Pseudounitary symmetry and the Gaussian pseudounitary ensemble of random matrices

Employing the currently discussed notion of pseudo-Hermiticity, we define a pseudounitary group. Further, we develop a random matrix theory that is invariant under such a group and call this ensemble of pseudo-Hermitian random matrices the pseudounitary ensemble. We obtain exact results for the nearest-neighbor level-spacing distribution for (2 x 2) PT-invariant Hamiltonian matrices that have forms, approximately Sln(1/S) near zero spacing for three independent elements and approximately S for four independent elements. This shows a level repulsion in a marked distinction with an algebraic form S(beta) in the Wigner surmise. We believe that this paves the way for a description of varied phenomena in two-dimensional statistical mechanics, quantum chromodynamics, and so on.