  We are dealing with some spectral properties of a phase space localization
operator PR corresponding to the indicator function of a disk of radius R < 1.
The localization procedure is achieved with respect to a set of negative
binomial states (NBS) labeled by points of the complex unit disk D and
depending on a parameter 2B > 1. We derive a formula expressing PR as function
of the pseudo harmonic oscillator whose potential function depends on B. The
phase space content outside the localization domain is estimated in terms of
the photon counting probability distribution associated with the NBS. By using
the coherent states transform attached to NBS, we transfer the action of the
operator PR to a Bergman space AB(D) of analytic functions on D satisfying a
growth condition depending on B and we explicitly give its integral kernel
whose limit as R goes to 1 coincides with the reproducing kernel of AB(D). This
leads to a natural generalization of this Hilbert space with respect to the
parameter R.
