Spin number coherent states and the problem of two coupled oscillators

From the definition of the standard Perelomov coherent states we introduce the Perelomov number coherent states for any $su(2)$ Lie algebra. With the displacement operator we apply a similarity transformation to the $su(2)$ generators and construct a new set of operators which also close the $su(2)$ Lie algebra, being the Perelomov number coherent states the new basis for its unitary irreducible representation. We apply our results to obtain the energy spectrum, the eigenstates and the partition function of two coupled oscillators. We show that the eigenstates of two coupled oscillators are the $SU(2)$ Perelomov number coherent states of the two dimensional harmonic oscillator with an appropriate choice of the coherent state parameters.