  The analytic Langlands correspondence was developed by Etingof, Frenkel and
Kazhdan in arXiv:1908.09677, arXiv:2103.01509, arXiv:2106.05243,
arXiv:2311.03743. For a curve $X$ and a group $G$ over a local field $F$, in
the tamely ramified setting one considers the variety $\operatorname{Bun}_G$ of
stable $G$-bundles on $X$ with Borel reduction at a finite subset $S\subset X$
of points. On one side of this conjectural correspondence there are Hecke
operators on $L^2(\operatorname{Bun}_G)$, the Hilbert space of
square-integrable half-densities on $\operatorname{Bun}_G$; on the other side
there are certain opers with regular singularities at $S$. In this paper we
prove the main conjectures of analytic Langlands correspondence in the case $G
= \operatorname{PGL}_2$, $X=\mathbb{P}^1_{\mathbb{C}}$ with wild ramification,
i.e. when several points in $S$ are collided together.
