Limiting absorption principle on Riemannian scattering (asymptotically conic) spaces, a Lagrangian approach

TL;DR

The paper proves a limiting absorption principle for Laplace-like operators on asymptotically conic manifolds using a Lagrangian distribution framework. By conjugating the spectral family with $e^{i\sigma/x}$, the outgoing radial set is moved to the zero section, enabling second microlocal/b-analytic techniques to establish Fredholm properties on spaces encoding Lagrangian regularity. The results cover nonzero energies, provide vector-valued extensions, and include a high-energy semiclassical regime with uniform estimates, building a comprehensive framework for resolvent analysis on scattering spaces. This approach yields precise mapping properties on $H_{\mathrm{sc}}^{s,r}$, $H_{\mathrm b}^{\tilde r,l}$, and $H_{\mathrm{sc,b}}^{s,r,l}$, and it clarifies the interplay between radial points, normal operators, and Lagrangian regularity in the limiting absorption context.

Abstract

We use a Lagrangian perspective to show the limiting absorption principle on Riemannian scattering, i.e. asymptotically conic, spaces, and their generalizations. More precisely we show that, for non-zero spectral parameter, the `on spectrum', as well as the `off-spectrum', spectral family is Fredholm in function spaces which encode the Lagrangian regularity of generalizations of `outgoing spherical waves' of scattering theory, and indeed this persists in the `physical half plane'.

Figures (4)

• Figure 1: Second microlocalized Euclidean space $\mathbb{R}^{n}$. The left hand side is the fiber-compactified sc-cotangent bundle, $\overline{{}^{{\mathrm{sc}}} T^*}\overline{\mathbb{R}^n}=\overline{\mathbb{R}^n}\times\overline{(\mathbb{R}^n)^*}$, the right hand side is its blow-up at the boundary of the zero section. The (interior of the) front face of the blow-up, shown by the curved arcs, can be identified with ${}^{{\mathrm b}} T^*_{\partial\overline{\mathbb{R}^n}}\overline{\mathbb{R}^n}$. The characteristic set of $\hat{P}(\sigma)$, $\sigma\neq 0$, discussed in Section \ref{['sec:operator']}, is also shown, both from the compactified perspective, as $\Sigma$, which is a subset of the boundary, and from the conic perspective, here conic in the base (i.e. the dilations are in the $\mathbb{R}^n_z$, spatial, factor), as $\mathrm{Char}(\hat{P})$. On the second microlocal figure on the right, the characteristic set within the boundary lies at the lift of the fibers of the sc-cotangent bundle over the boundary; from the b-perspective, it thus corresponds to symbolic behavior, and lies at fiber infinity. The fiber of cotangent bundle over the origin, i.e. $\{0\}\times\overline{(\mathbb{R}^n)^*}$, is also indicated; this is only special from the conic (dilation) perspective, in which it is the analogue of the zero section in standard microlocal analysis.
• Figure 2: The second microlocal space, on the right, obtained by blowing up the corner of $\overline{{}^{{\mathrm b}} T^*}X$, shown on the left.
• Figure 3: The support of $\chi$ on the second microlocal space, indicated by the rectangular box. The characteristic set is the circular curve tangent to the $\mu$ axis at the b-face, given by the sc-zero section.
• Figure 4: The support of $\chi$ on the second microlocal space, indicated by the rectangular box. The characteristic set is the circular curve tangent to the $\mu$ axis at the b-face, given by the sc-zero section.

Theorems & Definitions (47)

• Theorem 1.1
• Remark 1.2
• Remark 1.3
• Remark 1.4
• Theorem 1.5
• Remark 1.6
• Proposition 3.1
• Remark 3.2
• proof
• Lemma 3.3