Note that the Laplace-Bessel differential operator
Why $v>-\frac{1}{2}$ in the Laplace-Bessel differential operator
$$\begin{equation*} l_{v}=\frac{\mathrm{d}^{2}}{\mathrm{~d} t^{2}}+\frac{2 v+1}{t} \frac{\mathrm{d}}{\mathrm{d} t} ; \quad v \in \mathbb{R}, v>-\frac{1}{2} \tag{1.1} \end{equation*} $$