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

Abstract

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 $q$ are assumed continuous on each edge, and $p$ strictly positive on $\Gamma$. The problem is to find a solution $f:\Gamma \to \mathbb{R}$ to the problem $\mathcal{L}[f] = h$ with $2m$ additional conditions at the nodes of $\Gamma$. 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 $\l_1,\ldots,\l_{2m}$ acting on the space of admissible functions. A novel formula is given for the Green's function $G:\Gamma\times \Gamma \to \mathbb{R}$ associated to this problem. Namely, the solution to the semi-homogenous problem $\mathcal{L}[f] = h$, $\l_i[f] =0$ for $i=1,\ldots,2m$ is given by $f(x) = \int_\Gamma G(x,y) h(y)\,d\rho$.