¹Universidad Nacional de Colombia, Medellín, Colombia. Email: jmramirezo@unal.edu.co

Let Γ be geometric tree graph with m edges and consider the second order Sturm-Liouville operator L[u]=(-pu')'+qu acting on functions that are continuous on all of Γ, 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 Γ. The problem is to find a solution f:Γ → R to the problem L[f] = h with 2m additional conditions at the nodes of Γ. These node conditions include continuity at internal nodes, and jump conditions on the derivatives of f with respect to a positive measure ρ. Node conditions are given in the form of linear functionals \l₁,…,\l_2m acting on the space of admissible functions. A novel formula is given for the Green's function G:Γ\times Γ → R associated to this problem. Namely, the solution to the semi-homogenous problem L[f] = h, \l_i[f] =0 for i=1,…,2m is given by f(x) = \int_Γ G(x,y) h(y)\,dρ.

Key words: Problema Sturm-Liouville en grafo, función de Green.

Sea Γ un grafo tipo árbol con m aristas y considere el operador de Sturm-Liouville L[u]=(-pu')'+qu definido en el espacio de funciones continuas en Γ y continuamente diferenciables dos veces al interior de cada arista de Γ. Las funciones p y q se suponen continuas en cada arista, y p es estrictamente positiva en todo Γ. El problema consiste en hallar la solución f : Γ → R al problema dado por L[f] = h mas 2m condiciones en los nodos de Γ: en los nodos internos se especifican continuidad de f y condiciones de salto para las derivadas de f con respecto a una medida ρ. Estas condiciones de nodo se expresan en la forma de funcionales lineales \l₁,…,\l_2m actuando sobre el espacio de funciones admisibles para L. Se presenta una nueva fórmula para la función de Green G:Γ\times Γ → R asociada con este problema. Es decir, se expresa la solución del problema semi-homogéneo L[f] = h, \l_i[f] =0 para i=1,…,2m como f(x) = \int_Γ G(x,y) h(y)\,dρ.

Palabras clave: Sturm-Liouville problems on graphs, Green's function.

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