RESUMO
We investigate the level density σ(x) and the level-spacing distribution p(s) of random matrices M = AF ≠ M{}, where F is a (diagonal) inner product and A is a random, real, symmetric or complex, Hermitian matrix with independent entries drawn from a probability distribution q(x) with zero mean and finite higher moments. Although not Hermitian, the matrix M is self-adjoint with respect to F and thus has purely real eigenvalues. We find that the level density σ{F}(x) is independent of the underlying distribution q(x) and solely characterized by F, and therefore generalizes the Wigner semicircle distribution σ{W}(x). We find that the level-spacing distributions p(s) are independent of q(x), and are dependent upon both the inner product F and whether A is real or complex, and therefore generalize the Wigner surmise for level spacing. Our results suggest F-dependent generalizations of the well-known Gaussian Orthogonal Ensemble and Gaussian Unitary Ensemble classes.
