Random-matrix approach to RPA equations. I

We study the RPA equations in their most general form by taking the matrix elements appearing in the RPA equations as random. This yields either a unitary or an orthogonally invariant random-matrix model that does not appear in the Altland-Zirnbauer classification. The average spectrum of the model is studied with the help of a generalized Pastur equation. Two independent parameters govern the behaviour of the system: the strength α2 of the coupling between positive- and negative-energy states and the distance between the origin and the centers of the two semicircles that describe the average spectrum for α2 = 0, the latter measured in units of the equal radii of the two semicircles. With increasing α2, positive- and negative-energy states become mixed and ever more of the spectral strength of the positive-energy states is transferred to those at negative energy, and vice versa. The two semicircles are deformed and pulled toward each other. As they begin to overlap, the RPA equations yield non-real eigenvalues: The system becomes unstable. We determine analytically the critical value of the strength for the instability to occur. Several features of the model are illustrated numerically.