Unitary random-matrix ensemble with governable level confinement

A family of unitary α ensembles of random matrices with governable confinement potential V(x)∼‖x[Formula Presented] is studied employing exact results of the theory of nonclassical orthogonal polynomials. The density of levels, two-point kernel, locally rescaled two-level cluster function, and smoothed connected correlations between the density of eigenvalues are calculated for strong (α>1) and border (α=1) level confinement. It is shown that the density of states is a smooth function for α>1, and has a well-pronounced peak at the band center for α≤1. The case of border level confinement associated with transition point α=1 is reduced to the exactly solvable Pollaczek random-matrix ensemble. Unlike the density of states, all the two-point correlators remain (after proper rescaling) universal down to and including α=1.