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Decay for the wave and Schrödinger evolutions on manifolds with conical ends. II. (English) Zbl 1187.35032
This article is prolongation of [the authors, ibid. 362, No. 1, 19–52 (2010; Zbl 1185.35046)], where all cases of \(d+n>1\) are discussed. Here the author obtains the accelerated local decay estimate for \(0<\sigma=\sqrt{2\mu_n^2+(d-1)^2/4}-\frac{d-1}{2}\), and for all \(t\geq 1\), the inequality
\[ \|w_{\sigma}e^{it\triangle_{{\mathcal M}}} Y_n f\|_{L^{\infty}({\mathcal M})}\leq C(n,{\mathcal M}, \sigma)t^{-\frac{d+1}{2}-\sigma}\|w^{-1}_{\sigma}f \|_{L^{1}({\mathcal M})} \]
is proved where \(w_{\sigma}(x)=\langle x\rangle ^{-\sigma}\). Also a detailed scattering analysis of the Schrödinger operators of the form \(-\partial^2_{\xi}+(\nu^2-\frac{1}{4})\langle \xi \rangle ^{-2}+U(\xi)\) on the line is given, where \(U\) is real-valued and smooth with \(U^{(\ell)}(\xi)=O(\xi^{-3-\ell})\) for all \(\ell \geq 0\) as \(\xi \to \pm \infty\) and \(\nu >0\), together with the estimation of oscillatory integrals by (non)stationary phase .

MSC:
35J10 Schrödinger operator, Schrödinger equation
58J05 Elliptic equations on manifolds, general theory
58J60 Relations of PDEs with special manifold structures (Riemannian, Finsler, etc.)
35P25 Scattering theory for PDEs
35B45 A priori estimates in context of PDEs
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