We consider the Nonlinear Schrödinger (NLS) equation with third-order dispersion and prove that the Gaussian measure with covariance ${(1-{\partial }_x^2)}^{-\alpha }$ on $L^2(\mathbf{T})$ is quasi-invariant for the associated flow for $\alpha >1/2$. This is sharp and improves a previous result obtained in [20] where the values $\alpha >3/4$ 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 [4] and [5]. The arguments are general and can be used to other Hamiltonian equations.

中文翻译：

---

T上三次色散的三次NLS传输高斯测度的拟不变性

我们考虑了具有三阶色散的非线性薛定（（NLS）方程，并证明了具有协方差的高斯测度 ${（\mathrm{1个}-{\partial }_X^{\mathrm{2个}}）}^{-\alpha }$ 上 ${\mathrm{大号}}^{\mathrm{2个}}（\mathbf{Ť}）$ 对于的相关流是准不变的 $\alpha >\mathrm{1个}/\mathrm{2个}$。这很清晰，可以改善先前在[20]中获得的结果，其中$\alpha >3/4$获得了。而且，我们的方法是完全不同且更简单的，它基于Radon-Nikodym导数的显式公式。我们以与[4]和[5]相同的精神为后者获得一个明确的公式。这些参数是通用的，可用于其他哈密顿方程。