We consider supersymmetric matrix Hamiltonians. The existence of a zero-energy bound state, in particular for the $d=9$ model, is of interest in M-theory. While we do not quite prove its existence, we show that the decay at infinity such a state would have is compatible with normalizability (and hence existence) in $d=9$. Moreover, it would be unique. Other values of $d$, where the situation is somewhat different, shall also be addressed. The analysis is based on a Born-Oppenheimer approximation. This seminar...