BCAM Course | Scattering Theory for the Non Linear Schrödinger Equations

The main aim of the course is to present on one hand the main known results about scattering theory for NLS and on the other hand some basic open questions. The field is very hot and has attracted in the last decades the attention of several outstanding mathematicians. We shall present several techniques developed around this reach field ranging from the interaction Morawetz estimates to the conformal transformations.

LECTURE 1 Strichartz estimates, local and global existence of solutions, definition of scattering, lens and pseudoconformal transformations

LECTURE 2 Proof of no scattering in the long range regime, proof of scattering in L^2 for data in \Sigma in the short range regime

LECTURE 3 Scattering in H^1 for data in \Sigma in the short range regime

LECTURE 4 Scattering in H^1 for data in H^1 in the inter-critical regime via interaction Morawetz estimates

LECTURE 5 Scattering in H^1 for data in H^1 in the inter-critical regime via Kenig-Merle approach

Literature

1. Burq, Georgiev, Tzvetkov, Visciglia$H^1$ scattering for mass-subcritical NLS with short-range nonlinearity and datum in $\Sigma$, Annales Henri Poincar\'e, 24 (2023), 1355-1376.

2. Cazenave, Semilinear Schroedinger Equations, Lecture Notes, Courant Institute.

3. Cazenave, Fang, Xie, Scattering for the focusing energy-subcritical NLS, Science China Mathematics, 54 (2011) n. 10, 2037-2062.

4. Colliander, Kell, Staffilani, Tao, Takaoka, Global existence and scattering for rough solutions of a NLS on $\R^3$, 54 (2004) n.8 , 987-1014.

5. Linares-Ponce, Introduction to Nonlinear Dispersive Equations. Universitext.

6. Planchon, Vega, Bilinear Virial Identities and Applications, Annales scientifiques de l' ENS, 42 (2009) n.2, 261-290.

7. Tsutsumi, Yajima, The asymptotic behavior of NLS, Bulletin of the American Mathematical Society, 11 (1984) n. 1, 186--188.

8. Visciglia, On the decay of solutions to a class of defocusing NLS, Mathematical Research Letters (16) 2009, 919-926.