Asymptotic behavior for nonlinear Schr\"{o}dinger equations with critical time-decaying harmonic potential

TL;DR

This paper investigates the asymptotic behavior of solutions to nonlinear Schrödinger equations with critical time-decaying harmonic potentials, highlighting the role of logarithmic nonlinearities in the critical case.

Contribution

It characterizes the threshold for nonlinearities in the critical case using logarithmic terms, extending understanding beyond polynomial nonlinearities.

Findings

Dispersive estimates with weak decay for time-decaying harmonic oscillators

Threshold power of nonlinearity changes between short and long range

Logarithmic nonlinearities are essential in the critical case

Abstract

Time-decaying harmonic oscillators yield dispersive estimates with weak decay, and change the threshold power of the nonlinearity between the short and the long range. In the non-critical case for the time-decaying harmonic oscillator, this threshold can be characterized by polynomial nonlinearities. However, in the critical case, it is difficult to characterize the threshold using only polynomial terms, and thus we use logarithmic nonlinear terms.