Remarks on Regular Approximations to the Robin Aharonov-Bohm Hamiltonian

In the space ${\mathbb{R}}^3$, for Robin parameter $L>0$, it is shown that there is a family of Schr"odinger operators with penetrable toroidal solenoid and variable conductivity that approximates the magnetic Aharonov-Bohm operator with a Robin boundary condition at the solenoid (border). It is also shown that approximations via smooth potentials and then a barrier in the solenoid interior give Dirichlet boundary conditions. The approximations are in the strong resolvent sense and obtained through the $\mathrm{\Gamma }$-convergence technique, and they hold for the more general setting of smooth, closed and compact surfaces and continuous and bounded magnetic potentials.