on $\Omega$ and for all $u\in C^{\infty}(\Omega,\bf C)$; here $\nabla$ is the Levi-Civita connection assocated to the metric $g$ of $\Omega$. The operator $$\label{magnetic laplacian}
\Delta_A=(\nabla^A)^{\star}\nabla ^A$$ is called the [*magnetic Laplacian*]{} associated to the magnetic potential $A$, and the smooth two form $$B=dA$$ is the associated [*magnetic field*]{}. We will consider Neumann magnetic conditions, that is: $$\label{mneumann}
\nabla^A_Nu=0\quad\text{on}\quad{\partial}\Omega,$$ where $N$ denotes the inner unit normal. Then, it is well-known that $\Delta_A$ is self-adjoint, and admits a discrete spectrum $$0\le {\lambda}_1(\Delta_A)\le {\lambda}_2(\Delta_A) \le ... \to \infty.$$

The above is a particular case of a more general situation, where $E\to M$ is a complex line bundle with a hermitian connection $\nabla^E$, and where the magnetic Laplacian is defined as $\Delta_E=(\nabla^E)^{\star}\nabla ^E$.

The spectrum of the magnetic Laplacian is very much studied in analysis (see for example [@BDP] and the references therein) and in relation with physics. For *Dirichlet boundary conditions*, lower estimates of its fundamental tone have been worked out, in particular, when $\Omega$ is a planar domain and $B$ is the constant magnetic field; that is, when the function $\star B$ is constant on $\Omega$ (see for example a Faber-Krahn type inequality in [@Er1] and the recent[@LS] and the references 