Unitary random-matrix ensemble with governable level confinement

A family of unitary alpha ensembles of random matrices with governable confinement potential V (x) similar to \x\(alpha) 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 (alpha > 1) and border (alpha = 1) level confinement. It is shown that the density of states is a smooth function for alpha > 1, and has a well-pronounced peak at the band center for alpha less than or equal to 1. The case of border level confinement associated with transition point alpha = 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 alpha = 1.