We obtain the precise condition on the potentials of Yang-Mills theories in
0+1 dimensions and D0 brane quantum mechanics ensuring the discretness 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.Comment: 12pages, Latex, minor changes, more explanatory comment
