The isotropic harmonic oscillator in dimension 3 separates in several different coordinate systems. Separating in a particular coordinate system defines a system of three Poisson commuting integrals and, correspondingly, three commuting operators, one of which is the Hamiltonian. We show that the Lagrangian fibration defined by the Hamiltonian, the z component of the angular momentum, and a quartic integral obtained from separation in prolate spheroidal coordinates has a non-degenerate focus-focus point, and hence, non-trivial Hamiltonian monodromy for sufficiently large energies. The joint spectrum defined by the corresponding commuting quantum operators has non-trivial quantum monodromy implying that one cannot globally assign quantum numbers to the joint spectrum.
REFERENCES
1.
D. M.
Fradkin
, Am. J. Phys.
33
, 207
(1965
).2.
V. I.
Arnold
, Mathematical Methods of Classical Mechanics
, Graduate Texts in Mathematics (Springer
, Berlin, Heidelberg, New York
, 1978
), Vol. 60.3.
F.
Fassò
, Acta Appl. Math.
87
, 93
(2005
).4.
5.
A. S.
Miščenko
and A. T.
Fomenko
, Funct. Anal. Appl.
12
, 113
(1978
).6.
P.
Dazord
, T.
Delzant
et al., J. Differ. Geom.
26
, 223
(1987
).7.
K.
Schwarzschild
, Sitz. Ber. Kgl. Preuss. Akad. d. Wiss. Berlin
1916
, 548
.8.
E. G.
Kalnins
, J. M.
Kress
, and W.
Miller
, Jr, J. Math. Phys.
47
, 043514
(2006
).9.
E. P.
Wigner
, Group Theory and its Application to the Quantum Mechanics of Atomic Spectra
(Academic Press
, New York
, 1959
) [translation by J. J.
Griffin
of E. P.
Wigner
, Gruppentheorie und ihre Anwendung auf die Quantenmechanik der Atomspektren (Vieweg
, Braunschweig
, 1931
)].10.
D. J.
Griffiths
, Introduction to Quantum Mechanics
, 2nd ed. (Cambridge University Press
, Cambridge
, 2016
).11.
J. J.
Duistermaat
, Commun. Pure Appl. Math.
33
, 687
(1980
).12.
H. R.
Dullin
and H.
Waalkens
, Phys. Rev. Lett.
120
, 020507
(2018
).13.
H.
Waalkens
, H. R.
Dullin
, and P. H.
Richter
, Physica D
196
, 265
(2004
).14.
N. N.
Nekhoroshev
, D. A.
Sadovskií
, and B. I.
Zĥilinskií
, Ann. Henri Poincaré
7
, 1099
(2006
).15.
D.
Sadovskií
and B.
Zĥilinskií
, Ann. Phys.
322
, 164
(2007
).16.
B.
Zhilinskií
, “Monodromy and complexity of quantum systems
,” in The Complexity of Dynamical Systems: A Multi-Disciplinary Perspective
, edited by J.
Dubbeldam
, K.
Green
, and D.
Lenstra
(John Wiley & Sons
, 2011
), pp. 159
–181
.17.
R. H.
Cushman
and J. J.
Duistermaat
, Bull. Am. Math. Soc.
19
, 475
(1988
).18.
S.
Vũ Ngọc
, Commun. Pure Appl. Math.
53
, 143
(2000
).19.
D. A.
Sadovskií
and B. I.
Zĥilinskií
, Phys. Lett. A
256
, 235
(1999
).20.
B.
Zhilinskií
, “Hamiltonian monodromy as lattice defect
,” in Topology in Condensed Matter
, edited by M. I.
Monastyrsky
(Springer Berlin Heidelberg
, Berlin, Heidelberg
, 2006
), pp. 165
–186
.21.
R. H.
Cushman
, H. R.
Dullin
, A.
Giacobbe
, D. D.
Holm
, M.
Joyeux
, P.
Lynch
, D. A.
Sadovskíi
, and B.
Zĥilinskiíi
, Phys. Rev. Lett.
93
, 024302
(2004
).22.
M. S.
Child
, “Quantum monodromy and molecular spectroscopy
,” in Advances in Chemical Physics
, edited by S. A.
Rice
(John Wiley & Sons
, 2008
), pp. 39
–94
.23.
E.
Assémat
, K.
Efstathiou
, M.
Joyeux
, and D.
Sugny
, Phys. Rev. Lett.
104
, 113002
(2010
).24.
R. H.
Cushman
and D. A.
Sadovskií
, Europhys. Lett.
47
, 1
(1999
).25.
K.
Efstathiou
, D.
Sadovskií
, and B.
Zĥilinskií
, Proc. R. Soc. A
463
, 1771
(2007
).26.
P.
Cejnar
, M.
Macek
, S.
Heinze
, J.
Jolie
, and J.
Dobeš
, J. Phys. A: Math. Gen.
39
, L515
(2006
).27.
M.
Caprio
, P.
Cejnar
, and F.
Iachello
, Ann. Phys.
323
, 1106
(2008
).28.
H. R.
Dullin
and H.
Waalkens
, Phys. Rev. Lett.
101
, 070405
(2008
).29.
N.
Martynchuk
and H.
Waalkens
, Regular Chaotic Dyn.
21
, 697
(2016
).30.
N.
Martynchuk
, H.
Dullin
, K.
Efstathiou
, and H.
Waalkens
, Nonlinearity
32
(4
), 1296
–1326
(2019
).31.
D.
Sugny
, A.
Picozzi
, S.
Lagrange
, and H. R.
Jauslin
, Phys. Rev. Lett.
103
, 034102
(2009
).32.
C.
Chen
, M.
Ivory
, S.
Aubin
, and J.
Delos
, Phys. Rev. E
89
, 012919
(2014
).33.
N. C.
Leung
and M.
Symington
, J. Symplectic Geom.
8
, 143
(2010
).34.
M.
Symington
, in Proceedings of Symposia in Pure Mathematics
(American Mathematical Society
, 2003
), Vol. 71, p. 51
.35.
I. N.
Kozin
, D. A.
Sadovskií
, and B. I.
Zhilinskií
, Spectrochim. Acta, Part A
61
, 2867
(2005
).36.
V.
Lanchares
, A. I.
Pascual
, J.
Palacián
, P.
Yanguas
, and J. P.
Salas
, Chaos
12
, 87
(2002
).37.
S.
Ferrer
and J.
Gárate
, “On perturbed 3D elliptic oscillators: A case of critical inclination in galactic dynamics
,” in New Trends for Hamiltonian Systems and Celestial Mechanics
, edited by E. A.
Lacomba
and J.
Llibre
(World Scientific
, Singapore
, 1996
), pp. 179
–197
.38.
J.
Moser
, Commun. Pure Appl. Math.
23
, 609
(1970
).39.
K.
Efstathiou
and D. A.
Sadovskií
, Nonlinearity
17
, 415
(2004
).40.
C. A.
Coulson
and A.
Joseph
, Proc. Phys. Soc.
90
, 887
(1967
).41.
R. H.
Cushman
and L. M.
Bates
, Global Aspects of Classical Integrable Systems
(Birkhäuser
, Basel, Boston, Berlin
, 1997
).42.
R. H.
Cushman
, S.
Ferrer
, and H.
Hanßmann
, Nonlinearity
12
, 389
(1999
).43.
S.
Ferrer
, J.
Palacián
, and P.
Yanguas
, J. Nonlinear Sci.
10
, 145
(2000
).44.
L.
Michel
and B. I.
Zhilinskií
, Phys. Rep.
341
, 173
(2001
).45.
R. H.
Cushman
and D. A.
Sadovskií
, Physica D
142
, 166
(2000
).46.
K.
Efstathiou
, in Metamorphoses of Hamiltonian Systems with Symmetries
, Lecture Notes in Mathematics, edited by J.-M.
Morel
, F.
Takens
, and B.
Teissier
(Springer
, Berlin, Heidelberg, New York
, 2005
) p. 149
.47.
K.
Efstathiou
and D. A.
Sadovskií
, Rev. Mod. Phys.
82
, 2099
(2010
).48.
J. J.
Duistermaat
and G. J.
Heckman
, Inventiones Math.
69
, 259
(1982
).49.
V.
Ivrii
, Bull. Math. Sci.
6
, 379
(2016
).50.
L.
Grondin
, D. A.
Sadovskií
, and B. I.
Zhilinskií
, Phys. Rev. A
65
, 012105
(2001
).51.
A.
Giacobbe
, R. H.
Cushman
, D. A.
Sadovskií
, and B. I.
Zhilinskií
, J. Math. Phys.
45
, 5076
(2004
).52.
T.
Delzant
, Bull. Soc. Math. France
116
, 315
(1988
).53.
S.
Vũ Ngọc
, Adv. Math.
208
, 909
(2007
).54.
DLMF, “NIST Digital Library of Mathematical Functions,” http://dlmf.nist.gov/, Release 1.0.19 of 2018-06-22 (
2018
), F. W. J.
Olver
, A. B.
Olde Daalhuis
, D. W.
Lozier
, B. I.
Schneider
, R. F.
Boisvert
, C. W.
Clark
, B. R.
Miller
, and B. V.
Saunders
.55.
D. A.
Sadovskií
and B. I.
Zhilinskií
, Phys. Lett. A
256
, 235
(1999
).56.
A.
Pelayo
and S.
Vũ Ngọc
, Inventiones Math.
177
, 571
(2009
).57.
K.
Efstathiou
and N.
Martynchuk
, J. Geom. Phys.
115
, 104
(2017
), FDIS 2015: Finite Dimensional Integrable Systems in Geometry and Mathematical Physics.© 2019 Author(s).
2019
Author(s)
You do not currently have access to this content.
