The cubic-quintic nonlinear Schr\"odinger equation with inverse-square potential

TL;DR

This paper proves scattering for a three-dimensional nonlinear Schr"odinger equation with combined cubic and quintic nonlinearities and an inverse-square potential, extending known results to this more complex setting.

Contribution

It establishes scattering in the mass-energy region where the virial functional is positive for the cubic-quintic NLS with inverse-square potential, a novel extension of prior work.

Findings

Scattering established in specified mass-energy region.

Extension of scattering results to inverse-square potential setting.

Analysis of virial functional positivity region.

Abstract

We consider the nonlinear Schr\"odinger equation in three space dimensions with a focusing cubic nonlinearity and defocusing quintic nonlinearity and in the presence of an external inverse-square potential. We establish scattering in the region of the mass-energy plane where the virial functional is guaranteed to be positive. Our result parallels the scattering result of \cite{R.Killip, T. Oh, O. Pocovnicu, and M. Visan} in the setting of the standard cubic-quintic NLS.