9 TABLE I: LLL spectra of three spinless bosons interacting via a repulsive contact interaction gδ ( z i − z j ). 2nd column: Dimensions of the EXD and the nonspurious TI (in parenthesis) spaces (the EXD space is spanned by uncorrelated permanents of Darwin-Fock orbitals). 4th to 6th columns: Matrix elements [in units of g/ ( π Λ $^{2}$), Λ = √ ¯ h/ ( mω 0 )] of the contact interaction between the correlated RVM states { k, m } [see Eq. (36)]. The total angular momentum L = 3 k + 2 m . Last three columns: Energy eigenvalues from the RVM diagonalization of the associated matrix of dimension D $^{TI}$( L ). There is no nonspurious state with L = 1. The full EXD spectrum at a given L is constructed by including, in addition to the listed TI eigenvalues [ D $^{TI}$( L ) in number], all the energies associated with angular momenta smaller than L . An integer in square brackets indicates the energy ordering in the full EXD spectrum (including both spurious and TI states). Seven decimal digits are displayed, but the energy eigenvalues from the RVM diagonalization agree with the corresponding EXD TI ones within machine precision. L D $^{EXD}$( D $^{TI}$) { k, m } Matrix elements Energy eigenvalues (RVM diag. or EXD $^{TI}$) 0 1(1) { 0,0 } 1.5000000 1.5000000[1] 2 2(1) { 0,1 } 0.7500000 0.7500000[1] 3 3(1) { 1,0 } 0.3750000 0.3750000[1] 4 4(1) { 0,2 } 0.5625000 0.5625000[2] 5 5(1) { 1,1 } 0.4687500 0.4687500[2] 6 7(2) { 2,0 } 0.0468750 0.1482318 { 0,3 } 0.1482318 0.4687500 0.0000000[1] 0.5156250[4] 7 8(1) { 1,2 } 0.4921875 0.4921875[4] 8 10(2) { 2,1 } 0.0937500 0.1960922 { 0,4 } 0.1960922 0.4101562 0.0000000 0.5039062[6] 12 19(3) { 4,0 } 7.3242187 × 10 − 4 1.0863572 × 10 − 2 1.5742811 × 10 − 2 { 2,3 } 1.0863572 × 10 − 2 0.1611328 0.2335036 { 0,6 } 1.5742811 × 10 − 2 0.2335036 0.3383789 0.0000000 0.0000000 0.5002441[13] in Eq. (8) and λ > 1 is an integer positive number. Vi- brational excitations of a similar form, i.e., functions (and certain other variants), have been used earlier to approximate part of the LLL spectra. Such earlier en- deavors provided valuable insights, but overall they re- mained inconclusive; for electrons over the maximum density droplet [with magic L = L $_{0}$], see Refs. [8] and [9]; for electrons over the ν = 1 / 3 ( ν = L $_{0}$/ L ) Jastrow- Laughlin trial function [with magic L = 3 $_{0}$], see Ref. [10]; and for bosons in the range 0 ≤ L ≤ N , see Refs. [22, 25, 26]. The advantage of Q$_{λ}$ [50] (compared to ˜ Q$_{λ}$ ) is that it is translationally invariant (TI) [5, 25], a property shared with both Φ RBM L and Φ REM L . In the following, we will discuss illustrative cases, which will show that the RVM functions of Eq. (33) provide a correlated basis (RVM ba- sis) that spans the TI subspace [5, 25, 30] of nonspurious states in the LLL spectra. III. MOLECULAR DESCRIPTION OF LLL SPECTRA A. Three spinless bosons Only the (0 , 3) molecular configuration and the dipolar λ = 2 vibrations are at play (as checked numerically), i.e., the full TI spectra at any L are spanned by the wave TABLE I provides the systematics of the molecular description for the beginning (0 ≤ L ≤ 12) of the LLL spectrum. There are several cases when the TI subspace has dimension one and the exact solution Φ exact coincides with a single { k, m } state. For L = 0 the exact solution coincides with Φ RBM 0 = 1 ( Q 0 λ = 1); this is the only case when an LLL state has a Gross-Pitaevskii form, i.e., it is a single (normalized) permanent [see Eq. (33)] given by | 0 〉 as defined in Eq. (6). where 1 ≤ M ≤ 3 denotes the number of different single- particle angular momenta in the triad ( l$_{1}$, l$_{2}$, l$_{3}$ ) and the p$_{k}$ ’s are the multiplicities of each one of these different angular momenta. C$_{b}$ ( l$_{1}$, l$_{2}$, l$_{3}$ ) = ( 3 ∏ i =1 l$_{i}$ ! ) − 1 ( M ∏ k =1 p$_{k}$ ! ) − 1 ×   ∑ 1 ≤ i<j ≤ 3 cos [ 2 π ( l$_{i}$ − l$_{j}$ ) 3 ]   , (38) Φ RBM L (0 , 3) = l$_{1}$ + l$_{2}$ + l$_{3}$ = L ∑ 0 ≤ l$_{1}$ ≤ l$_{2}$ ≤ l$_{3}$ C$_{b}$ ( l$_{1}$, l$_{2}$, l$_{3}$ ) Perm[ z l$_{1}$ 1 , z l$_{2}$ 2 , z l$_{3}$ 3 ] , (37) ˜ Q$_{λ}$ = N ∑ i =1 z λ i (35) where the coefficients C$_{b}$ ( . . . ) are given by: Following Eq. (24), a simplified analytic expression for the (0,3) RBM can be derived, i.e., with k, m = 0 , 1 , 2 , . . . , and L = 3 k +2 m ; these states are always orthogonal. Φ RBM 3 k (0 , 3) Q m 2 ⇒ { k, m } , (36)