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Resolvent at low energy and Riesz transform for Schroedinger operators on asymptotically conic manifolds. II

Colin Guillarmou, Andrew Hassell

TL;DR

This work advances the low-energy resolvent analysis for Schrödinger operators on asymptotically conic manifolds by accommodating zero modes and zero-resonances. Through a refined blown-up space M^2_{k,sc} and a polyhomogeneous pseudodifferential calculus, the authors construct a detailed resolvent parametrix with multiple face-models, yielding explicit leading terms and compatibility across asymptotic regimes. They establish precise L^p bounds for the Riesz transform that depend on dimension and zero-mode data, showing generic bounds reduce to (n/(n-2), n/3) while allowing larger ranges when zero modes decay faster; they also analyze non-polyhomogeneous behaviour in resonance cases and compare with Jensen–Kato. These results illuminate the long-time heat kernel behaviour and spectral theory of P = Δ_g + V on noncompact spaces, with implications for geometric scattering and PDE analytic techniques on manifolds with ends.

Abstract

Let $(M^\circ, g)$ be an asymptotically conic manifold, in the sense that $M^\circ$ compactifies to a manifold with boundary $M$ in such a way that $g$ becomes a scattering metric on $M$. A special case of particular interest is that of asymptotically Euclidean manifolds, where $\partial M = S^{n-1}$ and the induced metric at infinity is equal to the standard metric. We study the resolvent kernel $(P + k^2)^{-1}$ and Riesz transform of the operator $P = Δ_g + V$, where $Δ_g$ is the positive Laplacian associated to $g$ and $V$ is a real potential function $V$ that is smooth on $M$ and vanishes to some finite order at the boundary. In the first paper in this series we made the assumption that $n \geq 3$ and that $P$ has neither zero modes nor a zero-resonance and showed (i) that the resolvent kernel is conormal to the lifted diagonal and polyhomogeneous at the boundary on a blown up version of $M^2 \times [0, k_0]$, and (ii) the Riesz transform of $P$ is bounded on $L^p(M^\circ)$ for $1 < p < n$, and that this range is optimal unless $V \equiv 0$ and $M^\circ$ has only one end. In the present paper, we perform a similar analysis assuming again $n \geq 3$ but allowing zero modes and zero-resonances. We find the precise range of $p$ for which the Riesz transform (suitably defined) of $P$ is bounded on $L^p(M)$ when zero modes (but not resonances, which make the Riesz transform undefined) are present. Generically the Riesz transform is bounded for $p$ precisely in the range $(n/(n-2), n/3)$, with a bigger range possible if the zero modes have extra decay at infinity.

Resolvent at low energy and Riesz transform for Schroedinger operators on asymptotically conic manifolds. II

TL;DR

This work advances the low-energy resolvent analysis for Schrödinger operators on asymptotically conic manifolds by accommodating zero modes and zero-resonances. Through a refined blown-up space M^2_{k,sc} and a polyhomogeneous pseudodifferential calculus, the authors construct a detailed resolvent parametrix with multiple face-models, yielding explicit leading terms and compatibility across asymptotic regimes. They establish precise L^p bounds for the Riesz transform that depend on dimension and zero-mode data, showing generic bounds reduce to (n/(n-2), n/3) while allowing larger ranges when zero modes decay faster; they also analyze non-polyhomogeneous behaviour in resonance cases and compare with Jensen–Kato. These results illuminate the long-time heat kernel behaviour and spectral theory of P = Δ_g + V on noncompact spaces, with implications for geometric scattering and PDE analytic techniques on manifolds with ends.

Abstract

Let be an asymptotically conic manifold, in the sense that compactifies to a manifold with boundary in such a way that becomes a scattering metric on . A special case of particular interest is that of asymptotically Euclidean manifolds, where and the induced metric at infinity is equal to the standard metric. We study the resolvent kernel and Riesz transform of the operator , where is the positive Laplacian associated to and is a real potential function that is smooth on and vanishes to some finite order at the boundary. In the first paper in this series we made the assumption that and that has neither zero modes nor a zero-resonance and showed (i) that the resolvent kernel is conormal to the lifted diagonal and polyhomogeneous at the boundary on a blown up version of , and (ii) the Riesz transform of is bounded on for , and that this range is optimal unless and has only one end. In the present paper, we perform a similar analysis assuming again but allowing zero modes and zero-resonances. We find the precise range of for which the Riesz transform (suitably defined) of is bounded on when zero modes (but not resonances, which make the Riesz transform undefined) are present. Generically the Riesz transform is bounded for precisely in the range , with a bigger range possible if the zero modes have extra decay at infinity.

Paper Structure

This paper contains 42 sections, 14 theorems, 223 equations, 1 figure.

Table of Contents

  1. Introduction
  2. The blown-up space $M^2_{k, \textrm{sc}}$ and the calculus $\Psi_k^{m, \mathcal{E}}(M; \tilde{\Omega}_b^{1/2})$.
  3. The space $M^2_{k, \textrm{sc}}$.
  4. Operator calculus
  5. Resolvent kernel for asymptotically Euclidean manifolds
  6. Term at $\textrm{sc}$ and $\textrm{bf}_0$ and the singularity at the diagonal
  7. Terms at zf
  8. Terms at $\textrm{rb}_0$ and $\textrm{lb}_0$.
  9. Additional term when $n=3$ and $m\geq 2$
  10. The case $n=5$ with $m=0$
  11. Error term and resolvent
  12. Proof of Lemma \ref{['complicated']}
  13. Resolvent kernel for asymptotically conic manifolds
  14. Additional term when $n=3$
  15. Terms at $\textrm{zf}$ when $n=5$, $m' = 0$
  16. ...and 27 more sections

Key Result

Theorem 1.1

Let $n \geq 3$, and let $P = \Delta_g$ be the Laplacian on an asymptotically conic manifold $(M,g)$. First we assume that $M$ has one end. Let $\Delta_{\partial M}$ be the Laplacian on the boundary of $M$ for the metric $h(0)$ given by (metricconic), let $\lambda_1$ be its first non-zero eigenvalue, while if $\nu_1\geq n/2$, then Next let $P = \Delta_g + V$ with $V$ as above, and suppose that eit

Figures (1)

  • Figure 1: The blow-up manifold $M^2_{k,\textrm{sc}}$.

Theorems & Definitions (23)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Remark 1.4
  • Remark 1.5
  • Theorem 2.1: Melrose's Relative Index Theorem, APS
  • Theorem 2.2: Regularity of solutions to $P_b u = f$
  • Proposition 2.3
  • Lemma 3.1
  • Remark 3.2
  • ...and 13 more