Energy level statistics and localization in sparsed banded random matrix ensemble

The matrix elements of the quantum Hamiltonian, which corresponds to the classical KAM system, H=H 0 + epsilon V (where H 0 is integrable) in an energy ordered eigenbasis of H 0 are considered. Their statistical properties are comprised in the definition of the sparsed banded random matrix ensemble (SBRME). In analysing the spectral statistics of this ensemble the authors find the power law level repulsion and the level spacing distribution which is well described by the Brody distribution. The numerical study of SBRME shows two interesting scaling laws: (i) the connection between the scaling variable x= alpha b 3/2 ( alpha =mean diagonal increment, b=bandwidth) and the level repulsion parameter beta as deduced from the Brody distribution, and (ii) the connection between the same scaling variable x and an expression which contains two types of localization length (entropy and geometric). The universal aspects and the importance of SBRME for the Hamiltonian systems in the transition region between integrability and chaos are discussed.