On the Spectral properties of Multi-branes, M2 and M5 branes

In this note we summarize some of the properties found in several papers. We characterize spectral properties of the quantum mechanical hamiltonian of theories with fermionic degrees of freedom beyond semiclassical approximation. We obtain a general class of bosonic polynomial potentials for which the Schr\"oedinger operator has a discrete spectrum. This class includes all the scalar potentials in membrane, 5-brane, p-branes, multiple M2 branes, BLG and ABJM theories. We also give a sufficient condition for discreteness of the spectrum for supersymmmetric and non supersymmetric theories with a fermionic contribution. We characterize then the spectral properties of different theories: the BMN matrix model, the supermembrane with central charges and a bound state of $N$ D2 with $m$ D0. We show that, while the first two models have a purely discrete spectrum with finite multiplicity, the latter has a continuous spectrum starting from a constant given in terms of the monopole charge.