  We study a set of truncated matrices, given by Smith~\cite{Smith2005}, in
connection to an identification criterion for the ground state in our proposed
quantum adiabatic algorithm for Hilbert's tenth problem. We identify the origin
of the trouble for this truncated example and show that for a suitable choice
of some parameter it can always be removed. We also argue that it is only an
artefact of the truncation of the underlying Hilbert spaces, through showing
its sensitivity to different boundary conditions available for such a
truncation. It is maintained that the criterion, in general, should be
applicable provided certain conditions are satisfied. We also point out that,
apart from this one, other criteria serving the same identification purpose may
also be available.
