Wavefunction structure in quantum many-fermion systems with k-body interactions: conditional q-normal form of strength functions

For finite quantum many-particle systems modeled with say m fermions in N single particle states and interacting with k-body interactions $k leq m$, the wavefunction structure is studied using random matrix theory. Hamiltonian for the system is chosen to be H=H₀(t) + λV(k) with the unperturbed H₀(t) Hamiltonian being a t-body operator and V(k) a k-body operator with interaction strength λ. Representing H₀(t) and V(k) by independent Gaussian orthogonal ensembles (GOE) of random matrices in t and k fermion spaces respectively, first four moments, in m-fermion spaces, of the strength functions F_κ(E) are derived; strength functions contain all the information about wavefunction structure. With E denoting the H energies or eigenvalues and κ denoting unperturbed basis states with energy E_κ, the F_κ(E) give the spreading of the κ states over the eigenstates E. It is shown that the first four moments of F_κ(E) are essentially same as that of the conditional q-normal distribution given in: P.J. Szabowski, Electronic Journal of Probability {\bf 15}, 1296 (2010). This naturally gives asymmetry in F_κ(E) with respect to E as E_κ increases and also the peak value changes with E_κ. Thus, the wavefunction structure in quantum many-fermion systems with k-body interactions follows in general the conditional q-normal distribution.