Given a hyperbolic surface $X$, one considers the multiset of lengths of closed primitive geodesics. This multiset is called the length spectrum $\mathcal{L}(X)$.

Question: is $\mathcal{L}(X)$ a spectrum of some (differential) operator? In what context does this operator naturally arise?

I hope that there are not only folklore arguments of a philosophical nature, but specific constructions leading to such a term.