Spectre du Laplacien singulier associ\'e aux m\'etriques canoniques sur ${\mathbb{P}}^1$

We construct a singular differential operator attached to a class of singular metrics on the line bundles over the complex projective space, ${\mathbb{P}}^1$. This operator extends the classical notion of the generalized Laplacian. We prove that this operator admits a infinite, discrete and positive spectrum. We compute it explicitly.