The spectral properties of $su(2)$ Hamiltonians are studied for energies near the critical classical energy $\varepsilon_c$ for which the corresponding classical dynamics presents hyperbolic points (HP). A general method leading to an algebraic relation for eigenvalues in the vicinity of $\varepsilon_c$ 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 $\varepsilon_c$, are also computed and shown to be in agreement with the numerical results.