  In this work we consider the existence and uniqueness of the ground state of
the regularized Hamiltonian of the Supermembrane in dimensions $D= 4,\,5,\,7$
and 11, or equivalently the $SU(N)$ Matrix Model. That is, the 0+1 reduction of
the 10-dimensional $SU(N)$ Super Yang-Mills Hamiltonian. This ground state
problem is associated with the solutions of the inner and outer Dirichlet
problems for this operator, and their subsequent smooth patching (glueing) into
a single state. We have discussed properties of the inner problem in a previous
work, therefore we now investigate the outer Dirichlet problem for the
Hamiltonian operator. We establish existence and uniqueness on unbounded
valleys defined in terms of the bosonic potential. These are precisely those
regions where the bosonic part of the potential is less than a given value
$V_0$, which we set to be arbitrary. The problem is well posed, since these
valleys are preserved by the action of the $SU(N)$ constraint. We first show
that their Lebesgue measure is finite, subject to restrictions on $D$ in terms
of $N$. We then use this analysis to determine a bound on the fermionic
potential which yields the coercive property of the energy form. It is from
this, that we derive the existence and uniqueness of the solution. As a
by-product of our argumentation, we show that the Hamiltonian, restricted to
the valleys, has spectrum purely discrete with finite multiplicity. Remarkably,
this is in contrast to the case of the unrestricted space, where it is well
known that the spectrum comprises a continuous segment. We discuss the relation
of our work with the general ground state problem and the question of
confinement in models with strong interactions.
