Sturm-Liouville problems on graphs with Robin boundary conditions
Yuri Latushkin, Vyacheslav Pivovarchik, Alesia Supranovych
TL;DR
This work investigates Sturm-Liouville problems on graphs under Robin-Kirchhoff boundary conditions, developing a determinant-based framework to encode spectral data via the characteristic function \\phi(\\lambda,b_1,...,b_p). The authors prove a key expansion \\phi(\\lambda,b_1,...,b_p) = \\phi(\\lambda,0,...,0) + \\sum_i b_i \\phi_i(\\lambda) + \\sum_{i<j} b_i b_j \\phi_{ij}(\\lambda) + \\cdots + (\\prod_i b_i) \\phi_{12...p}(\\lambda) and relate these to Dirichlet-standard auxiliary problems, with explicit forms for equi-length graphs. For trees and zero potential, they derive the eigenvalue sequences and refined asymptotics, showing how the first-order correction depends on the sum of Robin coefficients. They then establish an inverse problem result: given $2^p-1$ distinct Robin eigenvalues, the Robin parameters $b_i$ are uniquely recoverable by solving a linear-draction system derived from the determinant expansion, aided by sine-type function analysis to guarantee the existence of a suitable eigenvalue. The findings advance spectral and inverse problem techniques for quantum graphs and may inform design of boundary-control in networked Sturm-Liouville systems.
Abstract
We study characteristic functions and describe asymptotics of the eigenvalues for the spectral Sturm-Liouville problem on graphs equipped with Robin-Kirhhoff boundary conditions. Also, we show how to recover the coefficients in the Robin conditions for the quantum graphs provided the shape of the graphs and some Robin eigenvalues are known.