BCAM Severo Ochoa Course on Scattering Results for Schrödinger Operators

Speaker: Luccas Campos

Dates: 8–19 September 2025

Schedule: Mondays, Wednesdays, and Fridays from 14:00 to 16:00

Scattering Theory for the Non Linear Schrödinger Equations

The main goal of this course is to present the state-of-the-art of the scattering theory for Schrödinger equations, together with some open questions and research directions. The field remains very prolific and the techniques have been increasing in intricacy (and beauty!) in the last years, attracting the attention of renowned mathematicians around the globe.

We shall present different approaches to scattering, starting from the concentration-compactness-rigidity method, pioneered by Kenig and Merle - which has become standard in the study of global well-posedness and scattering in the context of dispersive and wave equations - and its adaptation to the focusing 3d cubic case in H^1 by Holmer and Roudenko, which solved the radial case, later on joined by Duyckaerts to tackle the nonradial case.

Albeit powerful, this approach requires building some heavy machinery in order to obtain the desired space-time bounds. We then present an alternative approach, based on Tao's scattering criterion, involving the so-called virial/Morawetz estimates by Dodson and Murphy. In the radial, intercritical case in H^1, these estimates provide a simple and short proof of scattering, relying mainly on a local coercivity bound below the ground state, on the local almost-conservation of the mass and on the combination of dispersive and Strichartz estimates. We will also see how this result can be carried on for spatially inhomogeneous versions of the NLS equation in the nonradial case.

The virial/Morawetz estimates can also be upgraded (although painfully) to the nonradial case and employed in the intercritical and in the critical cases. We will comment on the work of Killip and Visan for the focusing, energy-critical NLS equation in dimensions five and higher, in the nonradial setting, and in the works of Dodson in dimension four, in the mass-critical NLS and in the corresponding defocusing cases.

References

• Arora, A. K., Dodson, B., & Murphy, J. (2020). Scattering below the ground state for the 2D radial nonlinear Schrödinger equation. Proceedings of the American Mathematical Society, 148(4), 1653–1663.
• Campos, L. (2021). Scattering of radial solutions to the inhomogeneous nonlinear Schrödinger equation. Nonlinear Analysis, 202, 112118.
• Campos, L., & Cardoso, M. (2022). A Virial-Morawetz approach to scattering for the non-radial inhomogeneous NLS. Proceedings of the American Mathematical Society, 150(1), 221–236.
• Dodson, B. (2016). Global well-posedness and scattering for the mass critical nonlinear Schrödinger equation. Journal of the American Mathematical Society, 29(4), 1063–1097.
• Dodson, B. (2017). Global well-posedness and scattering for the focusing, energy-critical nonlinear Schrödinger problem in dimension 4 for initial data below a ground state threshold. Analysis & PDE, 10(7), 1605–1631.
• Dodson, B. (2019). Defocusing nonlinear Schrödinger equations. Cambridge University Press.
• Dodson, B., & Murphy, J. (2017). A new proof of scattering below the ground state for the 3D radial focusing cubic NLS. Proceedings of the American Mathematical Society, 145(10), 4337–4343.
• Duyckaerts, T., Holmer, J., & Roudenko, S. (2008). Scattering for the non-radial 3D cubic nonlinear Schrödinger equation. Annales de l'Institut Henri Poincaré (C) Analyse Non Linéaire, 26(3), 1025–1044.
• Holmer, J., & Roudenko, S. (2008). A sharp condition for scattering of the radial 3D cubic nonlinear Schrödinger equation. Communications in Mathematical Physics, 282(2), 435–467.
• Kenig, C. E., & Merle, F. (2006). Global well-posedness, scattering and blow-up for the energy-critical, focusing, non-linear Schrödinger equation in the radial case. Inventiones Mathematicae, 166(1), 145–205.
• Killip, R., & Visan, M. (2010). The focusing energy-critical nonlinear Schrödinger equation in dimensions five and higher. American Journal of Mathematics, 132(2), 361–424.
• Tao, T. (2004). On the asymptotic behavior of large radial data for a focusing non-linear Schrödinger equation. Dynamics of Partial Differential Equations, 1(1), 1–48.