## Classification of asymptotic profiles for nonlinear Schrödinger equations with small initial data.(English)Zbl 1033.81034

The authors consider the asymptotic states as time goes to infinity of solutions of the cubic nonlinear Schrödinger equation (with a Delta-bounded, decaying potential) when the linear Hamiltonian allows two bound states and eigenvalues satisfying some resonance conditions. They show that in this situation exactly three types of behavior can occur, namely that either the solutions vanish, either they converge to a nonlinear ground state, or they converge to a nonlinear exited state. It is needed that the initial datum should be small in $$H^1$$ Also, they obtain upper and lower bounds for the solutions. The proofs use outgoing estimates (introduced by the authors [Int. Math. Res. Not. 2002, 1629–1673 (2002; Zbl 1011.35120)] of the dispersive waves which measure the relevant time-direction dependent information of the waves.

### MSC:

 81Q10 Selfadjoint operator theory in quantum theory, including spectral analysis 81Q05 Closed and approximate solutions to the Schrödinger, Dirac, Klein-Gordon and other equations of quantum mechanics 35Q55 NLS equations (nonlinear Schrödinger equations)

### Keywords:

nonlinear Schrödinger equation; resonance conditions

Zbl 1011.35120
Full Text: