  We continue to develop the analytic Langlands program for curves over local
fields initiated in arXiv:1908.09677, arXiv:2103.01509 following a suggestion
of Langlands and a work of Teschner. Namely, we study the Hecke operators
introduced in arXiv:2103.01509 in the case of P^1 over a local field with
parabolic structures at finitely many points for the group PGL(2). We establish
most of the conjectures of arXiv:1908.09677, arXiv:2103.01509 in this case.
