Random-matrix approach to RPA equations. I

摘要

We study the RPA equations in their most general form by takingthe 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 classi-fication. The average spectrum of the model is studied with thehelp of a generalized Pastur equation. Two independent parame-ters govern the behaviour of the system: the strength a2 of the cou-pling between positive- and negative-energy states and thedistance between the origin and the centers of the two semicirclesthat describe the average spectrum forα~2=0, the latter measuredin units of the equal radii of the two semicircles. With increasing2, positive- and negative-energy states become mixed and evermore of the spectral strength of the positive-energy states is trans-ferred to those at negative energy, and vice versa. The two semicir-cles are deformed and pulled toward each other. As they begin tooverlap, the RPA equations yield non-real eigenvalues: The systembecomes unstable. We determine analytically the critical value ofthe strength for the instability to occur. Several features of themodel are illustrated numerically.