Resonances of a two-state semiclassical Schrödinger Hamiltonians

Abstract

The goal of this article is to generalize some travels made in (Helffer, B. and Sjöstrand, J., 1986, Résonances en limite semi-classique. Mémoires de la Société Mathématique de France Sér. 2; Hunziker, W., 1986, Distorsion analyticity, and molecular resonance curves. Ann. I. H. P. (section Physique Théorique), 45, 339–358.; Messirdi, B., 1994, Asymptotique de Born-Oppenheimer pour la prédissociation moléculaire (cas de potentiels réguliers). Annales de l'IHP, Section Physique Théorique, 61(3), 255–292.; Messirdi, B., 1993, Asymptotique de Born-Oppenheimer pour la prédissociation moléculaire. Thèse de Doctorat de l'Université de Paris 13) to multidimensional two-state perturbed and semiclassical systems. More precisely, we study the Schrödinger Hamiltonians on where V is flat at infinity, R is a bounded symmetric differential operator of first order and h tends to 0+, in the case where resonances appear. Using a microlocal estimates on resolvents of P(h) and the so-called Grushin problem the spectral study of the distorded operator of P(h) is reduced to the resolution of an analytic algebraic equation. Under the assumptions that V(x) admits a relatively compact well U and the Dirichlet realization of P(h) near U has a simple eigenvalue, it is then showed that P(h) has a unique resonance. We obtain also that the width of this resonance is exponentially small as h tends to zero.