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$L^2$ restriction bounds for analytic continuations of quantum ergodic Laplace eigenfunctions

John A. Toth, Xiao Xiao

TL;DR

The paper develops a 2-microlocal quantum ergodic restriction (2MQER) theory for analytic continuations of QE Laplace eigenfunctions on Grauert tubes by analyzing the FBI transform $T_{hol}(h)$. It proves that the restricted data $T_{Σ}u_h$ satisfies a semiclassical asymptotic formula with a density $q$ on $Σ$, captured by a pseudodifferential operator ${ olinebreak[4]{ ext{P}}}_{Σ,a}(h)$, and identifies the 2-microlocal mass distribution through the integral over $S^*M∩Σ$. The work then derives sharp $L^2$-restriction bounds for $igl\|T_{Σ}u_higr\|_{L^2(Σ)}$, including a universal upper bound $igo(h^{-1/2})$, a positive lower bound, and improved upper bounds under transversality-type geometric conditions, with weighted $L^2$ estimates for the complexified eigenfunctions. Collectively, these results advance understanding of how QE eigenfunctions behave under analytic continuation and restriction, providing tools for spectral geometry and nodal analysis in the analytic setting.

Abstract

We prove a quantum ergodic restriction (QER) theorem for real hypersurfaces $Σ\subset X,$ where $X$ is the Grauert tube associated with a real-analytic, compact Riemannian manifold. As an application, we obtain $h$ independent upper and lower bounds for the $L^2$ - restrictions of the FBI transform of Laplace eigenfunctions restricted to $Σ$ satisfying certain generic geometric conditions.

$L^2$ restriction bounds for analytic continuations of quantum ergodic Laplace eigenfunctions

TL;DR

The paper develops a 2-microlocal quantum ergodic restriction (2MQER) theory for analytic continuations of QE Laplace eigenfunctions on Grauert tubes by analyzing the FBI transform . It proves that the restricted data satisfies a semiclassical asymptotic formula with a density on , captured by a pseudodifferential operator , and identifies the 2-microlocal mass distribution through the integral over . The work then derives sharp -restriction bounds for , including a universal upper bound , a positive lower bound, and improved upper bounds under transversality-type geometric conditions, with weighted estimates for the complexified eigenfunctions. Collectively, these results advance understanding of how QE eigenfunctions behave under analytic continuation and restriction, providing tools for spectral geometry and nodal analysis in the analytic setting.

Abstract

We prove a quantum ergodic restriction (QER) theorem for real hypersurfaces where is the Grauert tube associated with a real-analytic, compact Riemannian manifold. As an application, we obtain independent upper and lower bounds for the - restrictions of the FBI transform of Laplace eigenfunctions restricted to satisfying certain generic geometric conditions.

Paper Structure

This paper contains 16 sections, 11 theorems, 176 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Plan of the paper
  3. Some Background
  4. The Cauchy-Riemann equation
  5. Complexified heat kernel and the FBI transform
  6. Grauert tubes and analytic $h$-pseudodifferential calculus
  7. Fermi coordinates near a hypersurface $\Sigma \subset B_{\tau}^*M$
  8. FBI transform
  9. Complexified heat operator on closed, compact manifolds
  10. Localization of the semiclassical wave front of $T_{\Sigma} u_h$
  11. Semiclassical wavefront
  12. Sharpness of (\ref{['wf1']})
  13. The wavefront set $WF_h(T_{\Sigma} u)$
  14. 2MQER: Proof of Theorem \ref{['thm1']}
  15. $L^2$-restriction bounds for $\| T_{\Sigma} u_h \|_{L^2(\Sigma)}$: proof of Theorem \ref{['bounds']}
  16. ...and 1 more sections

Key Result

Theorem 1

Let $(M,g)$ be a compact $C^{\omega}$ Riemannian manifold with Grauert tube $M_{\tau}^{{\mathbb C}},$$\Sigma \subset M_{\tau}^{{\mathbb C}}$ a compact, separating hypersurface and $\{ u_h \}$ any sequence of $L^2$-normalized QE Laplace eigenfunctions on $M.$ Then, for any $a \in C^{\infty}(\Sigma)$,

Figures (2)

  • Figure 1: Blue: $\{f\geq 0\}$. Gray: Admissible region of $(\theta,\phi)$.
  • Figure 2: $\phi$ attains its max and min when $J\nu\in \text{span} (\nu,\nabla\rho)$.

Theorems & Definitions (26)

  • Theorem 1
  • Definition 1
  • Theorem 2
  • Theorem 3
  • Proposition 4
  • proof
  • Example
  • Proposition 5
  • proof
  • Remark 1
  • ...and 16 more