  The primary objective of this paper is to generalize the results of
[arXiv:2111.03548] to the case of quasi-smooth Berkovich curves by establishing
a connection between the spectrum and the radii of convergence. To achieve
this, we investigate the continuity and variation of the spectrum of a p-adic
linear differential equation. Our findings demonstrate that the spectrum can be
governed by any controlling graph of the radii of convergence. Furthermore, by
analyzing the shape of the spectrum, we prove that approximating the connection
enables an accurate estimation of spectral radii of convergence. As a result,
we obtain a decomposition theorem with respect to the spectrum for differential
equations defined over a quasi-smooth curve at a generic point x. This
decomposition refines the one provided by the spectral radii of convergence.
Previous works [Dwork, Robba (Trans. Am. Math. Soc. 231.1 (1-46));
arXiv:1308.0859] have shown that decomposition with respect to the spectral
radii can be extended to a neighborhood of x. We prove that this can also hold
for the decomposition with respect to the spectrum.
