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Microlocal propagation near radial points and scattering for symbolic potentials of order zero

Andrew Hassell, Richard Melrose, Andras Vasy

TL;DR

This paper develops the scattering and spectral theory for $H=\Delta_g+V$ on a compact manifold with boundary equipped with a scattering metric $g$, with $V\in C^\infty(X)$ real, extending previous results from two dimensions to a broader class that includes perturbations of the Euclidean Laplacian by potentials homogeneous of degree zero near infinity. The approach is a microlocal analysis at radial points: reducing the bicharacteristic flow to a model normal form using scattering Fourier integral operators, constructing test modules and forward microlocalizers with positive-commutator estimates, and classifying energies into nonresonant, effectively nonresonant, and effectively resonant regimes. The paper yields a microlocal Morse decomposition of global eigenfunctions, an interval-uniform asymptotic completeness framework, a unitary map $M_+(\,\sigma\,)$ relating the resolvent to outgoing data, and a precise description of the $S$-matrix and Poisson operators, tying local radial data to the global spectral picture. Overall, the work provides a unified microlocal framework for propagation near radial points in geometric scattering with order-zero potentials, enabling rigorous control of resonant phenomena, asymptotic completeness, and the link between local radial analysis and global scattering data.

Abstract

In this paper, the scattering and spectral theory of $H=\Delta_g+V$ is developed, where $\Delta_g$ is the Laplacian with respect to a scattering metric $g$ on a compact manifold $X$ with boundary and $V\in C^\infty(X)$ is real; this extends our earlier results in the two-dimensional case. Included in this class of operators are perturbations of the Laplacian on Euclidean space by potentials homogeneous of degree zero near infinity. Much of the particular structure of geometric scattering theory can be traced to the occurrence of radial points for the underlying classical system; a general framework for microlocal analysis at these points forms the main part of the paper.

Microlocal propagation near radial points and scattering for symbolic potentials of order zero

TL;DR

This paper develops the scattering and spectral theory for on a compact manifold with boundary equipped with a scattering metric , with real, extending previous results from two dimensions to a broader class that includes perturbations of the Euclidean Laplacian by potentials homogeneous of degree zero near infinity. The approach is a microlocal analysis at radial points: reducing the bicharacteristic flow to a model normal form using scattering Fourier integral operators, constructing test modules and forward microlocalizers with positive-commutator estimates, and classifying energies into nonresonant, effectively nonresonant, and effectively resonant regimes. The paper yields a microlocal Morse decomposition of global eigenfunctions, an interval-uniform asymptotic completeness framework, a unitary map relating the resolvent to outgoing data, and a precise description of the -matrix and Poisson operators, tying local radial data to the global spectral picture. Overall, the work provides a unified microlocal framework for propagation near radial points in geometric scattering with order-zero potentials, enabling rigorous control of resonant phenomena, asymptotic completeness, and the link between local radial analysis and global scattering data.

Abstract

In this paper, the scattering and spectral theory of is developed, where is the Laplacian with respect to a scattering metric on a compact manifold with boundary and is real; this extends our earlier results in the two-dimensional case. Included in this class of operators are perturbations of the Laplacian on Euclidean space by potentials homogeneous of degree zero near infinity. Much of the particular structure of geometric scattering theory can be traced to the occurrence of radial points for the underlying classical system; a general framework for microlocal analysis at these points forms the main part of the paper.

Paper Structure

This paper contains 25 sections, 44 theorems, 287 equations.

Table of Contents

  1. Introduction
  2. Previous results
  3. Results and structure of the paper
  4. Results used from HMV1
  5. Notation
  6. Radial points
  7. Microlocal normal form
  8. Scattering Fourier integral operators
  9. Normal form
  10. Effective resonance and nonresonance
  11. Parameter-dependent normal form
  12. Microlocal solutions
  13. Test Modules
  14. Effectively nonresonant operators
  15. Effectively resonant operators
  16. ...and 10 more sections

Key Result

Lemma 2.3

At a non-degenerate radial point for p, where dp=\lambda \alpha, the linearization A' acting on T_{q}\Sigma is such that is in the Lie algebra of the symplectic group with respect to \omega _{q}:

Theorems & Definitions (123)

  • Definition 2.1
  • Definition 2.2
  • Lemma 2.3
  • proof
  • Definition 2.4
  • Lemma 2.5
  • Remark 2.6
  • proof
  • Lemma 2.7
  • Remark 2.8
  • ...and 113 more