On discrete graphs with purely discrete spectrum, the summed eigenvalue difference between the Laplacian with and without a potential equals the weighted sum of the potential.
A modified local Weyl law and spectral comparison results for $\delta'$-coupling conditions
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abstract
We study Schr\"odinger operators on compact finite metric graphs subject to $\delta'$-coupling conditions. Based on a novel modified local Weyl law, we derive an explicit expression for the limiting mean eigenvalue distance of two different self-adjoint realisations on a given graph. Furthermore, using this spectral comparison result, we also study the limiting mean eigenvalue distance comparing $\delta'$-coupling conditions to so-called anti-Kirchhoff conditions, showing divergence and thereby confirming a numerical observation in [arXiv:2212.12531]. .
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Spectral comparison results for Laplacians on discrete graphs
On discrete graphs with purely discrete spectrum, the summed eigenvalue difference between the Laplacian with and without a potential equals the weighted sum of the potential.