Nearest-neighbor distributions and tunneling splittings in interacting many-body two-level boson systems

We study the nearest-neighbor distributions of the k -body embedded ensembles of random matrices for n bosons distributed over two-degenerate single-particle states. This ensemble, as a function of k , displays a transition from harmonic-oscillator behavior (k=1) to random-matrix-type behavior (k=n) . We show that a large and robust quasidegeneracy is present for a wide interval of values of k when the ensemble is time-reversal invariant. These quasidegenerate levels are Shnirelman doublets which appear due to the integrability and time-reversal invariance of the underlying classical systems. We present results related to the frequency in the spectrum of these degenerate levels in terms of k and discuss the statistical properties of the splittings of these doublets.