Bohr-Sommerfeld Quantization Rules Revisited: The Method of Positive Commutators

Abdelwaheb Ifa (1) , Hanen Louati (2, 3) , Michel L. Rouleux (2, 3)

1 UTM - Université de Tunis El Manar
2 CPT - Centre de Physique Théorique - UMR 7332
3 CPT - E8 Dynamique quantique et analyse spectrale

We revisit the well known Bohr-Sommerfeld quantization rule (BS) of order 2 for a self-adjoint 1-D h-Pseudo-differential operator within the algebraic and microlocal framework of Helffer and Sjöstrand; BS holds precisely when the Gram matrix consisting of scalar products of some WKB solutions with respect to the " flux norm " is not invertible. The interest of this procedure lies in its possible generalization to matrix-valued Hamiltonians, like Bogoliubov-de Gennes Hamiltonian. It is simplified in the scalar case by using action-angle variables.

Domaines: Mathématiques [math]; Physique mathématique [math-ph]

Abdelwaheb Ifa, Hanen Louati, Michel L. Rouleux. Bohr-Sommerfeld Quantization Rules Revisited: The Method of Positive Commutators. Journal of Mathematical Sciences the University of Tokyo, In press, J. Math. Sci. Univ. Tokyo, 25, p.91-137. ⟨hal-01783488⟩