Spectral analysis of polynomial potentials and its relation with ABJ/M-type theories

We obtain a general class of polynomial potentials for which the Schrödinger 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 provide a proof of the discreteness of the spectrum of the associated Schrödinger operators. This is the first step in order to analyze BLG and ABJM supersymmetric theories from a non-perturbative point of view.