A modified local Weyl law and spectral comparison results for ${\delta }^{\mathrm{\prime }}$-coupling conditions

We study Schrödinger operators on compact finite metric graphs subject to ${\delta }^{\mathrm{\prime }}$-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 ${\delta }^{\mathrm{\prime }}$-coupling conditions to so-called anti-Kirchhoff conditions, showing divergence and thereby confirming a numerical observation in [arXiv:2212.12531]. .