REGULAR BOHR-SOMMERFELD QUANTIZATION RULES FOR A h-PSEUDO-DIFFERENTIAL OPERATOR : THE METHOD OF POSITIVE COMMUTATORS

摘要

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 Sjöstrand ; BS holds precisely when the Gram matrix consisting of scalar products of WKB solutions with respect to the " flux norm " is not invertible.