Singular spectrum Laplacian associated with the canonical metric on \(\mathbb P^1\) and the generalized theory of Fourier-Bessel series (Q488015)

In this paper, the author considers a class of singular metrics (i.e., continuous but not \(C^\infty\)) on the line bundles over \(\mathbb{P}^1\), the complex projective space of dimension 1. On this space, he shows that one can associate a singular operator extending the classical construction of the Laplacian. Moreover, he shows that this Laplacian possesses a spectrum which can be computed explicitly: it is discrete, infinite and positive. These computations are used to define a zeta function and a holomorphic torsion associated with these singular metrics with applications to Arakelov geometry.