Spectral analysis for a degenerate tree and applications

The energy in network of strings subject to the feedback in a root is decreasing to zero if the network is non-degenerate and this decrease is never exponential. We study in this paper the case of degenerate network (i.e. the ratio of two lengths of strings is a rational number) with a feedback in the root of the network. Under this assumption, we determine the energy limit and we identify the best rate of decay with the spectral abscissa of the underlying semi-group generator. We identify the spectral abscissa as in terms of the largest modulus of the roots of a polynomial.