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Bounds for quasimodes with polynomially narrow bandwidth on surfaces of revolution

Ambre Chabert

TL;DR

This work addresses bounds for spectral projectors P_{λ,δ} on polynomially narrow frequency intervals on compact surfaces of revolution, establishing a sharp L^2→L^∞ bound ∥P_{λ,δ}∥ ≲ λ^{1/2} δ^{1/2} for δ ≥ λ^{-1/32} away from the poles. The authors adapt Sogge’s microlocal approach to the setting of quantum completely integrable (QCI) manifolds, leveraging Colin de Verdière’s construction to reduce the problem to explicit oscillatory integrals bounded through careful phase analysis. The method hinges on expressing P_{λ,δ} via a smoothed joint functional calculus in commuting operators Q_1, Q_2, constructing parametrices (Hörmander, antipodal, and bicharacteristic-length), and applying refined stationary-phase and finite-type degeneracy estimates to those oscillatory integrals. The results provide a polynomial improvement over the generic eigenfunction bounds in this QCI context, illustrate the essential role of the equator and antipodal geometry, and suggest pathways to extend these techniques to broader quantum integrable settings and linked Weyl remainder phenomena.

Abstract

Given a compact surface of revolution with Laplace-beltrami operator $Δ$, we consider the spectral projector $P_{λ,δ}$ on a polynomially narrow frequency interval $[λ-δ,λ+ δ]$, which is associated to the self-adjoint operator $\sqrt{-Δ}$. For a large class of surfaces of revolution, and after excluding small disks around the poles, we prove that the $L^2 \to L^{\infty}$ norm of $P_{λ,δ}$ is of order $λ^{\frac{1}{2}} δ^{\frac{1}{2}}$ up to $δ\geq λ^{-\frac{1}{32}}$. We adapt the microlocal approach introduced by Sogge for the case $δ= 1$, by using the Quantum Completely Integrable structure of surfaces of revolution introduced by Colin de Verdière. This reduces the analysis to a number of estimates of explicit oscillatory integrals, for which we introduce new quantitative tools.

Bounds for quasimodes with polynomially narrow bandwidth on surfaces of revolution

TL;DR

This work addresses bounds for spectral projectors P_{λ,δ} on polynomially narrow frequency intervals on compact surfaces of revolution, establishing a sharp L^2→L^∞ bound ∥P_{λ,δ}∥ ≲ λ^{1/2} δ^{1/2} for δ ≥ λ^{-1/32} away from the poles. The authors adapt Sogge’s microlocal approach to the setting of quantum completely integrable (QCI) manifolds, leveraging Colin de Verdière’s construction to reduce the problem to explicit oscillatory integrals bounded through careful phase analysis. The method hinges on expressing P_{λ,δ} via a smoothed joint functional calculus in commuting operators Q_1, Q_2, constructing parametrices (Hörmander, antipodal, and bicharacteristic-length), and applying refined stationary-phase and finite-type degeneracy estimates to those oscillatory integrals. The results provide a polynomial improvement over the generic eigenfunction bounds in this QCI context, illustrate the essential role of the equator and antipodal geometry, and suggest pathways to extend these techniques to broader quantum integrable settings and linked Weyl remainder phenomena.

Abstract

Given a compact surface of revolution with Laplace-beltrami operator , we consider the spectral projector on a polynomially narrow frequency interval , which is associated to the self-adjoint operator . For a large class of surfaces of revolution, and after excluding small disks around the poles, we prove that the norm of is of order up to . We adapt the microlocal approach introduced by Sogge for the case , by using the Quantum Completely Integrable structure of surfaces of revolution introduced by Colin de Verdière. This reduces the analysis to a number of estimates of explicit oscillatory integrals, for which we introduce new quantitative tools.

Paper Structure

This paper contains 109 sections, 94 theorems, 1051 equations, 33 figures.

Table of Contents

  1. Introduction
  2. Results
  3. Earlier Works
  4. The global and pointwise Weyl laws
  5. The $L^p$ norm of eigenfunctions
  6. Spectral projectors on thin frequency intervals
  7. Earlier results for surfaces of revolution
  8. Method of proof
  9. Acknowledgments
  10. The Classical and Quantum Complete Integrability : a presentation of Colin de Verdière's construction and some developments
  11. Notations, definitions, first results
  12. Notations and a reminder on cotangent bundles
  13. Notations and definitions for surfaces of revolution
  14. The model of ellipsoids of revolution
  15. A simple geometric description of the geodesics
  16. ...and 94 more sections

Key Result

Theorem 1

In the set of simple symmetric smooth surfaces of revolution, there is an open set $\mathfrak{S}$, which contains all the ellipsoids of revolution which are close enough to, but not equal to the round sphere, such that, if $\mathcal{S} \in \mathfrak{S}$, $\varepsilon > 0$, and if $K_{\varepsilon}$ i

Figures (33)

  • Figure 1: Two examples of surfaces of revolution
  • Figure 2: Oblate and oblong ellipsoids of revolution
  • Figure 3: Description of the geodesics by their Clairaut integral
  • Figure 4: The geodesics in action-angle coordinates
  • Figure 5: The curves $\gamma$ and $\gamma_0$
  • ...and 28 more figures

Theorems & Definitions (187)

  • Theorem 1: Main result
  • Corollary 1.1
  • Remark 1.1
  • Theorem 2: Sogge
  • Theorem 3: Colin de Verdière
  • Proposition 1.1
  • Definition 2.1
  • Lemma 2.1
  • Definition 2.2
  • Definition 2.3
  • ...and 177 more