The flux norm, Bohr-Sommerfeld Quantization Rules and the scattering problem for h $\Psi$DO's on the real line
Mathematical Physics
2026-01-13 v3 math.MP
Abstract
We revisit the well known Bohr-Sommerfeld quantization rule (BS) of order 2 for a self-adjoint 1-D h-Pseudo-differential operator within the algebraic and microlocal framework of Helffer and Sjoestrand; BS holds precisely when Gram matrix consisting of scalar products of some WKB solutions with respect to the ``flux norm'' (or microlocal Wronskian) is not invertible. We simplify somewhat our previous proof [A. Ifa H. Louati and M. Rouleux. Bohr-Sommerfeld Quantization Rules Revisited: the Method of Positive Commutators. J. Math. Sci. Univ. Tokyo, 25(2):2018] by working in spatial representation only, as in complex WKB theory for Schroedinger operator. We consider also the scattering problem.
Keywords
Cite
@article{arxiv.2407.18893,
title = {The flux norm, Bohr-Sommerfeld Quantization Rules and the scattering problem for h $\Psi$DO's on the real line},
author = {Abdelwaheb Ifa and Michel Rouleux},
journal= {arXiv preprint arXiv:2407.18893},
year = {2026}
}