Nonlinear Schr\"{o}dinger equation for the twisted Laplacian in the critical case

TL;DR

This paper establishes the well-posedness of the nonlinear Schrödinger equation associated with the twisted Laplacian on complex space in the critical case, extending previous subcritical results through a truncation approach.

Contribution

It proves well-posedness for the critical nonlinear Schrödinger equation with the twisted Laplacian, using a truncation method to handle the nonlinearities at the critical exponent.

Findings

Established well-posedness in the critical case.

Extended previous subcritical results to the critical case.

Used truncation method to pass from truncated to original problem.

Abstract

We prove well-posedness of solution to the nonlinear Schr\"{o}dinger equation associated to the twisted Laplacian on $\C^n$ for a general class of nonlinearities including power type with subcritical case $0\leq \alpha<\frac{2}{n-1}$, see Ratnakumar, Sohani (J. Funct. Anal. 2013). In this paper, we consider critical case $\alpha=\frac{2}{n-1}$ with $n\geq 2$. Our approach is based on truncation of the given nonlinearity $G$, which is used by Cazenave Weissler (1989). We obtain solution for the truncated problem. We obtain solution to the original problem by passing to the limit.