On the Analytic Langlands Corrrespondence for $\operatorname{PGL}_2$ in Genus 0 with Wild Ramification

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.