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 $k$-body interactions ($k \leq m$), the wavefunction structure is studied using random matrix theory. Hamiltonian for system chosen to be $H=H_0(t) + \lambda V(k)$ unperturbed $H_0(t)$ being a $t$-body operator $V(k)$ interaction strength $\lambda$. Representing by independent Gaussian orthogonal ensembles (GOE) of matrices $t$ $k$ fermion spaces respectively, first four moments, $m$-fermion spaces, functions $F_\kappa(E)$ are derived; contain all information about structure. With $E$ denoting $H$ energies or eigenvalues $\kappa$ basis energy $E_\kappa$, give spreading over eigenstates $E$. It shown that moments essentially same as conditional $q$-normal distribution given in: P.J. Szabowski, Electronic Journal Probability {\bf 15}, 1296 (2010). This naturally gives asymmetry respect $E_\kappa$ increases also peak value changes $E_\kappa$. Thus, many-fermion follows general distribution.