Bohr-Sommerfeld Quantization Rules for 1-D Semiclassical Pseudo-Differential Operator: the Method of Microlocal Wronskian and Gram Matrix
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.
Keywords
Cite
@article{arxiv.2508.05586,
title = {Bohr-Sommerfeld Quantization Rules for 1-D Semiclassical Pseudo-Differential Operator: the Method of Microlocal Wronskian and Gram Matrix},
author = {Abdelwaheb Ifa},
journal= {arXiv preprint arXiv:2508.05586},
year = {2025}
}
Comments
32 pages, no figures. Theoretical article on second-order Bohr-Sommerfeld quantization for 1-D h-PDOs using microlocal Wronskian and Gram matrix methods. arXiv admin note: text overlap with arXiv:1702.06758