Wave front set for solutions to Schrödinger equations. (English) Zbl 1155.35014
Summary: We consider solutions to Schrödinger equation on \(\mathbb R^d\) with variable coefficients. Let \(H\) be the Schrödinger operator and let \(u(t)=e ^{-itH}u_0\) be the solution to the Schrödinger equation with the initial condition \(u_0\in L^2(\mathbb R^d)\). We show that the wave front set of \(u(t)\) in the nontrapping region can be characterized by the wave front set of \(e ^{-itH_{0}}u_{0}\), where \(H_{0}\) is the free Schrödinger operator. The characterization of the wave front set is given by the wave operator for the corresponding classical mechanical scattering (or equivalently, by the asymptotics of the geodesic flow).
MSC:
| 35J10 | Schrödinger operator, Schrödinger equation |
| 35Q55 | NLS equations (nonlinear Schrödinger equations) |
References:
| [1] | Craig, W.; Kappeler, T.; Strauss, W., Microlocal dispersive smoothing for the Schrödinger equation, Comm. Pure Appl. Math., 48, 769-860 (1996) · Zbl 0856.35106 |
| [2] | Dimassi, M.; Sjöstrand, J., Spectral Asymptotics in the Semi-Classical Limit, London Math. Soc. Lecture Note Ser., vol. 268 (1999), Cambridge Univ. Press: Cambridge Univ. Press Cambridge · Zbl 0926.35002 |
| [3] | Doi, S., Smoothing effects for Schrödinger evolution equation and global behavior of geodesic flow, Math. Ann., 318, 355-389 (2000) · Zbl 0969.35029 |
| [4] | Doi, S., Dispersion of singularities of solutions for Schrödinger equations, Comm. Math. Phys., 250, 473-505 (2004) · Zbl 1112.35051 |
| [5] | Hassel, A.; Wunsch, J., The Schrödinger propagator for scattering metrics, Ann. of Math. (2), 162, 487-523 (2005) · Zbl 1126.58016 |
| [6] | Hörmander, L., Analysis of Linear Partial Differential Operators, vols. I-IV (1985), Springer-Verlag: Springer-Verlag Berlin · Zbl 0601.35001 |
| [7] | Ito, K.; Nakamura, S., Singularities of solutions to Schrödinger equation on scattering manifold, preprint, November 2007 |
| [8] | Martinez, A., An Introduction to Semiclassical and Microlocal Analysis, Universitext (2002), Springer-Verlag: Springer-Verlag New York · Zbl 0994.35003 |
| [9] | Martinez, A.; Nakamura, S.; Sordoni, V., Analytic smoothing effect for the Schrödinger equation with long-range perturbation, Comm. Pure Appl. Math., 59, 1330-1351 (2006) · Zbl 1122.35027 |
| [10] | Martinez, A.; Nakamura, S.; Sordoni, V., Analytic wave front for solutions to Schrödinger equation, preprint, June 2007 |
| [11] | Nakamura, S., Propagation of the homogeneous wave front set for Schrödinger equations, Duke Math. J., 126, 349-367 (2005) · Zbl 1130.35023 |
| [12] | S. Nakamura, Semiclassical singularity propagation property for Schrödinger equations, J. Math. Soc. Japan, in press; S. Nakamura, Semiclassical singularity propagation property for Schrödinger equations, J. Math. Soc. Japan, in press |
| [13] | Reed, M.; Simon, B., The Methods of Modern Mathematical Physics, vols. I-IV (1980), Academic Press: Academic Press New York · Zbl 0459.46001 |
| [14] | Robbiano, L.; Zuily, C., Microlocal analytic smoothing effect for the Schrödinger equation, Duke Math. J., 100, 93-129 (1999) · Zbl 0941.35014 |
| [15] | Robbiano, L.; Zuily, C., Analytic theory for the quadratic scattering wave front set and application to the Schrödinger equation, Soc. Math. France, Astérisque, 283, 1-128 (2002) · Zbl 1029.35001 |
| [16] | Taylor, M., Pseudodifferential Operators, Princeton Math. Ser. (1981), Princeton Univ. Press: Princeton Univ. Press Princeton, NJ · Zbl 0453.47026 |
| [17] | Wunsch, J., Propagation of singularities and growth for Schrödinger operators, Duke Math. J., 98, 137-186 (1999) · Zbl 0953.35121 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
