Large time asymptotics of Schr\"odinger type equations with general data

Abstract: We consider a Schr\”odinger of equations with a general interaction term, which is linear or nonlinear, time dependent and including charge transfer potentials. Without the assumption of radial symmetry, we prove the global solutions are asymptotically given by a free wave and a weakly localized part. When the interaction term is quasi-periodic in time and localized in space, we also prove local decay estimates for the scattering states. The proof for asymptotic completeness is based on constructing in a new way the Free Channel Wave Operator, and further tools from the recent work of Baoping Liu and Avy Soffer[1,2]. The proof for local decay estimates is based on asymptotic completeness. It is an alternative to Mourre's method for time dependent problems. This new method also allows us to study Klein-Gordon equations, N-body Quantum and general 3-body Quantum systems. These are joint works with Avy Soffer.

[1] Liu, B., Soffer, A. (2020). A General Scattering theory for Nonlinear and Non-autonomous Schr\”odinger Type Equations-A Brief description. arXiv preprint arXiv:2012.14382. 

[2] Liu, B., Soffer, A. (2021) The Large Time Asymptotics of Nonlinear Multichannel Schr\”odinger Equations”. Submitted