Quasi-Invariance of Gaussian Measures Transported by the Cubic NLS with Third-Order Dispersion on $\mathbf{T}$

We consider the Nonlinear Schr"odinger (NLS) equation and prove that the Gaussian measure with covariance $(1-{\mathrm{\partial }}_x^2{)}^{-\alpha }$ on $L^2(\mathbf T)$isquasi-invariantfortheassociatedflowfor$\alpha>1/2$. This is sharp and improves a previous result obtained in \cite{OTT} where the values $\alpha >3\mathrm{/}4$ were obtained. Also, our method is completely different and simpler, it is based on an explicit formula for the Radon-Nikodym derivative. We obtain an explicit formula for this latter in the same spirit as in \cite{Cruz1} and \cite{Cruz2}. The arguments are general and can be used to other Hamiltonian equations.