Intrinsic moment of inertia of membranes as bounds for the mass gap of Yang-Mills theories

We obtain the precise condition on the potentials of Yang-Mills theories in 0+1 dimensions and D0-brane quantum mechanics ensuring the discreteness of the spectrum. It is given in terms of a moment of inertia of the membrane. From it we obtain a bound for the mass gap of any D+1 Yang-Mills theory in the slow-mode regime. In particular we analyze the physical case D=3. The quantum mechanical behavior of the theories, concerning its spectrum, is determined by harmonic oscillators with frequencies given by the inertial tensor of the membrane. We find a class of quantum mechanic potential polynomials of any degree, with classical instabilities that at quantum level have purely discrete spectrum.