Green's functions for Sturm-Liouville problems on directed tree graphs

Let $\mathrm{\Gamma }$ be geometric tree graph with $m$ edges and consider the second order Sturm-Liouville operator $\text{L}[u]=(-p{u}^{\mathrm{\prime }}{)}^{\mathrm{\prime }}+qu$ acting on functions that are continuous on all of $\mathrm{\Gamma }$, and twice continuously differentiable in the interior of each edge. The functions $p$ and $q$ are assumed uniformly continuous on each edge, and $p$ strictly positive on $\mathrm{\Gamma }$. The problem is to find a solution $f:\mathrm{\Gamma }\to \mathbb{R}$ to the problem $\text{L}[f]=h$ with $2m$ additional conditions at the nodes of $\mathrm{\Gamma }$. These node conditions include continuity at internal nodes, and jump conditions on the derivatives of $f$ with respect to a positive measure $\rho$. Node conditions are given in the form of linear functionals ${\text{l}}_1,\mathrm{.}\mathrm{.}\mathrm{.},{\text{l}}_{2m}$ acting on the space of admissible functions. A novel formula is given for the Green’s function $G:\Gamma\times \Gamma \to \R$ associated to this problem. Namely, the solution to the semi-homogenous problem $\text{L}[f]=h$, ${\text{l}}_i[f]=0$ for $i=1,\mathrm{.}\mathrm{.}\mathrm{.},2m$ is given by $f(x)={\int }_{\mathrm{\Gamma }}G(x,y)h(y)\text{ud}\rho$.