Potential-free $L^1$-estimates for positivity-preserving Riesz transform related to Schrödinger operator in dimension one

Let $V\geq 0$ be a locally integrable function on $\mathbb R$. Consider the Schrödinger operator $L=-\frac{d^2}{dx^2} +V$. We prove that for all $0<a\leq 1$, there is a constant $C_a$, independent of $V$, such that the Riesz transform type operator $V^aL^{-a}$ is bounded on $L^1(\mathbb R)$ and $\| V^aL^{-a}f\|_{L^1(\mathbb R)}\leq C_a\|f\|_{L^1(\mathbb R)}$.