content_image/352745.jpg content_image/352746.jpg content_image/352747.jpg
A related study [12], investigating the origin of many-body forces from truncation of the model space, also found an empirical fit dominated by a few linear combinations of matrix elements. While much of the fit was dominated by the monopole interactions, even better agreement was brought about using a contact interaction motivated by its usage in mean-field calculations [13, 14] and effective field theory[15]. 

In investigating the character and origin of the dominant matrix elements, it is useful to ask if there is anything special about the nuclear interaction. One way to ask this question is to compare with interactions drawn from the two-body random ensemble (TBRE), which despite their arbitrary nature are known to echo some features of real nuclear spectra [16– 19]. In this paper we conduct a sensitivity analysis of the low-lying spectra of random interactions and compare against a standard empirical interaction, USDB. We find that for more measures all the interactions are nearly indistinguishable, at least on a statistical level. 

## II. METHODOLOGY AND RESULTS 

Our methodology follows previous work [1, 5, 12]; we work in the 1 s -0 d valence space with an inert $^{16}$O core. Given an input set of two-body matrix elements (we leave aside single-particle energies and any A -dependent scaling), which we write as a vector ⃗ v , we can calculate the eigenvalues E$_{α}$ ( ⃗ v ) of the many-body Hamiltonian. For this work the label α ranged over all nuclides with 0 ≤ Z$_{valence}$ ≤ N$_{valence}$ ≤ 10 and took the ground state binding energy and the first five excitation energies. 

If one has a target spectrum E 0 $_{α}$, say from experiment, then the goal of the fit is to minimize 

<img src='content_image/352745.jpg'>

(for simplicity we leave off the experimental uncertainty in each state). Expanding to first order 

then minimizing (2) yields 

<img src='content_image/352746.jpg'>

<img src='content_image/352747.jpg'>

