Local smoothing for the Schrödinger equation on a multi-warped product manifold with inflection-transmission trapping

Christianson, Hans; Nowak, Derrick

doi:10.1090/proc/15615

Local smoothing for the Schrödinger equation on a multi-warped product manifold with inflection-transmission trapping
HTML articles powered by AMS MathViewer

by Hans Christianson and Derrick Nowak PDF

Proc. Amer. Math. Soc. 150 (2022), 213-229 Request permission

Abstract:

Geodesic trapping is an obstruction to dispersive estimates for solutions to the Schrödinger equation. Surprisingly little is known about solutions to the Schrödinger equation on manifolds with degenerate trapping, since the conditions for degenerate trapping are not stable under perturbations. In this paper we extend some of the results of Christianson and Metcalfe [Indiana Univ. Math. J. 63 (2014), pp. 969–992] on inflection-transmission type trapping on warped product manifolds to the case of multi-warped products. The main result is that the trapping on one cross section does not interact with the trapping on other cross sections provided the manifold has only one infinite end and only inflection-transmission type trapping.

References

N. Burq, Smoothing effect for Schrödinger boundary value problems, Duke Math. J. 123 (2004), no. 2, 403–427 (English, with English and French summaries). MR 2066943, DOI 10.1215/S0012-7094-04-12326-7
Hans Christianson, Semiclassical non-concentration near hyperbolic orbits, J. Funct. Anal. 246 (2007), no. 2, 145–195. MR 2321040, DOI 10.1016/j.jfa.2006.09.012
Hans Christianson, Dispersive estimates for manifolds with one trapped orbit, Comm. Partial Differential Equations 33 (2008), no. 7-9, 1147–1174. MR 2450154, DOI 10.1080/03605300802133907
Hans Christianson, Quantum monodromy and nonconcentration near a closed semi-hyperbolic orbit, Trans. Amer. Math. Soc. 363 (2011), no. 7, 3373–3438. MR 2775812, DOI 10.1090/S0002-9947-2011-05321-3
Hans Christianson, High-frequency resolvent estimates on asymptotically Euclidean warped products, Comm. Partial Differential Equations 43 (2018), no. 9, 1306–1362. MR 3915489, DOI 10.1080/03605302.2018.1517786
Walter Craig, Thomas Kappeler, and Walter Strauss, Microlocal dispersive smoothing for the Schrödinger equation, Comm. Pure Appl. Math. 48 (1995), no. 8, 769–860. MR 1361016, DOI 10.1002/cpa.3160480802
Hans Christianson and Jason Metcalfe, Sharp local smoothing for warped product manifolds with smooth inflection transmission, Indiana Univ. Math. J. 63 (2014), no. 4, 969–992. MR 3263918, DOI 10.1512/iumj.2014.63.5323
P. Constantin and J.-C. Saut, Local smoothing properties of dispersive equations, J. Amer. Math. Soc. 1 (1988), no. 2, 413–439. MR 928265, DOI 10.1090/S0894-0347-1988-0928265-0
Hans Christianson and Jared Wunsch, Local smoothing for the Schrödinger equation with a prescribed loss, Amer. J. Math. 135 (2013), no. 6, 1601–1632. MR 3145005, DOI 10.1353/ajm.2013.0047
Kiril Datchev, Local smoothing for scattering manifolds with hyperbolic trapped sets, Comm. Math. Phys. 286 (2009), no. 3, 837–850. MR 2472019, DOI 10.1007/s00220-008-0684-1
Shin-ichi Doi, Remarks on the Cauchy problem for Schrödinger-type equations, Comm. Partial Differential Equations 21 (1996), no. 1-2, 163–178. MR 1373768, DOI 10.1080/03605309608821178
Shin-ichi Doi, Smoothing effects of Schrödinger evolution groups on Riemannian manifolds, Duke Math. J. 82 (1996), no. 3, 679–706. MR 1387689, DOI 10.1215/S0012-7094-96-08228-9
Per Sjölin, Regularity of solutions to the Schrodinger equation, Duke Math. J. 55 (1987), no. 3, 699–715.
Luis Vega, Schrödinger equations: pointwise convergence to the initial data, Proc. Amer. Math. Soc. 102 (1988), no. 4, 874–878. MR 934859, DOI 10.1090/S0002-9939-1988-0934859-0