Be $S^1$ the unit circle and consider the following linear operator $\mathcal{L}$ defined as : $$\forall u:S^1 \rightarrow \mathbb{R} \ \ \ \mathcal{L}u(x) = u''(x) + A(x)u(x) $$
where $A :S^1 \rightarrow [0, +\infty[$.
From the Sturm-Liouville theory with periodic boundary conditions we know there exists a countable discrete set of eigenvalues $\{\lambda\}_{n \ge 0}$ which can be ordered in a decreasing way such that $\lambda_0$ > $\lambda_1$ > $\lambda_2$ >...
Is there a method to find the first eigenvalue $\lambda_0$ defined above as the maximum of the eigenvalues ?