Degenerate operators on the half-line

We study elliptic and parabolic problems governed by the singular elliptic operators $$y^{\alpha}\left(D_{yy}+\frac{c}{y}D_y\right)-V(y),\qquad\alpha \in\mathbb R$$ in $\mathbb R_+$, where $V$ is a potential having non-negative real part.