Remarks on Time-dependent Schrödinger Equations, Bound States, and Coherent States
It has been well known from the beginning of quantum theory that there exist deep connections between the time evolution of a classical Hamiltonian system and the bound states for the Schrödinger equation, in particular in the semiclassical régime. These connections are well understood for integrable systems (Bohr—Sommerfeld quantization rules). But for more intricate systems (like classically chaotic Hamiltonian) the mathematical analysis of the bound states is much more difficult and there are few rigorous mathematical results. In this paper our goal is to revisit some of these results and to show that they can be proven, and sometimes improved, by using essentially two technics: the Wigner—Weyl calculus and the propagation of observables on one side, the propagation of coherent states on the other side. We want to emphasize that in our approach we get rather explicit estimates in terms of classical dynamics
The main ideas explained here, in particular the use of coherent states, are the results of several year of collaboration with Monique Combescure.
KeywordsCoherent State Trace Formula Classical Flow Quantum Ergodicity Liouville Measure
Unable to display preview. Download preview PDF.
- 9.De Bièvre S., Robert D., Semiclassical propagation and the log he-1 time-barier, I. M. R. N. 12 (2003), 667–696.Google Scholar
- 10.Dozias S., thesis, DMI. ENS., Paris, 1994.Google Scholar
- 12.Folland G.B., Harmonic Analysis in Phase Space, Annals of Math. Studies, Vol. 122, Princeton University Press, 1989.Google Scholar
- 15.Helfer B., Martinez A., Robert D., Ergodicité et limite semi-classique, CMP 109 (1987), 313–326.Google Scholar
- 20.Paul T., Semi-classical methods with emphasis on coherent states, in IMA Volumes in Mathematics and Applications , J. Rauch and B. Simon, eds., Vol. 95, Springer, 1997, 51–97.Google Scholar
- 21.Paul T., Uribe A., Sur la formule semi-classique des traces, Note CRAS 313:I (1991), 217–222.Google Scholar
- 24.Robert D., Autour de l’Approximation Semi-classique, Birkhäuser, Prog. Math., Vol. 68., 1987.Google Scholar
- 25.Robert D., Remarks on asymptotic solutions for time dependent Schrödinger equations. in Optimal control and partial differential equations, IOS Press, 188–197, (2001).Google Scholar
- 26.Sunada T., Quantum ergodicity, in Progress in Inverse Spectral Geometry, Andersson, Stig I. et al., eds., Birkhäuser, Basel Trends in Mathematics (1997), 175–196.Google Scholar
- 29.Zelditch S., Quantum mixing, JFA 140 (1996), 68–86; and Quantum Dynamics from the Semi-classical Point of View , unpublished lectures, Institut Borel, II1P, Paris, 1996.Google Scholar