  We report an analytical study of the vibrational spectrum of the simplest
model of jamming, the soft perceptron. We identify two distinct classes of soft
modes. The first kind of modes are related to isostaticity and appear only in
the close vicinity of the jamming transition. The second kind of modes instead
are present everywhere in the glass phase and are related to the hierarchical
structure of the potential energy landscape. Our results highlight the
universality of the spectrum of normal modes in disordered systems, and open
the way towards a detailed analytical understanding of the vibrational spectrum
of low-temperature glasses.

Disordered Systems and Neural Networks;Soft Condensed Matter;Statistical Mechanics