therein, also for Neumann boundary condition). The case when the potential $A$ is a closed $1$-form is particularly interesting from the physical point of view (Aharonov-Bohm effect), and also from the geometric point of view. For Dirichlet boundary conditions, there is a serie of papers for domains with a pole, when the pole approaches the boundary (see [@AFNN; @NNT] and the references therein). Last but not least, there is a Aharonov-Bohm approach to the question of nodal and minimal partitions, see chapter 8 of [@BH].

For *Neumann boundary conditions*, we refer in particular to the paper [@HHHO], where the authors study the multiplicity and the nodal sets corresponding to the ground state ${\lambda}_1$ for non-simply connected planar domains with harmonic potential (see the discussion below).

Let us also mention the recent article [@LLPP] (chapter 7) where the authors establish a *Cheeger type inequality* for ${\lambda}_1$; that is, they find a lower bound for ${\lambda}_1(\Delta_A)$ in terms of the geometry of $\Omega$ and the potential $A$. In the preprint [@ELMP], the authors approach the problem via the Bochner method.

Finally, in a more general context (see [@BBC]) the authors establish a lower bound for ${\lambda}_1(\Delta_A)$ in terms of the *holonomy* of 