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Analytic Microlocal Bohr-Sommerfeld Expansions

Antide Duraffour

TL;DR

The paper develops an exponentially sharp Bohr-Sommerfeld quantization for self-adjoint Weyl-quantized pseudodifferential operators on $L^2(\mathbb{R})$ with holomorphic symbols in a regime where the energy sets are regular and discrete. By conjugating the operator with the Bargmann transform to a complex-analytic pseudodifferential framework, it constructs analytic WKB quasimodes around complex energy curves and then patches them globally using a $\overline{\partial}$-lemma, yielding spectral approximations with residuals of order $O(e^{-\mathscr{C}/\hbar})$. A central outcome is the Maslov correction, shown to contribute a $\pi\hbar$ shift in the action, producing a refined quantization condition $\mathcal{A}(\lambda)+\pi\hbar+O(\hbar^2)\in 2\pi\hbar\mathbb{Z}$ and ensuring simple eigenvalues. The approach unifies holomorphic-symbol analysis, Bargmann-space techniques, and microlocal WKB theory to produce exponentially sharp spectral descriptions and provides a pathway to multi-component energy sets via patching. The results advance a rigorous, exponentially precise spectral theory beyond classical Schrödinger settings and have potential implications for quantum tunneling analyses in broader 1D Hamiltonian contexts.

Abstract

This article is devoted to Gevrey-analytic estimates in ___, of the Bohr-Sommerfeld expansion of the eigenvalues of self-adjoint pseudo-differential operators acting on L^2(R) in the regular case. We consider an interval of energies in which the spectrum of P is discrete and such that the energy sets are regular connected curves. Under some assumptions on the holomorphy of the symbol p, we will use the isometry between L^2(R) and the Bargmann space to obtain an exponentially sharp description of the spectrum in the energy window . More precisely it is possible to build exponentially sharp WKB quasimodes in the Bargmann space. A precise examination of the principal symbols will provide an interpretation to the Maslov correction $π$___ in the Bargmann space.

Analytic Microlocal Bohr-Sommerfeld Expansions

TL;DR

The paper develops an exponentially sharp Bohr-Sommerfeld quantization for self-adjoint Weyl-quantized pseudodifferential operators on with holomorphic symbols in a regime where the energy sets are regular and discrete. By conjugating the operator with the Bargmann transform to a complex-analytic pseudodifferential framework, it constructs analytic WKB quasimodes around complex energy curves and then patches them globally using a -lemma, yielding spectral approximations with residuals of order . A central outcome is the Maslov correction, shown to contribute a shift in the action, producing a refined quantization condition and ensuring simple eigenvalues. The approach unifies holomorphic-symbol analysis, Bargmann-space techniques, and microlocal WKB theory to produce exponentially sharp spectral descriptions and provides a pathway to multi-component energy sets via patching. The results advance a rigorous, exponentially precise spectral theory beyond classical Schrödinger settings and have potential implications for quantum tunneling analyses in broader 1D Hamiltonian contexts.

Abstract

This article is devoted to Gevrey-analytic estimates in ___, of the Bohr-Sommerfeld expansion of the eigenvalues of self-adjoint pseudo-differential operators acting on L^2(R) in the regular case. We consider an interval of energies in which the spectrum of P is discrete and such that the energy sets are regular connected curves. Under some assumptions on the holomorphy of the symbol p, we will use the isometry between L^2(R) and the Bargmann space to obtain an exponentially sharp description of the spectrum in the energy window . More precisely it is possible to build exponentially sharp WKB quasimodes in the Bargmann space. A precise examination of the principal symbols will provide an interpretation to the Maslov correction ___ in the Bargmann space.
Paper Structure (32 sections, 39 theorems, 182 equations, 2 figures)

This paper contains 32 sections, 39 theorems, 182 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Mathematical Formulation of the Problem
  3. Assumptions on the symbol
  4. Statement of the Main Theorem
  5. Scheme of the Proof
  6. Bargmann Transform and complex pseudo-differential operators
  7. The Bargmann Transform
  8. Complex pseudo-differential Operators
  9. Analysis of the geometry
  10. Action-Angle Lemma in the $L^2(\mathbb{R})$ setting
  11. Holomorphy of the action and eikonal equations
  12. Index of the energy sets
  13. Construction of global WKB quasimodes
  14. Normal form in the smooth setting
  15. Microlocal WKB Expansions
  16. ...and 17 more sections

Key Result

Theorem 1.4

There exist a classical analytic symbol $\lambda(\hbar,I) = A^{-1}(I-\pi \hbar) + \mathcal{O}(\hbar^2)$ defined for $I$ in a small neighbourhood of $A([E_1,E_2])$, a constant $\mathscr{C} > 0$ and a function $\mathcal{O}(e^{-\frac{\mathscr{C}}{\hbar}})$ such that, for $\hbar$ small enough and the eigenvalues of $P$ in $(E_1,E_2)$ are simple.

Figures (2)

  • Figure 1: Illustration of the action-angle change of variables.
  • Figure 2: Illustration of Patching.

Theorems & Definitions (86)

  • Definition 1.1: Tubular Neighbourhood
  • Remark 1.2
  • Remark 1.3
  • Theorem 1.4
  • Theorem 2.1: Real Weyl Quantization on $H_{\Phi_0}(\mathbb{C})$
  • Lemma 2.2: Uncertainty Principle
  • proof
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • ...and 76 more