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Decay for the wave and Schrödinger evolutions on manifolds with conical ends. I. (English) Zbl 1185.35046
For \(d\)-dimensional (\(d\geq 1\)) compact Riemannian manifold \(\Omega \subset \mathbb{R}^N\) and \((d+1)\)-dimensional Riemannian manifold \(\mathcal{M}:=\{(x,r(x)\omega):x\in \mathbb{R},\omega\in \Omega\} \) with \(r>0\) smooth metric \(ds^2=(1+r'{}^2(x))dx^2+r^2(x)ds^2_{\Omega}\) and conical ends \(r(x)=|x|+O(x^{-1})\) at \(x\to \pm \infty \) the Hamiltonian flow on them exhibits trapping. The author has obtained the dispersive estimates for the Schrödinger evolution \(e^{it\triangle_{\mathcal{M}}}\) and the wave evolution \(e^{it\sqrt{-\triangle_{\mathcal{M}}}}\) for data of the form \(f(x,\omega)=Y_n(\omega)u(x)\), where \(Y_n\) are eigenfunctions of \(-\Delta_{\Omega}\) corresponding to eigenvalues \(\mu_n^2\).
In the part I of the article the case \(d=1, Y_0=1\) is investigated. Two main results are obtained here:
(A) A detailed scattering analysis of Schrödinger operators of the form \(-\partial^2_{\xi}+V(\xi)\) on the line where \(V(\xi)\) has inverse square behavior at infinity.
(B) 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
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