Green's functions for sturm-liouville problems on directed tree graphs

Let $\Gamma$ be geometric tree graph with $m$ edges and consider the second order Sturm-Liouville operator $\mathcal{L}[u]=(-pu')'+qu$ acting on functions that are continuous on all of $\Gamma$, and twice continuously differentiable in the interior of each edge. The functions $p$ and $q$ a...