$ at all points of $\Omega$. This implies that $$\label{uoft}
u=u(t)$$ depends only on $t$.

### Third step : spectrum of circles and Riemannian products {#sectioncircle}

In this section, we give an expression for the eigenfunctions of the magnetic Laplacian on a circle with a Riemannian metric $g$ and a closed potential $A$. Of course, we know that any metric $g$ on a circle is always isometric to the canonical metric $g_{\rm can}=\,dt^2$, where $t$ is arc-length. But our problem in this proof is to reconstruct the global metric of the cylinder and to show that it is a product, and we cannot suppose a priori that the restricted metric of each level set of $\psi$ is the canonical metric. The same is true for the restricted potential: we know that it is Gauge equivalent to a potential of the type $a\,dt$ for a scalar $a$, but we cannot suppose a priori that it is of that form.

We refer to Appendix \[riemannian circle\] for the complete proof of the following fact.

\[circle\] Let $(M,g)$ be the circle of length $L$ endowed with the metric $
g=\theta(t)^2\,dt^2
$ where $t\in [0,L]$ and $\theta(t)$ is a positive function, periodic of period 