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High-energy eigenfunctions of point perturbations of the Laplacian

Santiago Verdasco

TL;DR

The paper studies high-frequency eigenfunctions of the Laplacian perturbed by a finite set of point scatterers on a compact manifold, modeling these as self-adjoint extensions $\Delta_L$ parameterized by a Lagrangian subspace. It builds a quasimode theory from Green's functions and leverages diagonal and off-diagonal Weyl laws to control the perturbations, enabling precise statements about semiclassical defect measures. The main contribution is showing that, under a zero-looping-directions condition at the perturbation points, the defect measures are supported on $S^*M$ and invariant under the geodesic flow, revealing robust semiclassical behavior despite singular perturbations. This extends quantum-classical correspondence results to singular perturbations and clarifies how point scatterers influence high-frequency eigenfunction concentration and dynamics.

Abstract

In this paper, we explore the high-frequency properties of eigenfunctions of point perturbations of the Laplacian on a compact Riemannian manifold. More specifically, we are interested in understanding to what extent the high-frequency behavior of eigenfunctions depends on the global dynamics of the geodesic flow in the manifold, as is the case when smooth perturbations are present. Our main result proves that as soon as the Laplacian is perturbed by a finite family of Dirac masses placed at points whose set of looping directions has zero measure, semiclassical measures corresponding to high-frequency sequences of eigenfunctions are supported on the unit cosphere bundle and invariant under the geodesic flow. The main difficulty in establishing this result relies on the fact that point perturbations are unbounded operators that cannot be written as pseudodifferential operators, and therefore, the corresponding perturbed Laplacian does not have an unambiguously defined classical flow.

High-energy eigenfunctions of point perturbations of the Laplacian

TL;DR

The paper studies high-frequency eigenfunctions of the Laplacian perturbed by a finite set of point scatterers on a compact manifold, modeling these as self-adjoint extensions parameterized by a Lagrangian subspace. It builds a quasimode theory from Green's functions and leverages diagonal and off-diagonal Weyl laws to control the perturbations, enabling precise statements about semiclassical defect measures. The main contribution is showing that, under a zero-looping-directions condition at the perturbation points, the defect measures are supported on and invariant under the geodesic flow, revealing robust semiclassical behavior despite singular perturbations. This extends quantum-classical correspondence results to singular perturbations and clarifies how point scatterers influence high-frequency eigenfunction concentration and dynamics.

Abstract

In this paper, we explore the high-frequency properties of eigenfunctions of point perturbations of the Laplacian on a compact Riemannian manifold. More specifically, we are interested in understanding to what extent the high-frequency behavior of eigenfunctions depends on the global dynamics of the geodesic flow in the manifold, as is the case when smooth perturbations are present. Our main result proves that as soon as the Laplacian is perturbed by a finite family of Dirac masses placed at points whose set of looping directions has zero measure, semiclassical measures corresponding to high-frequency sequences of eigenfunctions are supported on the unit cosphere bundle and invariant under the geodesic flow. The main difficulty in establishing this result relies on the fact that point perturbations are unbounded operators that cannot be written as pseudodifferential operators, and therefore, the corresponding perturbed Laplacian does not have an unambiguously defined classical flow.
Paper Structure (10 sections, 22 theorems, 124 equations)

This paper contains 10 sections, 22 theorems, 124 equations.

Table of Contents

  1. Introduction
  2. Point-perturbations of Laplacian
  3. Description of point-perturbations
  4. Spectrum of A*
  5. Description of point-perturbations of Laplacian through Hermitian matrices
  6. Spectrum of Lap L
  7. Approximation of new eigenfunctions by quasimodes
  8. Proof of Theorem Gh L2norm
  9. Proof of Theorem Gh L2norm improvement
  10. Properties of semiclassical defect measures

Key Result

Theorem 1

Let $(M,g)$ be a closed Riemannian manifold of dimension $2$ or $3$ with positive Laplacian $\mathop{\mathrm{\Delta}}\nolimits$. Let $(\mathop{\mathrm{\Delta}}\nolimits_{L_n})_{n \in \mathbb{N}}$ be a sequence of point-perturbations of $\mathop{\mathrm{\Delta}}\nolimits$ on a fixed finite set $Q \su for some sequence $(\lambda_n)_{n \in \mathbb{N}}$ tending to infinity. Any semiclassical defect me

Theorems & Definitions (40)

  • Theorem 1
  • Remark 2
  • Corollary 3
  • Theorem 4
  • Lemma 5
  • proof
  • Lemma 6
  • proof
  • Proposition 7
  • proof
  • ...and 30 more