Uniform semiclassical estimates for the propagation of quantum observables. (English) Zbl 1069.35061
Summary: We prove here that the semiclassical asymptotic expansion for the propagation of quantum observables, for \(C^\infty\)-Hamiltonians growing at most quadratically at infinity, is uniformly dominated at any order by an exponential term whose argument is linear in time. In particular, we recover the Ehrenfest time for the validity of the semiclassical approximation. Furthermore, if the Hamiltonian and the initial observables are holomorphic in a complex neighborhood of the phase space, we prove that the quantum observable is an analytic semiclassical observable. Other results about the large time behavior of observables with emphasis on the classical dynamic are also given. In particular, precise Gevrey estimates are established for classically integrable systems.
MSC:
| 35Q40 | PDEs in connection with quantum mechanics |
| 35B40 | Asymptotic behavior of solutions to PDEs |
| 35C20 | Asymptotic expansions of solutions to PDEs |
| 35J10 | Schrödinger operator, Schrödinger equation |
| 81Q20 | Semiclassical techniques, including WKB and Maslov methods applied to problems in quantum theory |
Keywords:
Ehrenfest time; semiclassical approximation; Hamiltonian; quantum observable; semiclassical observable; Gevrey estimatesReferences:
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