## Bohr-Sommerfeld quantization rules revisited: The method of positive commutators

J. Math. Sci. Univ. Tokyo
Vol. 25 (2018), No. 2, Page 91-127.

Ifa, Abdelwaheb; Louati, Hanen; Rouleux, Michel
Bohr-Sommerfeld quantization rules revisited: The method of positive commutators
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. It is simplified by using action-angle variables. The interest of this procedure lies in its possible generalization to matrix-valued Hamiltonians, like Bogoliubov-de Gennes Hamiltonian.