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Eigenvalues of non self-adjoint Toeplitz operators near an elliptic critical value with analytic regularity

Nathan Réguer

Abstract

In this article, we determine the spectrum of real-analytic, non self-adjoint Toeplitz operators on compact K{ä}hler manifolds and on the complex plane, on neighbourhoods of critical values of the symbol. We consider specifically critical values of the symbol on which its Hessian is elliptic and we get asymptotic expansion on eigenvalues in a neighbourhood with quantisation conditions similar to Bohr-Sommerfeld. To do so, we recall and further develop analytic semiclassical tools, in particular the symbolic calculus of complex Fourier integral operators using contour deformation. We detail the well known case of operators with quadratic symbols, and we treat a general case through normal form reduction. Finally, we prove resolvent estimates on norms with weights that come from the non-real part of the symbol.

Eigenvalues of non self-adjoint Toeplitz operators near an elliptic critical value with analytic regularity

Abstract

In this article, we determine the spectrum of real-analytic, non self-adjoint Toeplitz operators on compact K{ä}hler manifolds and on the complex plane, on neighbourhoods of critical values of the symbol. We consider specifically critical values of the symbol on which its Hessian is elliptic and we get asymptotic expansion on eigenvalues in a neighbourhood with quantisation conditions similar to Bohr-Sommerfeld. To do so, we recall and further develop analytic semiclassical tools, in particular the symbolic calculus of complex Fourier integral operators using contour deformation. We detail the well known case of operators with quadratic symbols, and we treat a general case through normal form reduction. Finally, we prove resolvent estimates on norms with weights that come from the non-real part of the symbol.
Paper Structure (19 sections, 56 theorems, 391 equations, 4 figures)

This paper contains 19 sections, 56 theorems, 391 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Quadratic symbol
  3. Normal form
  4. Quantisation formula
  5. Analytic semiclassical tools
  6. Analytic pseudo spectrum
  7. Complex phases Fourier Integral Operators
  8. Symbolic calculus of Toeplitz operators
  9. Local study
  10. Normal form
  11. Condition on the Hessian
  12. Moser's trick
  13. Action integral
  14. Spectral description and resolvent estimate
  15. Multi-well symbol
  16. ...and 4 more sections

Key Result

Theorem 1.1

Let $M$ be either a compact Kähler manifold of complex dimension $1$ or $\mathbb{C}$, $\mathfrak{e}\in \mathbb{C}$, and $f$ a real-analytic function on $M$. We consider Hypothesis hyp_global to be satisfied. It means that $f$ is in a class of analytic symbols, and writing $f_0$ its principal symbol, for all $n,l\in\mathbb{N}^2$ and for $\hbar$ small enough. Moreover,there exists $\rho,C>0$ with $B

Figures (4)

  • Figure 1: On the left: spectrum of $H$, the yellow balls are the sets where the norm of the resolvent is greater than $\hbar^{-\infty}$. On the right: Spectrum of $H_{\theta}$, the yellow balls are now the $c$-analytic pseudo-spectrum, and the grey area is the classical range.
  • Figure 2: Spectrum of $T_N(f)$: the yellow balls form $\sigma_{c'}(T_N(f))$. For $r\neq s$ a ball at level $j=r$ is disjoint from the balls at level $s$, but on the same level a connected component can consist of multiple balls.
  • Figure 3: Examples of contours, for $F(x,y) = -\left( y-y_0(x)\right)^2$ and $y_0(x)\neq 0$.
  • Figure 4: Steps 1 and 2 of the proof of Proposition \ref{['prop_linear_contours']}.

Theorems & Definitions (123)

  • Theorem 1.1
  • Lemma 2.1
  • proof
  • Proposition 2.2: prav07 Proposition 2.1.1
  • Lemma 2.3
  • proof
  • Theorem 2.4
  • proof
  • Definition 3.1
  • Lemma 3.2: tref05 Theorem 4.3
  • ...and 113 more