[Paper Review] The scattering operator on asymptotically hyperbolic manifolds

Antônio Sá Barreto, Yiran Wang|arXiv (Cornell University)|Sep 8, 2016

Spectral Theory in Mathematical Physics26 references3 citations

TL;DR

This paper establishes a formula for the Schwartz kernel of the scattering operator on asymptotically hyperbolic manifolds (AHM) in terms of the fundamental solution of the wave operator, without requiring trapping assumptions. For non-trapping AHMs, it proves the scattering operator is a Fourier integral operator quantizing the scattering relation, refining prior results on radiation fields and scattering microlocal structure.

ABSTRACT

We obtain a formula for the Schwartz kernel of the scattering operator in terms of the Schwartz kernel of the fundamental solution of the wave operator on asymptotically hyperbolic manifolds. If there are no trapped geodesics, this formula is used to show that the scattering operator is a Fourier integral operator that quantizes the scattering relation.

Motivation & Objective

• To derive a general formula for the scattering operator's Schwartz kernel on asymptotically hyperbolic manifolds without trapping assumptions.
• To characterize the scattering operator as a Fourier integral operator quantizing the scattering relation in the non-trapping case.
• To refine the microlocal description of radiation fields on non-trapping AHMs, providing a uniform kernel description up to infinity.
• To extend and unify results on radiation fields and scattering relations in the context of asymptotically hyperbolic geometry.

Proposed method

• Derives the scattering operator's kernel from the wave operator's fundamental solution using microlocal analysis on the double space $\partial X \times_0 X$.
• Applies blow-ups at the front and boundary faces to resolve singularities and analyze asymptotic behavior of the wave kernel.
• Uses the resolvent's meromorphic continuation and the relation between $E_+$ and $R_+(\lambda)$ to connect the wave kernel to the scattering operator.
• Lifts the kernel via $\beta_{\partial}^*$ and $\beta_0^*$ maps to analyze behavior at infinity and on the boundary.
• Applies the theory of Lagrangian distributions and Fourier integral operators to show the scattering operator is quantized by the scattering relation.
• Employs the parametrix construction and microlocal techniques from previous works on AHM scattering to establish uniform kernel structure.

Experimental results

Research questions

• RQ1 How can the Schwartz kernel of the scattering operator on asymptotically hyperbolic manifolds be expressed in terms of the wave operator's fundamental solution?
• RQ2 What is the microlocal structure of the scattering operator on non-trapping asymptotically hyperbolic manifolds?
• RQ3 How does the scattering operator relate to the scattering relation in the non-trapping case?
• RQ4 Can the radiation field kernel be described uniformly up to the front face of the double space in non-trapping AHMs?
• RQ5 What is the precise role of the wave kernel's asymptotic behavior in determining the scattering operator's structure?

Key findings

• The scattering operator's Schwartz kernel is expressed as $\beta_{\partial}^*\mathcal{K}_{\mathcal{S}}(s,\tilde{m},s') = \mathcal{A}(s - s' - 2\log\rho_{\mathrm{ff},0}, \tilde{m})$, where $\mathcal{A}$ arises from the wave kernel's restriction.
• For non-trapping AHMs, the scattering operator is a Fourier integral operator quantizing the scattering relation, as shown via microlocal analysis of the kernel's wave front set.
• The radiation field kernel $\mathcal{K}_{\mathcal{R}_+}$ is uniformly described as $\beta_{1L}^*\mathcal{K}_{\mathcal{R}_+} = \mathcal{D}(s - \log\rho_{\mathrm{ff},L}, m) + \partial_s\mathcal{D}(s - \log\rho_{\mathrm{ff},L}, m)$, with $\mathcal{D}$ in $I^0$ classes over specific Lagrangian submanifolds.
• The result refines Sá Barreto and Wunsch's earlier work by extending the microlocal description of radiation fields to include the front face of $\partial X \times_0 X$.
• The formula for the scattering kernel via the wave kernel holds without trapping assumptions, generalizing previous results.
• The scattering matrix kernel is shown to be obtainable from the scattering operator via Fourier transform conjugation, consistent with earlier findings.