Comparing the spectrum of Schrödinger operators on quantum graphs

Abstract

We study Schrödinger operators on compact finite metric graphs subject to $\delta$-coupling and standard boundary conditions. We compare the $n$-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. obtained for domains in ${\mathbb{R}}^2$ to the setting of quantum graphs. This also leads to a generalization of related results previously and independently obtained in [arXiv:2212.09143] and [arXiv:2212.12531] for metric graphs. In addition, based on our main result, we introduce some notions of circumference for a (quantum) graph which might prove useful in the future.

10 pages

Cited by in corpus (4)

• A note on Ambarzumian's theorem for quantum graphs
• Some spectral comparison results on infinite quantum graphs
• A modified local Weyl law and spectral comparison results for ${\delta }^{\prime }$-coupling conditions
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