The flux norm, Bohr-Sommerfeld Quantization Rules and the scattering problem for h $\Psi$DO's on the real line

TL;DR

This paper revisits the Bohr-Sommerfeld quantization rule for 1-D self-adjoint h-Pseudo-differential operators, simplifying the proof and extending the analysis to the scattering problem within an algebraic and microlocal framework.

Contribution

It simplifies the proof of the Bohr-Sommerfeld quantization rule for 1-D h-Pseudo-differential operators and extends the analysis to the scattering problem using spatial representation.

Findings

BS rule holds when the Gram matrix of WKB solutions is not invertible.

Simplified proof using spatial representation and microlocal Wronskian.

Extended analysis to the scattering problem for h ΨDOs.

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.