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Random Matrix Theory with U(N) Racah Algebra for Transition Strengths

For finite quantum many-particle systems, a given 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), particle addition to or removal from a given system and so on. Working towards developing a complete statistical theory for transition strength densities (transition strengths multiplied by the density of states at the initial and final energies), we have started a program to derive formulas for the lower order bivariate moments of the strength densities generated by a variety of transition operators. In this paper results are presented for a transition operator that removes $k_0$ number of particle by considering $m$ spinless fermions in $N$ single particle states. The Hamiltonian that is $k$-body is represented by EGUE($k$) [embedded Gaussian unitary ensemble of $k$-body interactions] and similarly the transition operator by an appropriate independent EGUE. Employing the embedding $U(N)$ algebra, finite-$N$ formulas for moments up to order four are derived and they show that in general the smoothed (with respect to energy) bivariate transition strength densities take bivariate Gaussian form. Extension of these results to particle addition operator and beta decay type operators are discussed.