  By solving the Schr\"odinger equation one obtains the whole energy spectrum,
both the bound and the continuum states. If the Hamiltonian depends on a set of
parameters, these could be tuned to a transition from bound to continuum
states. The behavior of systems near the threshold, which separates
bound-states from continuum states, is important in the study of such
phenomenon as: ionization of atoms and molecules, molecule dissociation,
scattering collisions and stability of matter. In general, the energy is
non-analytic as a function of the Hamiltonian parameters or a bound-state does
not exist at the threshold energy. The overall goal of this chapter is to show
how one can predict, generate and identify new class of stable quantum systems
using large-dimensional models and the finite size scaling approach. Within
this approach, the finite size corresponds not to the spatial dimension but to
the number of elements in a complete basis set used to expand the exact
eigenfunction of a given Hamiltonian. This method is efficient and very
accurate for estimating the critical parameters, $\{\lambda_i\}$, for stability
of a given Hamiltonian, $H(\lambda_i)$. We present two methods of obtaining
critical parameters using finite size scaling for a given quantum Hamiltonian:
The finite element method and the basis set expansion method.
