Let $H : \operatorname{dom}(H) \subset L^2(\Omega) \rightarrow L^2(\Omega)$, where $dom(H) \subset H^2(\Omega)$, $\Omega \subset \mathbb{R}$ should be a bounded open interval(so 1-d setting(!)) and $H$ be a self-adjoint Schrödinger operator $H = -\frac{d^2}{dx^2} + V$ and $V \in C^{\infty}$. Now assume that the Schrödinger equation has a ground-state solution $\psi$ (with $\psi > 0$- not sure if we actually need this, but it should definitely simplify quite something). By rescaling the potential, we can get that $H \psi = 0.$

Then we define $W(x) :=\frac{\psi'(x)}{\psi(x)}$ and closed(!) operators $A = \frac{d}{dx}+ W $ and $A^* = -\frac{d}{dx}+W$.

Now, my question is: Given this situation, are the domains of the operators $A,A^*$ fixed? Or is there at least a standard way to construct an appropriate domain of $A$ by using the domain of $H$ ?