Scattering solutions to nonlinear Schrödinger equation with a long range potential

Abstract

In this paper, we consider a nonlinear Schrödinger equation with a repulsive inversepower potential. It is known that the corresponding stationary problem has a "radial" ground state. Here, the "radial" ground state is a least energy solution among radial solutions to the stationary problem. We prove that if radial initial data below the "radial" ground state has positive virial functional, then the corresponding solution to the nonlinear Schrödinger equation scatters. In particular, we can treat not only short range potentials but also long range potentials. Contents 1. Introduction 1 2. Preliminaries 6 2.1. Notation and definition 6 2.2. Some tools 6 3. Parameter independence of the splitting below the "radial" ground state 8 4. Well-posedness 9 4.1. Local well-posedness 9 4.2. Small data theory 11 4.3. Stability 13 4.4. Final state problem 16 5. Linear profile decomposition 17 6. Scattering 21 Acknowledgements 28 References 29

Mass-energy threshold dynamics for the focusing NLS with a repulsive inverse-power potential

“…Note that E V is the generation Hamiltonian of (NLS a ). The Cauchy problem for the present equation has been studied by Guo, Wang and Yao [8] (see also [10]), more precisely: for u 0 ∈ H 1 (R 3 ), there exist T * = T ( u 0 H 1 ) > 0 and a unique solution u ∈ C([0, T * ), H 1 (R 3 )) of the Cauchy problem (NLS a ). Furthermore, the solution satisfies the conservation of energy and mass E V (u(t)) = E V (u 0 ) and M (u(t)) = M (u 0 ), for all t ∈ [0, T * ), where…”

“…In [10], the authors have studied the global existence and scattering of solutions to (NLS a ) when the initial data has nonnegative virial functional P V , where…”