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Twisted Pseudodifferential Calculus and Application to the Quantum Evolution of Molecules
André Martinez and Vania Sordoni, Università di Bologna, Italy
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Memoirs of the American Mathematical Society
2009; 82 pp; softcover
Volume: 200
ISBN-10: 0-8218-4296-X
ISBN-13: 978-0-8218-4296-6
List Price: US$62 Individual Members: US$37.20
Institutional Members: US\$49.60
Order Code: MEMO/200/936

The authors construct an abstract pseudodifferential calculus with operator-valued symbol, suitable for the treatment of Coulomb-type interactions, and they apply it to the study of the quantum evolution of molecules in the Born-Oppenheimer approximation, in the case of the electronic Hamiltonian admitting a local gap in its spectrum. In particular, they show that the molecular evolution can be reduced to the one of a system of smooth semiclassical operators, the symbol of which can be computed explicitely. In addition, they study the propagation of certain wave packets up to long time values of Ehrenfest order.

• Introduction
• Assumptions and main results
• A modified operator
• Twisted $$h$$-admissible operators
• Twisted partial differential operators
• Construction of a quasi-invariant subspace
• Decomposition of the evolution for the modified operator
• Proof of Theorem 2.1
• Proof of Corollary 2.6
• Computing the effective Hamiltonian
• Propagation of wave-packets
• Application to polyatomic molecules
• Appendix A. Smooth pseudodifferential calculus with operator-valued symbol
• Appendix B. Propagation of the support
• Appendix C. Two technical lemmas
• Appendix. Bibliography