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the configuration - interaction shell model is a useful framework for a detailed understanding of low - energy nuclear structure @xcite .
the many - body basis is a large dimension ( @xmath0 ) set of slater determinants , which are antisymmeterized products of single - particle states .
one must truncate the single - particle states , corresponding to one or a few shells ( typically using the harmonic oscillator as an approximation to the mean - field ) ; the many - body basis may be further truncated . for phenomenological calculations
one writes the hamiltonian in terms of single - particle energies and two - body matrix elements , while for _ ab initio _ calculations one may extend this to three - body interactions @xcite .
the two - body matrix elements are the matrix elements of the residual interaction in the lab frame , @xmath1 where @xmath2 is the normalized , antisymmeterized product of particles in orbits labeled by @xmath3 and @xmath4 and coupled to good angular momentum @xmath5 and isospin @xmath6 .
if one starts from a translationally invariant interaction between particles , one can either compute the integral in the lab frame or start in the relative frame and then transform to the lab frame ; in either case there are correlations between the matrix elements , although they are not obvious to the casual observer
.    often for semi - phenomenological calculations , one starts from a `` realistic '' interaction , and then adjusts the two - body matrix elements until the rms error on a set of experimentally known energy levels is minimized @xcite . in the @xmath7-@xmath8 shell , such a semi - phenomenological interaction has been recently derived @xcite , improving on an earlier interaction@xcite .
it has been found that the fits are empirically dominated by a few linear combinations of matrix elements .
the physical meaning of those dominant combinations is not immediately obvious .
one might naively guess the linear correlations are due to an underlying translationally invariant interaction ( although a density dependence would destroy this ) .
somewhat more phenomenologically , it has been argued by appealing to mean - field properties that one can improve fits primarily through adjusting the monopole - monopole part of the interaction , that is , interaction terms that look like @xmath9 , where @xmath10 is the number of particles in the @xmath11th orbit .
this protocol for shifting monopole strengths has been successfully applied to several semi - empirical interactions @xcite    a related study @xcite , 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 @xcite and effective field theory@xcite .    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 @xcite . 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 .
our methodology follows previous work @xcite ; we work in the @xmath7-@xmath8 valence space with an inert @xmath12o core .
given an input set of two - body matrix elements ( we leave aside single - particle energies and any @xmath13-dependent scaling ) , which we write as a vector @xmath14 , we can calculate the eigenvalues @xmath15 of the many - body hamiltonian . for this work the label @xmath16 ranged over all nuclides with @xmath17 and took the ground state binding energy and the first five excitation energies .
if one has a target spectrum @xmath18 , say from experiment , then the goal of the fit is to minimize @xmath19 ( for simplicity we leave off the experimental uncertainty in each state ) .
expanding to first order @xmath20 then minimizing ( [ chi2 ] ) yields @xmath21 this equation is in the form @xmath22 where @xmath23 the derivatives come via the hellmann - feynman theorem @xcite @xmath24 where @xmath25 is the hamitonian operator whose strength is @xmath26 .
we then find the eigenvalues of @xmath27 ( which is equivalent to finding the squares of the eigenvalues in the singular - value decomposition of @xmath28 ) .
we do this for both the usdb interaction and for an ensemble of 100 sets of random two - body interactions , also called the two - body random ensemble ( tbre ) .
the results are shown for the tbre in fig .  1 , where we have separated out the sensitivity just for the binding energies ( ground state energies ) and the excitation energies .
although not shown , the equivalent svd eigenvalues for usdb are completely within the tbre results .
the lower curve is for ground states only , while the upper curve is for excitations energies relative to the ground state . clearly , and perhaps unsurprisingly , the ground state energies are predominantly sensitive to just a few linear combinations of matrix elements  significantly fewer than excitations energies .    fig .
1 characterizes _ eigenvalues _ of @xmath27 . the next step is to characterize the _ eigenvectors _
associated with the dominant eigenvalues , specifically by comparing with monopole and contact interactions .
to do so we first discuss a method for quantifying the overlap between two vector subspaces @xcite .
consider two vectors subspaces , @xmath29 and @xmath30 .
let @xmath31 be a matrix whose column vectors are the ( orthonormal ) basis vectors of @xmath29 , and similarly with @xmath32 . from these one constructs the overlap matrix @xmath33 .
note that if the subspaces are not of equal dimension then @xmath34 is not a square matrix . in any case
we do a singular value decomposition of @xmath34 ; the svd eigenvalue spectrum then is a measure of the overlap of the two spaces .
if the two spaces perfectly overlap then all eigenvalues are 1 , if just @xmath35 of the dimensions perfectly overlap than @xmath35 eigenvalues will be 1 and the rest zero .
note that this method is invariant under arbitrary choice of orthonormal bases .
we begin with the monopole - monopole interaction of the form @xmath36 , which has six unique terms , and thus six vectors or linear combinations of matrix elements , in the @xmath37-shell .
these we combine with the @xmath38 most dominant linear combinations that arise from the previous analysis ; somewhat arbitrarily we chose @xmath39 ( our results do not change qualitatively for other small values of @xmath38 ) .
the results , the svd eigenvalues of @xmath40 , are shown in fig .
2 . it is important to note we compute the spectrum separately for each randomly generated interaction and then compute the distribution .
the results for ground states and for excited states are similar , so we combine all states into a single calculation .
the eigenvalues for usdb are roughly @xmath41 higher than for the tbre , but otherwise qualitatively very similar .
we also compared for contact interactions ; we took only two terms , the @xmath42 channel and @xmath43 channel ( there being only the @xmath44-wave channel in relative coordinates ) .
these results we summarize in table 1 .
usdb & 0.60 & 0.58 & 0.62 + tbre & @xmath45 & @xmath46 & @xmath45 +    for comparison , the leading eigenvalue for the overlaps of usdb versus the six - term monopole is 0.94 , while that of the tbre versus monopole is @xmath47 .
there is somewhat more sensitivity to the monopole interaction than the contact interaction ; however , the reader should keep in mind that is not the whole story . recall that when fitting an interaction , the linearized equations are cast in the form @xmath48 .
our analysis in this paper is entirely with the eigenvalues of @xmath27 , but in any fit one must also look to @xmath49 ( which in practice is the deviation of the theoretical spectra from experiment ) . for example , in @xcite it was found that using a contact interaction brought better agreement than a monopole interaction .
one can understand this in terms of conjugate gradient methods @xcite : the direction of local steepest gradient may not in fact point towards the global minimum .    by our measures so far , both the tbre and the empirically - fit usdb look qualitatively similar .
therefore we take a final analysis by comparing the dominant linear combinations of the usdb with those from the tbre .
this is show in fig .
3 , using the same analysis as for fig .  2 . for comparison with the previous results ,
the leading eigenvalue is @xmath50 .
we have analyzed the sensitivity of the low - lying spectra of the random two - body ensemble of interactions to variations of the hamiltonian matrix elements ; by using singular value decomposition , we find the dominant linear combinations , which would be important in any fit to experimental data .
we found the svd eigenvalues follow a pattern remarkably similar to that shown by semi - realistic / semi - phenomenological interactions such as usdb .
we also analyzed the most dominant linear combinations of matrix elements by computing the overlap with monopole and contact interactions .
overall , both the tbre and the empirical usdb had qualitatively similar results .
the u.s .
department of energy supported this investigation through contracts de - fg02 - 96er40985 and de - fc02 - 09er41587 , and through subcontract b576152 by lawrence livermore national laboratory under contract de - ac52 - 07na27344 .
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the input to the configuration - interaction shell model includes many dozens or hundreds of independent two - body matrix elements . 
 previous studies have shown that when fitting to experimental low - lying spectra , the greatest sensitivity is to only a few linear combinations of matrix elements . here 
 we consider interactions drawn from the two - body random ensemble , or tbre , and find that the low - lying spectra are also most sensitive to only a few linear combinations of two - body matrix elements , in a fashion nearly indistinguishable from an interaction empirically fit to data . 
 we find in particular the spectra for both the random and empirical interactions are sensitive to similar matrix elements , which we analyze using monopole and contact interactions .