In the present paper, we build a combinatorial invariant, called the “spectral monodromy” from the spectrum of a single (non-self-adjoint) h-pseudodifferential operator with two degrees of freedom in the semi-classical limit. Our inspiration comes from the quantum monodromy defined for the joint spectrum of an integrable system of n commuting self-adjoint h-pseudodifferential operators, given by S. Vu Ngoc [“Quantum monodromy in integrable systems,” Commun. Math. Phys. 203(2), 465–479 (1999)]. The first simple case that we treat in this work is a normal operator. In this case, the discrete spectrum can be identified with the joint spectrum of an integrable quantum system. The second more complex case we propose is a small perturbation of a self-adjoint operator with a classical integrability property. We show that the discrete spectrum (in a small band around the real axis) also has a combinatorial monodromy. The main difficulty in this case is that we do not know the description of the spectrum everywhere, but only in a Cantor type set. In addition, we also show that the corresponding monodromy can be identified with the classical monodromy, defined by J. Duistermaat [“On global action-angle coordinates,” Commun. Pure Appl. Math. 33(6), 687–706 (1980)].
REFERENCES
1.
M. K.
Ali
, “The quantum normal form and its equivalents
,” J. Math. Phys.
26
(10
), 2565
–2572
(1985
).2.
V. I.
Arnold
, “Characteristic class entering into quantization conditions
,” Funkc. Anal. Priloz.
1
, 1
–13
(1967
).3.
M.
Audin
, Les systemes hamiltoniens et leur integrabilite
, Cours Specialisés [Specialized Courses] (Société Mathématique de France
, Paris
, 2001
), Vol. 8
.4.
D.
Bambusi
, S.
Graffi
, and T.
Paul
, “Normal forms and quantization formula
,” Commun. Math. Phys.
207
(1
), 173
–195
(1999
).5.
J.
Blank
, P.
Exner
, and M.
Havlíček
, Hilbert Space Operators in Quantum Physics
, 2nd ed., Theoretical and Mathematical Physics (Springer
, New York
, 2008
).6.
J.-B.
Bost
, “Tores invariants des systèmes dynamiques Hamiltoniens (d'après Kolmogorov, Arnold, Moser, Russmann, Zehnder, Herman, Poschel, : : :)
,” Asterisque
133-134
, 113
–157
(1986
), Seminar Bourbaki, 1984/1985, http://eudml.org/doc/110039.7.
H. W.
Broer
and G. B.
Huitema
, “A proof of the isoenergetic KAM-theorem from the “ordinary” one
,” J. Differ. Equations
90
(1
), 52
–60
(1991
).8.
H.
Boer
, “Do Diophantine vectors form a Cantor bouquet?
,” J. Differ. Equations Appl.
16
(5–6
), 433
–434
(2010
).9.
H.
Broer
, R.
Cushman
, F.
Fassò
, and F.
Takens
, “Geometry of KAM tori for nearly integrable Hamiltonian systems
,” Ergod. Theory Dyn. Syst.
27
(3
), 725
–741
(2007
).10.
D.
Burns
and R.
Hind
, “Symplectic geometry and the uniqueness of Grauert tubes
,” Geom. Funct. Anal.
11
(1
), 1
–10
(2001
).11.
S. E.
Cappell
, R.
Lee
, and E. Y.
Mimmer
, “On the Maslov index
,” Commun. Pure Appl. Math.
47
(2
), 121
–186
(1994
).12.
A.-M.
Charbonnel
, “Comportement semi-classique du spectre conjoint d'opérateurs pseudodifférentiels qui commutent
,” Asymptotic Anal.
1
(3
), 227
–261
(1988
).13.
L.
Charles
and S.
Vũ Ngọc
, “Spectral asymptotics via the semiclassical Birkhoff normal form
,” Duke Math. J.
143
(3
), 463
–511
(2008
).14.
R. H.
Cushman
and L. M.
Bates
, Global Aspects of Classical Integrable Systems
(Birkhuser Verlag
, Basel
, 1997
).15.
E. B.
Davies
, Spectral Theory and Differential Operators
, Cambridge Studies in Advanced Mathematics
Vol. 42
(Cambridge University Press
, Cambridge
, 1995
).16.
A.
Delshams
and P.
Gutiérrez
, “Effective stability and KAM theory
,” J. Differ. Equations
128
(2
), 415
–490
(1996
).17.
M.
Dimassi
and J.
Sjöstrand
, Spectral Asymptotics in the Semi-Classical Limit
, London Mathematical Society Lecture Note Series
Vol. 268
(Cambridge University Press
, Cambridge
, 1999
).18.
J. J.
Duitermaat
, “On global action-angle coordinates
,” Commun. Pure Appl. Math.
33
(6
), 687
–706
(1980
).19.
B.
Eckhardt
, “Birkhoff-Gustavson normal form in classical and quantum mechanics
,” J. Phys. A
19
(15
), 2961
–2972
(1986
).20.
J. V.
Egorov
, “The canonical transformations of pseudodifferential operators
,” Usp. Mat. Nauk
24
(5
), 235
–236
(1969
), http://www.mathnet.ru/php/archive.phtml?wshow=paper&jrnid=rm&paperid=5554&option_lang=eng.21.
B.
Fedosov
, Deformation Quantization and Index Theory
, Mathematical Topics
Vol. 9
(Akademie Verlag
, Berlin
, 1996
).22.
M.
Hitrik
, Lagrangian Tori and Spectra for Non-Selfadjoint Operators
, Séminaire: Équations aux Dérivées Partielles. 2005–2006 Based on joint works with J. Sjöstrand and S. Vu Ngọc, p. Exp. No. XXIV, 16.23.
M.
Hitrik
and J.
Sjöstrand
, “Non-selfadjoint perturbations of selfadjoint operators in 2 dimensions. I
,” Ann. Henri Poincare
5
(1
), 1
–73
(2004
).24.
M.
Hitrik
and J.
Sjöstrand
, “Nonselfadjoint perturbations of selfadjoint operators in two dimensions. II. Vanishing averages
,” Commun. Partial Differ. Equ.
30
(7–9
), 1065
–1106
(2005
).25.
M.
Hitrik
and J.
Sjöstrand
, “Non-selfadjoint perturbations of selfadjoint operators in two dimensions. IIIa. One branching point
,” Can. J. Math.
60
(3
), 572
–657
(2008
).26.
M.
Hitrik
and J.
Sjöstrand
, “Diophantine tori and Weyl laws for non-selfadjoint operators in dimension two
,” Commun. Math. Phys.
314
(2
), 373
–417
(2012
).27.
M.
Hitrik
, J.
Sjöstrand
, and S.
Vũ Ngọc
, “Diophantine tori and spectral asymptotics for nonselfadjoint operators
,” Am. J. Math.
129
(1
), 105
–182
(2007
).28.
H.
Hoffer
and E.
Zehnder
, Symplectic Invariants and Hamiltonian Dynamics
, The Floer Memorial Volume, Progress in Mathematics
Vol. 133
(Birkhuser
, Basel
, 1995
), pp. 525
–544
.29.
L.
Hörmander
, The Analysis of Linear Partial Differential Operators. I-IV
, Classics in Mathematics
(Springer-Verlag
, Berlin
, 1983
); Fourier Integral Operators.30.
S.-J.
Kan
, “Erratum to the paper: On rigidity of Grauert tubes over homogeneous Riemannian manifolds [J. Reine Angew. Math. 577 (2004), 213–233; mr2108219]
,” J. Reine Angew. Math.
596
, 235
(2006
).31.
T.
Kato
, Perturbation Theory for Linear Operators
, Classics in Mathematics
(Springer-Verlag
, Berlin
, 1995
), Reprint of the 1980 edition.32.
H.
Mineur
, “Quelques propriétés générales des équations de la mécanique
,” Bull. Astron. (2)
13
, 309
–328
(1948
).33.
J. E.
Moyal
, “Quantum mechanics as a statistical theory
,” Proc. Cambridge Philos. Soc.
45
, 99
–124
(1949
).34.
C. R.
De Oliveira
, Intermediate Spectral Theory and Quantum Dynamics
, Progress in Mathematical Physics
Vol. 54
(Birkhuser Verlag
, Basel
, 2009
).35.
G.
Popov
, “Invariant tori, effective stability, and quasimodes with exponentially small error terms. I-II. Birkhoff normal forms
,” Ann. Henri Poincare
1
(2
), 223
–279
(2000
).36.
J.
Pöschel
, “Integrability of Hamiltonian systems on Cantor sets
,” Commun. Pure Appl. Math.
35
(5
), 653
–696
(1982
).37.
M.
Reed
and B.
Simon
, Methods of Modern Mathematical Physics. IV. Analysis of Operators
(Academic Press [Harcourt Brace Jovanovich Publishers]
, New York
, 1978
).38.
D.
Robert
, Autour de lapproximation semi-classique
, Progress in Mathematics
Vol. 68
(Birkhuser Boston Inc.
, Boston, MA
, 1987
).39.
M. A.
Shubin
, Pseudodifferential Operators and Spectral Theory
, 2nd ed. (Springer-Verlag
, Berlin
, 2001
), Translated from the 1978 Russian original by Stig I. Andersson.40.
J.
Sjöstrans
, “Asymptotic distribution of eigenfrequencies for damped wave equations
,” Publ. Res. Inst. Math. Sci.
36
(5
), 573
–611
(2000
).41.
J.
Sjöstrans
, Asymptotic Distribution of Eigenfrequencies for Damped Wave Equations
, Journées Équations aux Dérivées Partielles (La Chapelle sur Erdre, 2000) (University of Nantes
, Nantes
, 2000
), p. Exp. No. XVI, 8.42.
J.
Sjöstrans
, “Eigenvalue distribution for non-self-adjoint operators with small multiplicative random perturbations
,” Ann. Fac. Sci. Toulouse Math. (6)
18
(4
), 739
–795
(2009
).43.
Y.
Colin de Verdière
, “Spectre conjoint d'opérateurs pseudo-différentiels qui commutent. II. Le cas intégrable
,” Math. Z.
171
(1
), 51
–73
(1980
).44.
Y.
Colin de Verdière
, Méthodes semi-classiques et théorie spectrale
(Cours de DEA
, 2006
).45.
S.
Vũ Ngọc
, “Quantum monodromy in integrable systems
,” Commun. Math. Phys.
203
(2
), 465
–479
(1999
).46.
S.
Vũ Ngọc
, Invariants symplectiques et semi-classiques des systémes intégrables avec singularités
, Séminaire: Équations aux Dérivées Partielles, 2000–2001, p. Exp. No. XXIV, 16.47.
S.
Vũ Ngọc
, “Quantum monodromy and Bohr-Sommerfeld rules
,” Lett. Math. Phys.
55
(3
), 205
–217
(2001
);Topological and Geometrical Methods
(Dijon
, 2000
).48.
S.
Vũ Ngọc
, Systèmes intégrables semi-classiques: du local au global, Panoramas et Synthèses
[Panoramas and Syntheses] (Société Mathématique de France
, Paris
, 2006
), Vol. 22
.49.
S.
Vũ Ngọc
, “Quantum Birkhoff normal forms and semiclassical analysis
,” Noncommutativity and Singularities
, Advanced Studies in Pure Mathematics
Vol. 55
(Mathematical Society of Japan
, Tokyo
, 2009
), pp. 99
–116
.50.
H.
Weyl
, The Thoery of Groups and Quantum Mechanics
(Library of Theoretical Physics
, Dover
, 1950
), Translated from the German edition.© 2014 AIP Publishing LLC.
2014
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