Resolvent at low energy III: The spectral measure

• Published in: Transactions of the American Mathematical Society 2013-11, Vol.365 (11), p.6103-6148
• Main Authors: GUILLARMOU, COLIN, HASSELL, ANDREW, SIKORA, ADAM
• Format: Article
• Language: English
• Subjects: Analysis of PDEs; Mathematics; Spectral Theory
• ISSN: 0002-9947; 1088-6850
• DOI: 10.1090/S0002-9947-2013-05849-7

Let M^\circ an asymptotically conic Riemaniann metric on M^\circ compactifies to a manifold with boundary M becomes a scattering metric on M be the positive Laplacian associated to g P = \Delta + V is a potential function obeying certain conditions. We analyze the asymptotics of the spectral measure dE(\lambda ) = (\lambda /\pi i) \big ( R(\lambda +i0) - R(\lambda - i0) \big ), where R(\lambda ) = (P - \lambda ^2)^{-1} \lambda \to 0 M^2 \times [0, \lambda _0) \cos (t \sqrt {P_+}) \sin (t \sqrt {P_+})/\sqrt {P_+}, as t \to \infty In future articles, this result on the spectral measure will be used to (i) prove restriction and spectral multiplier estimates on asymptotically conic manifolds, and (ii) prove long-time dispersion and Strichartz estimates for solutions of the Schrödinger equation on M is nontrapping.