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

TL;DR

This paper identifies a broad class of polynomial potentials with discrete spectra in Schrödinger operators, encompassing key theories like membrane, M2 branes, BLG, and ABJM, advancing non-perturbative analysis of these supersymmetric models.

Contribution

It introduces a general class of polynomial potentials ensuring discrete spectra, linking spectral theory with membrane and M-brane theories, and provides a proof of spectrum discreteness.

Findings

Established a class of polynomial potentials with discrete spectra.

Connected spectral properties to membrane and M-brane theories.

Provided a proof of spectrum discreteness for these potentials.

Abstract

We obtain a general class of polynomials for which the Schrodinger 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 Schrodinger operators. This a a first step in order to analyze BLG and ABJM supersymmetric theories from a non-perturbative point of view.