Bohr-Sommerfeld Quantization Rules for 1-D Semiclassical Pseudo-Differential Operator: the Method of Microlocal Wronskian and Gram Matrix

TL;DR

This paper refines the Bohr-Sommerfeld quantization rule for 1-D semiclassical operators using microlocal analysis, introducing a Gram matrix approach that simplifies the process and potentially extends to matrix Hamiltonians.

Contribution

It presents a microlocal Wronskian and Gram matrix method to establish the Bohr-Sommerfeld rule without traditional matching, enabling generalization to matrix-valued Hamiltonians.

Findings

Gram matrix non-invertibility characterizes BS rule

Method avoids traditional matching techniques

Potential extension to matrix Hamiltonians

Abstract

In this paper, we revisit the well known Bohr-Sommerfeld quantization rule (BS) of order 2 for a self-adjoint 1-D semiclassical pseudo-differential operator, within the algebraic and microlocal framework of B. Helffer and J. Sj\"{o}strand. BS holds precisely when the Gram matrix consisting of scalar products of certain 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 BdG Hamiltonian.