Weak Dispersive Estimates for Schrödinger Equations with Long Range Potentials
- 5 March 2009
- journal article
- research article
- Published by Taylor & Francis in Communications in Partial Differential Equations
- Vol. 34 (1) , 74-105
- https://doi.org/10.1080/03605300802683596
Abstract
We prove local smoothing estimates for the Schrödinger initial value problem with data in the energy space L 2(ℝ d ), d ≥ 2 and a general class of potentials. In the repulsive setting we have to assume just a power like decay (1 + |x|)−γ for some γ > 0. Also attractive perturbations are considered. The estimates hold for all time and as a consequence a weak dispersion of the solution is obtained. The proofs are based on similar estimates for the corresponding stationary Helmholtz equation and Kato H-smooth theory.Keywords
This publication has 15 references indexed in Scilit:
- Some dispersive estimates for Schrödinger equations with repulsive potentialsJournal of Functional Analysis, 2006
- Agmon-Kato-Kuroda theorems for a large class of perturbationsDuke Mathematical Journal, 2006
- Dispersion and Moment Lemmas RevisitedJournal of Differential Equations, 1999
- Morrey–Campanato Estimates for Helmholtz EquationsJournal of Functional Analysis, 1999
- SOME EXAMPLES OF SMOOTH OPERATORS AND THE ASSOCIATED SMOOTHING EFFECTReviews in Mathematical Physics, 1989
- Local smoothing properties of dispersive equationsJournal of the American Mathematical Society, 1988
- Regularity of solutions to the Schrödinger equationDuke Mathematical Journal, 1987
- Decay and scattering of solutions of a nonlinear Schrödinger equationJournal of Functional Analysis, 1978
- Time decay for the nonlinear Klein-Gordon equationProceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences, 1968
- Darstellung der Eigenwerte von? u+? u=0 durch ein RandintegralMathematische Zeitschrift, 1940