Comparing the spectrum of Schrödinger operators on quantum graphs

Author:

Bifulco Patrizio,Kerner Joachim

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.

Funder

Deutsche Forschungsgemeinschaft

Publisher

American Mathematical Society (AMS)

Subject

Applied Mathematics,General Mathematics

Reference14 articles.

1. Asymptotic behaviour and numerical approximation of optimal eigenvalues of the Robin Laplacian;Antunes, Pedro Ricardo Simão;ESAIM Control Optim. Calc. Var.,2013

2. R. Band, H. Schanz, and G. Sofer, Differences between Robin and Neumann eigenvalues on metric graphs, arXiv:2212.12531, 2022.

3. Heat-kernel and resolvent asymptotics for Schrödinger operators on metric graphs;Bolte, Jens;Appl. Math. Res. Express. AMRX,2015

4. The heat kernel on the diagonal for a compact metric graph;Borthwick, David;Ann. Henri Poincar\'{e},2023

5. Die Grundlehren der mathematischen Wissenschaften, Band 132;Kato, Tosio,1966

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