On a measure of the complexity of nuclear states and the onset of chaos

Through the S-matrix of the nuclear system, an effective nonhcrmitean hamiltonian is introduced. A matrix which measures the complexity of nuclear states from an E2 sum rule is introduced, Numerical calculations for the SU ( 3 )-U ( 5 ) transition of the interacting boson model are performed. The transition from order to chaos in nuclear spectra is discussed in terms of a participation index. A transition from order to chaos is obtained when the SU ( 3 ) symmetry is broken, in agreement with the nearest neighbour distributions, but fails to reflect the order corresponding to the U ( 5 ) symmetry. The transition from order to chaos in nuclear spectra is an intriguing question. A convenient measure of chaotic behaviour is given by the distribution of the average level distances between nearcst neighbours with the same spin and parity [ I-8 ]. Recently also a number of investigations of the behaviour of B (E2) values for regular and chaotic systems havc been made [ 9-12 ]. There have been attempts of establishing wave-function related measures of quantum chaos [13-16]. It is the purposc of this letter to introduce a quantitative measure for the degree of complexity of nuclear states and to study its behaviour for the transition from order to chaos. This measure uses the decay properties of nuclear states. It is based on the idea that a level has a complicated structure if it is connected with many other levels by strong decay width whereas it has a simple structure if it is connccted with only one or at most a few levels by strong decay widths. Due to their decay, nuclear levels are resonances rather than stationary states and are described by an effective hamiltonian which is in general nonhermitean. A convenient way to introduce an effective nonhermitean hamiltonian is through a suitable form of the S-matrix of the system. Such a form has been used by Mahaux and Weidenmiiller [ 17 ] and others [ 13,18-20 ]. The effective hamiltonian (which is obtaincd from this form of the S-matrix ) is