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

TL;DR

This paper studies the final state behavior of solutions to nonlinear Schrödinger equations with a time-decaying harmonic oscillator, identifying conditions for long-range nonlinearity and deriving decay estimates for asymptotic states.

Contribution

It determines the final state and decay estimates for NLS with a time-decaying harmonic potential, extending understanding of long-range interactions in such systems.

Findings

Identified the range of nonlinearity powers for long-range behavior.

Derived decay estimates for the asymptotic states.

Established the existence of final states for the system.

Abstract

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.