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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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