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Regular Bohr-Sommerfeld rules for non-self-adjoint Berezin--Toeplitz operators and complex Lagrangian states

Alix Deleporte, Yohann Le Floch

Abstract

We describe the eigenvalues and eigenvectors of real-analytic, non-self-adjoint Berezin--Toeplitz operators, up to exponentially small error, on complex one-dimensional compact manifolds, under the hypothesis of regularity of the energy levels. These results form a complex version of the Bohr-Sommerfeld quantization conditions; they hold under a hypothesis that the skew-adjoint part is small but can be of principal order with respect to the semiclassical parameter. To this end, we develop a calculus of Fourier Integral Operators and Lagrangian states associated with complex Lagrangians; these tools are of independent interest.

Regular Bohr-Sommerfeld rules for non-self-adjoint Berezin--Toeplitz operators and complex Lagrangian states

Abstract

We describe the eigenvalues and eigenvectors of real-analytic, non-self-adjoint Berezin--Toeplitz operators, up to exponentially small error, on complex one-dimensional compact manifolds, under the hypothesis of regularity of the energy levels. These results form a complex version of the Bohr-Sommerfeld quantization conditions; they hold under a hypothesis that the skew-adjoint part is small but can be of principal order with respect to the semiclassical parameter. To this end, we develop a calculus of Fourier Integral Operators and Lagrangian states associated with complex Lagrangians; these tools are of independent interest.

Paper Structure

This paper contains 21 sections, 36 theorems, 219 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Prequantum line bundles and their holomorphic sections
  3. Non-self-adjoint spectral asymptotics
  4. Complex semiclassical analysis
  5. Acknowledgements
  6. Berezin--Toeplitz quantization in real-analytic regularity
  7. Analytic symbol classes
  8. Asymptotics of the Bergman kernel and covariant Berezin--Toeplitz operators
  9. Complex Lagrangian states and Fourier Integral Operators
  10. Holomorphic extensions
  11. Lagrangian states
  12. Calculus of Fourier Integral Operators and Lagrangian sections
  13. Fourier Integral Operators in practice
  14. Local model
  15. Normal forms
  16. ...and 6 more sections

Key Result

Theorem 1

Let $(M,J,\omega)$ be a quantizable, compact, real-analytic Kähler manifold of complex dimension 1 and let $L\to M$ be a prequantum line bundle over $M$. Let $p:\mathbb{C}\times M\to \mathbb{C}$ be a real-analytic map, holomorphic in the first variable, and such that $p_0:x\mapsto p(0,x)$ is real-va such that the difference between one element of the spectrum and the corresponding Bohr-Sommerfeld

Figures (3)

  • Figure 1: Zeroth order approximation: the spectrum of the operator $T_k^{\text{cov}}(\varepsilon)$ from Equation \ref{['eq:Tk_S2']} (blue diamonds) and the approximate eigenvalues given by the solutions of Equation \ref{['eq:BS_princ_S2']} (red crosses) for $\varepsilon = 0.2$ at $k=20$ (above) and $k=100$ (below).
  • Figure 2: Comparison between the spectrum of the operator $S_k(\varepsilon)$ from Equation \ref{['eq:Sk_S2']} (blue diamonds) and the approximate eigenvalues given by the solutions of Equation \ref{['eq:BS_half_S2']} (red crosses) for $\varepsilon = 0.2$ at $k=20$ (above) and $k=100$ (below).
  • Figure 3: Zoom on a few eigenvalues in the $k=100$ plots displayed in Figures \ref{['fig:spectre_zix2_eps02']} (top) and \ref{['fig:spectre_zix2_eps02_half']} (bottom). Recall that for exposition reasons, the top figure displays eigenvalues of $T_k(\varepsilon)$ while the bottom one displays eigenvalues of $S_k(\varepsilon)$, but the important information here is the difference in the precision of the approximation thanks to the subprincipal correction.

Theorems & Definitions (79)

  • Theorem 1
  • Definition 2.1
  • Definition 2.2
  • Proposition 2.3
  • Proposition 2.4
  • Lemma 3.1
  • proof
  • Corollary 3.2
  • Lemma 3.3
  • proof
  • ...and 69 more