eigenvalue of the magnetic Laplacian on the circle with potential $A$ (see section \[riemannian circle\]). If $\Sigma_2$ and $\Sigma_1$ are two concentric circles of respective lengths $L$ and $L_{\epsilon} \to L$, the domain is a thin annulus with $\lambda_1 \to \dfrac{4\pi^2}{L^2} d(\Phi^A,{\bf Z})^2$ which shows that our estimate is sharp.

Our aim is to use these estimates on cylinders as a basis stone in order to study the same type of questions on compact surfaces of higher genus.

Proof of the main theorem
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Preliminary facts and notation {#preliminary}
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First, we recall the variational definition of the spectrum. Let $\Omega$ be a compact manifold with boundary and $\Delta_A$ the magnetic Laplacian with Neumann boundary conditions. One verifies that $$\int_{\Omega}(\Delta_Au)\bar u=\int_{\Omega}{\lvert{\nabla^Au}\rvert}^2,$$ and the associated quadratic form is then $$Q_A(u)=\int_{\Omega}{\lvert{\nabla^Au}\rvert}^2.$$ The usual variational characterization gives:

$$\lambda_1(\Omega,A)= \min\Big\{ \frac{Q_A(u)}{\Vert u\Vert^2}:\ u\in C^{1}(\Omega,\mathbb C) / \{0\}\Big\}$$

The following proposition (which is well-known) expresses the [*gauge invariance*]{} of the spectrum of the magnetic Laplacian.

\[basic facts\] a The spectrum of $\Delta_A$ is equal to the spectrum of $\Delta_{A+d\phi}$ for all smooth real valued functions $\phi$; in particular, when $A$ is exact, the spectrum of $\Delta_A$ reduces to that of the classical Laplace-Beltrami operator acting 