Spectrum of p-adic linear differential equations II: Variation of the spectrum

Abstract

This paper explores the spectrum of $p$-adic linear differential equations, extending previous findings to quasi-smooth Berkovich curves. It focuses on the relationship between the spectrum and the radii of convergence by investigating the continuity and variation of the spectrum for these equations. The study reveals that the spectrum can be influenced by the controlling graph of the radii of convergence and is continuous on the skeleton of annuli. Furthermore, the paper demonstrates that approximating the connection allows for an accurate estimation of spectral radii of convergence. Key results include a refined decomposition theorem with respect to the spectrum and its extension to neighborhoods of specific points. The work builds upon earlier research and introduces new approaches for understanding the spectral properties of differential equations defined over quasi-smooth curves.