Bohr-Sommerfeld quantization conditions for Schrodinger operator: the Method of Microlocal Wronskian and Gram Matrix

In this work, we present a new formulation of the well known Bohr-Sommerfeld quantization rule (BS) of order 2 for a Schrodinger operator within the algebraic and microlocal framework of B. Helffer and J. Sjostrand; BS holds precisely when the Gram matrix consisting of scalar products of some WKB solutions with respect to the "flux norm" is not invertible. This condition is obtained using the microlocal Wronskian and does not rely on traditional matching techniques. It is simplified by using action-angle variables. The interest of this procedure lies in its possible generalization to matrixvalued Hamiltonians, like Bogoliubov-de Gennes Hamiltonian.