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The first aim of this paper is to generalize the results of arXiv:2111.03548to the case of quasi-smooth curves, which consist on providing the link betweenthe spectrum and the radii of convergence. In order to do that, we study thecontinuity and the variation of the spectrum of a differential equation, andprove that it can be controlled by the controlling graph of the radii ofconvergence. Besides, using the shape of the spectrum, we prove that byapproximating a connection we approximate the radii of converge. All this leadto a decomposition theorem with respect to the spectrum, for differentialequations defined over a quasi-smooth curve. Moreover, this decomposition isfiner than the one provided by the radii of convergence.

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