 0  1  2  3  4  5  6  0  2  4  6  8  10 E/ ω 0 ω$_{0}$ -1/2 (1,0) (0,1) (2,0) (1,1) (3,0) (0,2) (2,1) (4,0) (1,2) (3,1) (5,0) FIG. 2: (Color online) Evolution of the excitation energies for two electrons in a one-dimensional harmonic trap. Solid lines denote exact excitation energies, labelled by ( n, m ) as explained in the text. Solid dots indicate the calculated eigen- values of the QCM equation of motion. Crosses on the right denote the strong-correlation limit of the eigenvalues, given by Eq. (113). with n , m non-negative integers. The degeneracy of the excited states is the number of integers less or equal n + 2 m with the same parity as n + 2 m , i.e. where [ y ] denotes the integer part of y . The displacement field associated with the ( n, m ) excitation is where λ$_{0}$ = √ ℓ 2 $_{cm}$+ ( ℓ$_{0}$/ 2) 2 = √ ℏ /mω$_{0}$ . The parity is ( − 1) n +2 $^{m}$− 1 and the number of nodes in n + 2 m − 1. The situation is quite different in the strongly corre- lated limit ( ω$_{0}$ → 0, or e 2 → ∞ ). The relative Hamilto- nian reduces to a harmonic oscillator of frequency $_{ω$_{0}$}$√$_{3}$ with equilibrium distance x$_{0}$ = (2 e $^{2}$/mω2 $_{0}$) 1 / 3 (we assume a ≪ x$_{0}$ ). The ground-state wave function for the rel- ative motion is a symmetric linear combination of two Gaussians of width $^{ℓ}$∞ = (2 ℏ $_{/}$√$_{3}$$_{mω}$$_{0}$$_{)}$ 1 / 2 centered at x$_{1}$ − x$_{2}$ = ± x$_{0}$ . The corresponding ground state den- sity n$_{0}$ ( x ) consists of two Gaussian peaks of the width $^{λ}$∞ = √ ℓ 2 $_{cm}$+ ( $^{ℓ}$∞/ 2) 2 = [ ℏ (1 + √$_{3)}$$_{/}$$_{2}$√$_{3}$$_{mω}$$_{0}$$_{]}$ 1 / 2 cen- tered at x = ± x$_{0}$/ 2. The excitation spectrum has the form Let us now see what our elastic equation of motion (61) predicts for this system. The kinetic part of the equilib- rium stress tensor T$_{0}$ ( x ) works out to be The phenomenon of different excited states produc- ing the same displacement field occurs also in the non- interacting limit: all the states with the same value of n + 2 m (e.g. (0 , 1) and (2 , 0)) have the same displace- ment field. But, in this case, the states with the same displacement field also have the same energy: therefore the noninteracting excitation energies can be accurately reproduced by a linear eigenvalue problem. Looking at the figures we observe that, in the strongly correlated regime , there are groups of excited states (e.g. { (0 , 2) , (2 , 0) } ; { (3 , 0) , (1 , 2) } ; { (2 , 1) , (0 , 3) } ; { (4 , 0) , (0 , 2) , (0 , 4) } , such that all the excited states within one group produce the same displacement field, up to a normalization constant. In general, states with a given value of n + m and the same parity of m have the same displacement field, but different energies. Clearly, this is a feature of the exact solution that cannot be reproduced by any linear eigenvalue problem with a frequency-independent kernel. From these figures we see that the displacement field of the (1 , 0) excitation, which corresponds to a rigid trans- lation of the center of mass, is uniform in space, while the displacement field of the (0 , 1) excitation, which cor- responds to the classical breathing mode, changes sign around the origin. The (1 , 0) and (0 , 1) excitations corre- spond to the classical phonon modes of a system of two localized particles. The remaining excitations are quan- tum mechanical in character, as can be surmised from the fact that their displacement fields (in the strongly corre- lated regime) have significant variation over the regions where the density has peaks, i.e. the places where the particles would be classically localized. These modes de- scribe the dynamics of the wave function of the localized electrons. The evolution of the lowest-lying energy levels with given value of the pair n, m as a function of ω$_{0}$ is shown by the solid lines in Fig. (2). Some of the displacement fields of the low-lying excitations in the strongly corre- lated regime (Eq. (109)) are shown in Fig. (3). The parity is ( − 1) $^{n}$− 1 and the number of nodes is 2( n + m − 1)+mod( n 1 , 2), where mod( n − 1 , 2) ≡ n − 1 (mod 2). and the degeneracy is completely removed. The displace- ment field is analytically found to be 14 u$_{nm}$ ( x ) ∝ $^{H$_{n}$$_{+}$$_{m}$}$− 1 ( x − x$_{0}$/ 2 $^{λ}$∞ ) θ ( x ) + ( − 1) $^{$^{m}$H$_{n}$$_{+}$$_{m}$}$− 1 ( x + x$_{0}$/ 2 $^{λ}$∞ ) θ ( − x ) . (109) lim $^{ω$_{0}$}$→ 0 E$_{nm}$ ω$_{0}$ = n + $_{m}$√$_{3}$$_{,}$ (108) u$_{nm}$ ( x ) ∝ $^{H$_{n}$$_{+2}$$_{m}$}$− $_{1}$( x/λ$_{0}$ ) , (107) D$_{nm}$ = 1 + [ n + 2 m 2 ] , (106)