Soumia Belmouhoub, Bekkai Messirdi, Tahar Bouguetaia

Semiclassical Resonances, Theory and Application to a General Diatomic Molecular Hamiltonian

We study in this paper resonances of Schrödinger operators. Resonance energies are accessible from a general class of complex distortions, they also coincide with the poles of the meromorphic continuation of the resolvent. We prove that in the Born--Oppenheimer approximation for diatomic molecules, this study can be reduced to the one of a matrix of semiclassical pseudodifferential operators with operator-valued symbols, without modifying the Hamiltonian near the collision set of nuclei. We consider here the case where two electronic levels cross, and where molecular resonances appear and can be well located. We also investigate the action of the effective Hamiltonian on WKB solutions and show that these resonances have an imaginary part exponentially small.

Mathematics Subject Classification: 35P15, 47A75, 47G30, 35S30, 35C20

Key words and phrases: Analytic dilation, analytic distorsion, resonances, Born--Oppenheimer approximation, effective Hamiltonian, pseudodifferential operators, width