Professor Joshua Feinberg: Density of eigenvalues in a quasi-Hermitian random matrix model: The case of indefinite metric
We discuss a model of random matrices which are quasi-hermitian with respect to a fixed deterministic metric B. This ensemble is comprised of N×N matrices H=AB, where A is a complex-Hermitian matrix drawn from a U(N)-invariant probability distribution (e.g., the GUE ensemble). For positive-definite B, the resulting spectrum is real, because H is similar to a Hermitian matrix. In this talk we shall concentrate on the average spectrum of this ensemble for indefinite-metric (analogous to the broken PT-symmetry phase), in which case H is no-longer similar to a Hermitian matrix, and therefore its spectrum becomes complex. We will present analytical and numerical results for this spectrum in the complex plane in the large-N limit, and explain its behavior as the number of negative eigenvalues (a finite fraction of N) of the metric B increases.