A phase space localization operator in negative binomial states

We study the spectral properties of the phase space localization operator \(P_{R}\), defined by the indicator function of a disk \(D_{R}\) of radius \(R<1.\) The localization is performed using a family of negative binomial states (NBS), labeled by points z in the unit disk \(\mathbb {D}\) and parameterized by \(\nu > {\frac{1}{2}}\). These states are intrinsically connected to the 1D pseudo-harmonic oscillator (PHO) through superpositions of its eigenfunctions. The superposition coefficients form an orthonormal basis of a weighted Bergman space \(\mathcal {A}^{\nu }\left( \mathbb {D}\right) \), which also emerges as the eigenspace of a 2D Schrödinger operator with a magnetic field (proportional to \(\nu \)) corresponding to the lowest hyperbolic Landau level. The eigenvalues \(\lambda _{j}^{\nu ,R}\) of \(P_{R}\) were obtained via a discrete spectral resolution within a shared eigenbasis for \(P_{R}\) and the PHO. By using these eigenvalues we obtain a closed-form expression for the variance of the particle count in \(D_{R}\) under the determinantal point process (DPP) defined by the weighted Bergman kernel. Beyond \(D_{R}\), the phase space content of \(P_{R}\) was estimated via the NBS photon-counting distribution, revealing a non-zero residual contribution. Using the coherent state transform associated with NBS, we mapped \(P_{R}\) to \(\mathcal {A}^{\nu }\left( \mathbb {D}\right) \) and we derive its explicit integral kernel \(K_{\nu ,R}\left( z,w\right) \), which converges to the Bergman kernel \(K_{\nu }\left( z,w\right) \) as \(R\rightarrow 1\).