Ergodic Properties of Local Spectral Density for a Conservative System of Coupled Quantum States

The shape and the inverse participation ratio (IPR) of local spectral density (LSD) are studied for a generic isolated system of coupled quantum states, the Hamiltonian of which is represented by a band random matrix with the disordered leading diagonal. We find for the matrices with arbitrary small band that the lack of ergodicity for LSD can be associated with an exponential increase in IPR with the ratio $v/\Delta_c$ ($v$ - the root of mean square for off-diagonal matrix elements, $\Delta_c$ - the energy spacing between directly coupled basis states). Criterions specifying transition to localization and ergodicity for LSD are considered.