Regular Bohr-Sommerfeld quantization rules for a h-pseudo-differential operator: The method of positive commutators

TL;DR

This paper revisits the Bohr-Sommerfeld quantization rule for 1-D pseudo-differential Hamiltonians, establishing its validity through the invertibility of a Gram matrix of WKB solutions within an algebraic and microlocal framework.

Contribution

It provides a new proof of the Bohr-Sommerfeld rule using positive commutator methods and the flux norm in the context of pseudo-differential operators.

Findings

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

The method employs positive commutators within an algebraic and microlocal framework.

The approach clarifies the conditions under which the quantization rule is valid.

Abstract

We revisit in this Note the well known Bohr-Sommerfeld quantization rule (BS) for 1-D Pseudo-differential self-adjoint Hamiltonians within the algebraic and microlocal framework of Helffer and Sj\"ostrand; BS holds precisely when the Gram matrix consisting of scalar products of WKB solutions with respect to the "flux norm" is not invertible.