Approximation of arbitrary vertex couplings by magnetic Schrödinger operators

Let $H_\varepsilon^\omega$ be the magnetic Schrödinger operator obtained by replacing the approximating graph by a network with fat edge width $\varepsilon$, replacing the $\delta$-couplings by constant potentials on segments of fat edge of length $\varepsilon$, and replacing the Laplacian on the added edges by a magnetic Laplacian. Set $d=\varepsilon^\alpha$, and let $H^\omega$ be the corresponding limiting operator on the graph. The vertex couplings are given by the conditions in

, with identification operator $J$. **Approximation conjecture.** If $\alpha>0$ is sufficiently small, the approximation result analogous to Theorem~ is valid in the described setting for any vertex coupling

with the same identification operator $J$. This would extend the established approximation results for $\delta$- and $\delta'_s$-couplings to arbitrary vertex couplings, including magnetic ones; the source does not provide a proof or a resolution of this proposed extension.