In the wake of a preceding article [31] introducing the Schrödinger–Virasoro group, we study its affine action on a space of (1+1)-dimensional Schrödinger operators with time- and space-dependent potential V periodic in time. We focus on the subspace corresponding to potentials that are at most quadratic in the space coordinate, which is in some sense the natural quantization of the space of Hill (Sturm–Liouville) operators on the one-dimensional torus. The orbits in this subspace have finite codimension, and their classification by studying the stabilizers can be obtained by extending Kirillov's results on the orbits of the space of Hill operators under the Virasoro group. We then explain the connection to the theory of Ermakov–Lewis invariants for time-dependent harmonic oscillators. These exact adiabatic invariants behave covariantly under the action of the Schrödinger–Virasoro group, which allows a natural classification of the orbits in terms of a monodromy operator on L2(ℝ) which is closely related to the monodromy matrix for the corresponding Hill operator.
@article{CML_2010__2_2_217_0,
author = {Unterberger, J\'er\'emie},
title = {A classification of periodic time-dependent generalized harmonic oscillators using a {Hamiltonian} action of the {Schr\"odinger{\textendash}Virasoro} group},
journal = {Confluentes Mathematici},
pages = {217--263},
publisher = {World Scientific Publishing Co Pte Ltd},
volume = {2},
number = {2},
year = {2010},
doi = {10.1142/S1793744210000168},
language = {en},
url = {http://archive.numdam.org/articles/10.1142/S1793744210000168/}
}
TY - JOUR AU - Unterberger, Jérémie TI - A classification of periodic time-dependent generalized harmonic oscillators using a Hamiltonian action of the Schrödinger–Virasoro group JO - Confluentes Mathematici PY - 2010 SP - 217 EP - 263 VL - 2 IS - 2 PB - World Scientific Publishing Co Pte Ltd UR - http://archive.numdam.org/articles/10.1142/S1793744210000168/ DO - 10.1142/S1793744210000168 LA - en ID - CML_2010__2_2_217_0 ER -
%0 Journal Article %A Unterberger, Jérémie %T A classification of periodic time-dependent generalized harmonic oscillators using a Hamiltonian action of the Schrödinger–Virasoro group %J Confluentes Mathematici %D 2010 %P 217-263 %V 2 %N 2 %I World Scientific Publishing Co Pte Ltd %U http://archive.numdam.org/articles/10.1142/S1793744210000168/ %R 10.1142/S1793744210000168 %G en %F CML_2010__2_2_217_0
Unterberger, Jérémie. A classification of periodic time-dependent generalized harmonic oscillators using a Hamiltonian action of the Schrödinger–Virasoro group. Confluentes Mathematici, Tome 2 (2010) no. 2, pp. 217-263. doi : 10.1142/S1793744210000168. http://archive.numdam.org/articles/10.1142/S1793744210000168/
[1] M. Abramowitz et al. , Handbook of Mathematical Functions ( Harri Deutsch , 1984 ) .
[2] J. Balog, L. Fehér and L. Palla, Int. J. Mod. Phys. A 13, 315 (1998), DOI: 10.1142/S0217751X98000147.
[3] F. A. Berezin and M. A. Shubin , The Schrödinger Equation ( Kluwer , 1991 ) .
[4] A. Bohm et al. , The Geometric Phase in Quantum Systems , Texts and Monographs in Physics ( Springer , 2003 ) .
[5] S. Bouquet and H. R. Lewis, J. Math. Phys. 37, 5509 (1996), DOI: 10.1063/1.531708.
[6] S. Bouquetet al., J. Math. Phys. 33, 591 (1992).
[7] A. Erdelyi et al. , Higher Transcendental Functions ( McGraw-Hill , 1953 ) .
[8] I. M. Gelfand and G. E. Shilov , Generalized Functions 1 ( Academic Press , 1964 ) .
[9] H. Goldstein , Classical Mechanics ( Addison-Wesley , 1959 ) .
[10] I. S. Gradshteyn and I. M. Ryzhik , Table of Integrals, Series, and Products ( Academic Press , 1980 ) .
[11] M. F. Guasti and H. Moya-Cessa, J. Phys. A 36, 2069 (2003), DOI: 10.1088/0305-4470/36/8/305.
[12] M. F. Guasti and H. Moya-Cessa, Phys. Lett. A 311, 1 (2003).
[13] L. Guieu, Sur la géométrie des orbites de la représentation coadjointe du groupe de Bott–Virasoro, PhD thesis, Université Aix-Marseille I, 1994 .
[14] L. Guieu and C. Roger, L’Algèbre et le Groupe de Virasoro: Aspects Géométriques et Algébriques, Généralisations (CRM, 2007) .
[15] V. Guillemin and S. Sternberg , Symplectic Techniques in Physics ( Cambridge Univ. Press , 1984 ) .
[16] G. Hagedorn, Ann. Phys. 269, 77 (1998).
[17] G. Hagedorn, M. Loss and J. Slawny, J. Phys. A 19, 521 (1986), DOI: 10.1088/0305-4470/19/4/013.
[18] M. Henkel, J. Stat. Phys. 75, 1023 (1994), DOI: 10.1007/BF02186756.
[19] M. Henkel, Nucl. Phys. B 641, 405 (2002), DOI: 10.1016/S0550-3213(02)00540-0.
[20] M. Henkel, Phase-ordering kinetics: Ageing and local scale-invariance , cond-mat/0503739 .
[21] A. Joye, Geometric and mathematical aspects of the adiabatic theorem of quantum mechanics, PhD thesis, Ecole Polytechnique Fédérale de Lausanne (1992) .
[22] B. Khesin and R. Wendt , The Geometry of Infinite-Dimensional Lie Groups , Series of Modern Surveys in Mathematics 51 ( Springer , 2008 ) .
[23] A. A. Kirillov, The Geometry of Moments, Lecture Notes in Math. 970 (Springer, 1982) pp. 101–123.
[24] V. F. Lazutkin and T. F. Pankratova, Funct. Anal. Appl. 9, 306 (1975), DOI: 10.1007/BF01075876.
[25] P. G. L. Leach and H. R. Lewis, J. Math. Phys. 23, 2371 (1982).
[26] H. R. Lewis and W. B. Riesenfeld, J. Math. Phys. 10, 1458 (1969), DOI: 10.1063/1.1664991.
[27] W. Magnus and S. Winkler , Hill’s Equation ( John Wiley and Sons , 1966 ) .
[28] U. Niederer, Helv. Phys. Act. 47, 167 (1974).
[29] P. B. E. Padilla, Ermakov–Lewis dynamic invariants with some applications, Master Thesis, Institutot de Fisica (Guanajuato, Mexico), available on , arXiv:math-ph/0002005 .
[30] J. R. Ray and J. L. Reid, Phys. Rev. A 26, 1042 (1982), DOI: 10.1103/PhysRevA.26.1042.
[31] C. Roger and J. Unterberger, Ann. Henri Poincaré 7, 1477 (2006), DOI: 10.1007/s00023-006-0289-1.
[32] M. Henkel and J. Unterberger, Nucl. Phys. B 660, 407 (2003).
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