Final state problem for nonlinear Schr\"{o}dinger equations with time-decaying harmonic oscillators

We consider the final-state problem for the nonlinear Schr\"{o}dinger equations (NLS) with a suitable time-decaying harmonic oscillator. In this equation, the power of nonlinearity $|u|^{\rho}u$ is included in the long-range class if $0 < \rho \leq 2/(n(1- \lambda))$ with $0 \leq \lambda <1/2$, which is determined by the harmonic potential and a coefficient of Laplacian. In this paper, we find the final state for this system and obtain the decay estimate for asymptotics.