Regular Bohr-Sommerfeld quantization rules for a h-pseudo-differential   operator: The method of positive commutators
  We revisit in this Note the well known Bohr-Sommerfeld quantization rule (BS) for 1-D Pseudo-differential self-adjoint Hamiltonians within the algebraic and microlocal framework of Helffer and Sj\"ostrand; BS holds precisely when the Gram matrix consisting of scalar products of WKB solutions with respect to the "flux norm" is not invertible. 
Abdelwaheb Ifa, Michel Rouleux
math-ph math.MP