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Adiabatic decoupling and time-dependent Born-Oppenheimer theory. (English) Zbl 1017.81014
In the recent paper [see ibid. 223, 583-626 (2001; Zbl 1161.81376)] G. Hagedorn and A. Joye constructed an exponentially accurate time-dependent Born-Oppenheimer approximation for molecular quantum mechanics. In the Hagedorn-Joye’s approach adiabatic and semiclassical limits are being taken simultaneously, and they are coupled.
The paper under review contains careful separation between the adiabatic decoupling of a given group of energy bands from their orthogonal subspace and the semiclassics within the energy bands. Band crossings are allowed. The adiabatic decoupling leads to an effective Schrödinger equation for the nuclei, including contributions from the Berry connection.

MSC:
81Q20 Semiclassical techniques, including WKB and Maslov methods applied to problems in quantum theory
81V55 Molecular physics
81Q70 Differential geometric methods, including holonomy, Berry and Hannay phases, Aharonov-Bohm effect, etc. in quantum theory
47N50 Applications of operator theory in the physical sciences
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