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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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[1] Milton Abramowitz and Irene A. Stegun , Handbook of mathematical functions with formulas, graphs, and mathematical tables, A Wiley-Interscience Publication, John Wiley & Sons, Inc., New York; John Wiley & Sons, Inc., New York, 1984. Reprint of the 1972 edition; Selected Government Publications. Irene A. Stegun , Pocketbook of mathematical functions, Verlag Harri Deutsch, Thun, 1984. Abridged edition of Handbook of mathematical functions edited by Milton Abramowitz and Irene A. Stegun; Material selected by Michael Danos and Johann Rafelski.
[2] Galtbayar Artbazar and Kenji Yajima, The \?^{\?}-continuity of wave operators for one dimensional Schrödinger operators, J. Math. Sci. Univ. Tokyo 7 (2000), no. 2, 221 – 240. · Zbl 0976.34071
[3] J. Bourgain, Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations. I. Schrödinger equations, Geom. Funct. Anal. 3 (1993), no. 2, 107 – 156. , https://doi.org/10.1007/BF01896020 J. Bourgain, Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations. II. The KdV-equation, Geom. Funct. Anal. 3 (1993), no. 3, 209 – 262. · Zbl 0787.35098
[4] N. Burq, P. Gérard, and N. Tzvetkov, Strichartz inequalities and the nonlinear Schrödinger equation on compact manifolds, Amer. J. Math. 126 (2004), no. 3, 569 – 605. · Zbl 1067.58027
[5] N. Burq, P. Gérard, and N. Tzvetkov, Bilinear eigenfunction estimates and the nonlinear Schrödinger equation on surfaces, Invent. Math. 159 (2005), no. 1, 187 – 223 (English, with English and French summaries). · Zbl 1092.35099
[6] Walter Craig, Thomas Kappeler, and Walter Strauss, Microlocal dispersive smoothing for the Schrödinger equation, Comm. Pure Appl. Math. 48 (1995), no. 8, 769 – 860. · Zbl 0856.35106
[7] P. Deift and E. Trubowitz, Inverse scattering on the line, Comm. Pure Appl. Math. 32 (1979), no. 2, 121 – 251. · Zbl 0388.34005
[8] Shin-ichi Doi, Smoothing effects for Schrödinger evolution equation and global behavior of geodesic flow, Math. Ann. 318 (2000), no. 2, 355 – 389. · Zbl 0969.35029
[9] P. Gérard, Nonlinear Schrödinger equations on compact manifolds, European Congress of Mathematics, Eur. Math. Soc., Zürich, 2005, pp. 121 – 139. · Zbl 1076.35115
[10] M. Goldberg and W. Schlag, Dispersive estimates for Schrödinger operators in dimensions one and three, Comm. Math. Phys. 251 (2004), no. 1, 157 – 178. · Zbl 1086.81077
[11] Andrew Hassell, Terence Tao, and Jared Wunsch, A Strichartz inequality for the Schrödinger equation on nontrapping asymptotically conic manifolds, Comm. Partial Differential Equations 30 (2005), no. 1-3, 157 – 205. · Zbl 1068.35119
[12] Andrew Hassell, Terence Tao, and Jared Wunsch, Sharp Strichartz estimates on nontrapping asymptotically conic manifolds, Amer. J. Math. 128 (2006), no. 4, 963 – 1024. · Zbl 1177.58019
[13] Arne Jensen and Tosio Kato, Asymptotic behavior of the scattering phase for exterior domains, Comm. Partial Differential Equations 3 (1978), no. 12, 1165 – 1195. · Zbl 0419.35067
[14] J.-L. Journé, A. Soffer, and C. D. Sogge, Decay estimates for Schrödinger operators, Comm. Pure Appl. Math. 44 (1991), no. 5, 573 – 604. · Zbl 0743.35008
[15] Jeffrey Rauch, Local decay of scattering solutions to Schrödinger’s equation, Comm. Math. Phys. 61 (1978), no. 2, 149 – 168. · Zbl 0381.35023
[16] Luc Robbiano and Claude Zuily, Strichartz estimates for Schrödinger equations with variable coefficients, Mém. Soc. Math. Fr. (N.S.) 101-102 (2005), vi+208 (English, with English and French summaries). · Zbl 1097.35002
[17] I. Rodnianski and T. Tao, Longtime decay estimates for the Schrödinger equation on manifolds, Mathematical aspects of nonlinear dispersive equations, Ann. of Math. Stud., vol. 163, Princeton Univ. Press, Princeton, NJ, 2007, pp. 223 – 253. · Zbl 1133.35022
[18] W. Schlag, Dispersive estimates for Schrödinger operators in dimension two, Comm. Math. Phys. 257 (2005), no. 1, 87 – 117. · Zbl 1134.35321
[19] W. Schlag, Dispersive estimates for Schrödinger operators: a survey, Mathematical aspects of nonlinear dispersive equations, Ann. of Math. Stud., vol. 163, Princeton Univ. Press, Princeton, NJ, 2007, pp. 255 – 285. · Zbl 1143.35001
[20] Hart F. Smith and Christopher D. Sogge, Global Strichartz estimates for nontrapping perturbations of the Laplacian, Comm. Partial Differential Equations 25 (2000), no. 11-12, 2171 – 2183. · Zbl 0972.35014
[21] Gigliola Staffilani and Daniel Tataru, Strichartz estimates for a Schrödinger operator with nonsmooth coefficients, Comm. Partial Differential Equations 27 (2002), no. 7-8, 1337 – 1372. · Zbl 1010.35015
[22] Daniel Tataru, Parametrices and dispersive estimates for Schrödinger operators with variable coefficients, Amer. J. Math. 130 (2008), no. 3, 571 – 634. · Zbl 1159.35315
[23] Ricardo Weder, \?^{\?}-\?^{\?} estimates for the Schrödinger equation on the line and inverse scattering for the nonlinear Schrödinger equation with a potential, J. Funct. Anal. 170 (2000), no. 1, 37 – 68. · Zbl 0943.34070
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