Random matrix theory for transition strengths: Applications and open questions

Abstract

Abstract. Embedded random matrix ensembles are generic models for describing statistical properties of finite isolated interacting quantum many-particle systems. A finite quantum system, induced by a transition operator, makes transitions from its states to the states of the same system or to those of another system. Examples are electromagnetic transitions (then the initial and final systems are same), nuclear beta and double beta decay (then the initial and final systems are different) and so on. Using embedded ensembles (EE), there are efforts to derive a good statistical theory for transition strengths. With m fermions (or bosons) in N meanfield single particle levels and interacting via two-body forces, we have with GOE embedding, the so called EGOE(1+2). Now, the transition strength density (transition strength multiplied by the density of states at the initial and final energies) is a convolution of the density generated by the mean-field one-body part with a bivariate spreading function due to the two-body interaction. Using the embedding U(N) algebra, it is established, for a variety of transition operators, that the spreading function, for sufficiently strong interactions, is close to a bivariate Gaussian. Also, as the interaction strength increases, the spreading function exhibits a transition from bivariate Breit-Wigner to bivariate Gaussian form. In appropriate limits, this EE theory reduces to the polynomial theory of Draayer, French and Wong on one hand and to the theory due to Flambaum and Izrailev for one-body transition operators on the other. Using spin-cutoff factors for projecting angular momentum, the theory is applied to nuclear matrix elements for neutrinoless double beta decay (NDBD). In this paper we will describe: (i) various developments in the EE theory for transition strengths; (ii) results for nuclear matrix elements for 130 Te and 136 Xe NDBD; (iii) important open questions in the current form of the EE theory.

Statistical Nuclear Spectroscopy with $q$-normal and bivariate $q$-normal distributions and $q$-Hermite polynomials

“…IV. SHELL MODEL ORBIT OCCUPANCIES USING q-NORMAL AND q-HERMITE POLYNOMIALS Shell model orbit occupancies are measurable and for example in the last decade there are several experiments by Schiffer and collaborators measuring proton and neutron orbit occupancies in nuclei that are candidates for neutrinoless double beta decay; see [16,78] and references therein. Similarly, they are also needed in many applications, see for example [23,79].…”

“…) into configuration partial densities such that the partial strength densities are always positive definite [78]. Then, proceeding as suggested in [38], for H = h + V , we have the bivariate convolution form…”

“…(39). An approximate method for J projection is again to use energy dependent spin-cutoff factors as described for example in [38,78]. There are additional complications as in general transition operators are tensor operators with respect to J and hence it is also important to consider transition strength densities with reduced matrix elements of transition operators; see [5] for discussion and some results related to this.…”