A class of non-weight modules over the Schr\"{o}dinger-Virasoro algebras

We construct and classify the free $U(\mathbb{C}L_0\oplus \mathbb{C}M_0\oplus\mathbb{C}Y_0)$-modules of rank $1$ over the Schr\"{o}dinger-Virasoro algebra $\mathfrak{sv}(s)$ for $s=0$.Moreover, we show that the class of free $U(\mathbb{C}L_0\oplus\mathbb{C}M_0)$-modules of rank $1$ over the Schr\"{o}dinger-Virasoro algebra $\mathfrak{sv}(s)$ for $s=\frac{1}{2}$ is nonexistent.