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 $Ł[u]=(-p{u}^{\prime }{)}^{\prime }+qu$ 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 uniformly 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 $Ł[f]=h$ 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 ${ł}_1,...,{ł}_{2m}$ 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 $Ł[f]=h$, ${ł}_i[f]=0$ for $i=1,...,2m$ is given by $f(x) = \int_ΓG(x,y) h(y) \ud ρ$.

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