  The spectral properties of $su(2)$ Hamiltonians are studied for energies near
the critical classical energy $\epsilon_c$ for which the corresponding
classical dynamics presents hyperbolic points (HP). A general method leading to
an algebraic relation for eigenvalues in the vicinity of $\epsilon_c$ is
obtained in the thermodynamic limit, when the semi-classical parameter
$n^{-1}=(2s)^{-1}$ goes to zero (where $s$ is the total spin of the system).
Two applications of this method are given and compared with numerics. Matrix
elements of observables, computed between states with energy near $\epsilon_c$,
are also computed and shown to be in agreement with the numerical results.
