysis of Landau-like Hamiltonians
,” J. Math. Phys.
46
(3
), 032107
(2005
).24.
Ghanmi
, A.
and Intissar
, A.
, “Construction of concrete orthonormal basis for (L2, Γ, χ)-theta functions associated to discrete subgroups of rank one in
,” J. Math. Phys.
54
(6
), 063514
(2013
).25.
Hall
, B. C.
, “Holomorphic methods in analysis and mathematical physics
,” in First Summer School in Analysis and Mathematical Physics (Cuernavaca Morelos, 1998)
, Contemporary Mathematics Vol. 260
(Aportaciones Matematicas, American Mathematical Society
, Providence, RI
, 2000
), pp. 1
–59
.26.
Landau
, L.
and Lifschitz
, E.
, Mécanique Quantique: Théorie Non Relativiste. Physique Théorique Tome III
(Editions Mir
, Moscow
, 1967
).27.
Matsumoto
, H.
and Ueki
, N.
, “Spectral analysis of Schrödinger operators with magnetic fields
,” J. Funct. Anal.
140
(1
), 218
–255
(1996
).28.
Mouayn
, Z.
, “Coherent state transforms attached to generalized Bargmann spaces on the complex plane
,” Math. Nachr.
284
(14–15
), 1948
–1954
(2011
).29.
Prieto
, C. T.
, “Holomorphic spectral geometry of magnetic Schrödinger operators on Riemann surfaces
,” Differ. Geom. Appl.
24
(3
), 288
–310
(2006
).30.
Segal
, I.
, in Lectures at the Summer Seminar on Applied Mathematics
, Boulder, CO
, 1960
.31.
Shigekawa
, I.
, “Eigenvalue problems for the Schrödinger operator with the magnetic field on a compact Riemannian manifold
,” J. Funct. Anal.
75
(1
), 92
–127
(1987
).32.
Shubin
, M.
, “Essential self-adjointness for semi-bounded magnetic Schrödinger operators on non-compact manifolds
,” J. Funct. Anal.
186
(1
), 92
–116
(2001
).33.
Stefanov
, A.
, “Strichartz estimates for the magnetic Schrödinger equation
,” Adv. Math.
210
(1
), 246
–303
(2007
).34.
Vasilevski
, N. L.
, “Poly-Fock spaces
,” in Differential Operators and Related Topics Vol. I (Odessa, 1997)
, Operator Theory: Advances and Applications Vol. 117
(Birkhäuser
, Basel
, 2000
), pp. 371
–386
.35.
Zhu
, K.
, Analysis on Fock Spaces
, Graduate Texts in Mathematics Vol. 263
(Springer
, New York
, 2012
).© 2025 Author(s). Published under an exclusive license by AIP Publishing.
2025
Author(s)
You do not currently have access to this content.