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.

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 Hp,q(z,z̄)
,”
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 (C,+)
,”
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
).
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