Comparing the spectrum of Schrödinger operators on quantum graphs

Abstract

We study Schrödinger operators on compact finite metric graphs subject to δ \delta -coupling and standard boundary conditions. We compare the n n -th eigenvalues of those self-adjoint realizations and derive an asymptotic result for the mean value of deviations. By doing this, we generalize recent results from Rudnick et al. [Comm. Math. Phys. 388 (2021), pp. 1603–1635] obtained for domains in R 2 \mathbb {R}^2 to the setting of quantum graphs. This also leads to a generalization of related results previously and independently obtained by Sofer [Spectral curves of quantum graphs with δ s \delta _s type vertex conditions, arXiv:2212.09143, 2022] and Band et al. [Differences between Robin and Neumann eigenvalues on metric graphs, arXiv:2212.12531, 2022] for metric graphs. In addition, based on our main result, we introduce some surface measures for a (quantum) graph which might prove useful in the future.