content_image/439200.jpg content_image/439212.jpg content_image/439211.jpg content_image/439210.jpg content_image/439216.jpg
<img src='content_image/439200.jpg'>

in Eq. (8) and λ > 1 is an integer positive number. Vi- brational excitations of a similar form, i.e., 

(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 

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 

## A. Three spinless bosons 

<img src='content_image/439212.jpg'>

functions 

with k, m = 0 , 1 , 2 , . . . , and L = 3 k +2 m ; these states are always orthogonal. 

Following Eq. (24), a simplified analytic expression for the (0,3) RBM can be derived, i.e., 

<img src='content_image/439211.jpg'>

where the coefficients C$_{b}$ ( . . . ) are given by: 

<img src='content_image/439210.jpg'>

where 1 ≤ ≤ 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. 

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). 

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