Pseudo-Hermitian β-Hermite Ensemble with Non-positive Metric

A β-Hermite ensemble which is pseudo-Hermitian with respect to a non-positive metric is introduced. It is shown that the introduction of the non-positive metric splits the spectrum in a fraction of eigenvalues that remains at the real axis, while the fraction of complex conjugate eigenvalues, with a negligible real part, goes to the imaginary axis. Therefore, it is like there is, practically, in the imaginary axis a spectrum dual to the real one at the real axis or, in physical terms, emission/absorption spectra. It is shown that the pseudo-Hermitian condition restricts the possible form the tridiagonal matrices may have to satisfy it with a nonpositive matric.