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

Abdelwaheb Ifa, Michel Rouleux

TL;DR

The paper develops a microlocal flux-norm framework for Bohr-Sommerfeld quantization of order $2$ for 1D $h$-pseudodifferential operators, showing BS rules emerge precisely when a Gram matrix built from microlocal WKB quasi-modes is non-invertible. It achieves a streamlined proof in the spatial representation, derives the generalized action through a flux-form analysis up to $\mathcal{O}(h^2)$, and expresses the BS condition as ${\cal S}_{h}(E)=2\pi n h$ with explicit $S_0,S_1,S_2$ components, including the subprincipal and higher corrections. The authors verify the Ansatz in the Schrödinger setting with analytic coefficients by reducing to Airy form and computing the order-$4$ WKB expansion, thereby establishing the phase/mode matching and connection formulas. They further extend the flux-norm method to semi-classical scattering over a compact barrier, deriving a monodromy matrix in $SU(1,1)$ and demonstrating flux conservation, while outlining how these ideas may extend to $2\times2$ systems and higher dimensions. Overall, the work provides a robust, representation-based route to BS quantization and scattering that complements traditional Weyl- and functional-calculus approaches and offers a pathway to more intricate microlocal systems.

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.

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

TL;DR

The paper develops a microlocal flux-norm framework for Bohr-Sommerfeld quantization of order for 1D -pseudodifferential operators, showing BS rules emerge precisely when a Gram matrix built from microlocal WKB quasi-modes is non-invertible. It achieves a streamlined proof in the spatial representation, derives the generalized action through a flux-form analysis up to , and expresses the BS condition as with explicit components, including the subprincipal and higher corrections. The authors verify the Ansatz in the Schrödinger setting with analytic coefficients by reducing to Airy form and computing the order- WKB expansion, thereby establishing the phase/mode matching and connection formulas. They further extend the flux-norm method to semi-classical scattering over a compact barrier, deriving a monodromy matrix in and demonstrating flux conservation, while outlining how these ideas may extend to systems and higher dimensions. Overall, the work provides a robust, representation-based route to BS quantization and scattering that complements traditional Weyl- and functional-calculus approaches and offers a pathway to more intricate microlocal systems.

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.
Paper Structure (10 sections, 5 theorems, 140 equations)

This paper contains 10 sections, 5 theorems, 140 equations.

Table of Contents

  1. Introduction
  2. Quasi-modes and BS mod $\mathcal{O}(h^{2})$ in the spatial representation
  3. WKB solutions mod $\mathcal{O}(h^{2})$ in the spatial representation
  4. Well normalized quasi-modes mod $\mathcal{O}(h^{2})$ in the spatial representation
  5. The homology class of the generalized action
  6. Bohr-Sommerfeld quantization rule
  7. Checking the Ansatz in the case of Schrödinger operator with analytic coefficients
  8. Reduction to Airy equation
  9. WKB solution of order 4
  10. Scattering over a barrier: case of a compactly supported smooth potential

Key Result

Theorem 1.1

Let $P(x,hD_x;h)$ be as in (1.3). With the notations and hypotheses stated above, for $h>0$ small enough there exists a smooth function ${\cal S}_{h}: I\to \mathbb{R}$, called the semi-classical action, with asymptotic expansion such that $E\in I$ is an eigenvalue of $P$ iff it satisfies the implicit equation (Bohr-Sommerfeld quantization condition) ${\cal S}_{h}(E)=2\pi nh$, $n\in \mathbb{Z}$. T

Theorems & Definitions (5)

  • Theorem 1.1
  • Lemma 2.1
  • Lemma 2.2
  • Proposition 2.1
  • Theorem 3.1