Nawel Latigui, Kaoutar Ghomari, Bekkai Messirdi

Theoretical and Numerical Results on Birkhoff Normal Forms and Resonances in the Born--Oppenheimer Approximation

abstract:
This paper mainly focuses on the Birkhoff normal form theorem for the Born-Oppenheimer Hamiltonians. Normal forms are accessible via those of the effective Hamiltonian obtained by the Grushin reduction method and the pseudodifferential calculus with operator-valued symbols. Resonance situations are discussed; the theoretical computations of Birkhoff normal form in the 1:1 resonance are written explicitly. Our approach gives compatible numerical results while using a computer program.

Mathematics Subject Classification: 58J40, 58K50, 47A55

Key words and phrases: Birkhoff normal form, Born-Oppenheimer approximation, effective Hamiltonian, resonance