 0  10  20  30  40  50  60  70  80  90  0  2  4  6  8  10  12  0  10  20  30  40  50  60  70  80  90  0  2  4  6  8  10  12 ω$_{+}$ (q) ω$_{-}$ (q) ω$_{+}$ (q) ω$_{-}$ (q) q q ω L ω$_{p}$ (a) (b) ω T ω$_{L}$ (q) ω$_{T}$ (q)  0.5  1  1.5  2  2.5  3  0 r s =1 r s =5  3  2.5  2  1.5  1  0.5  0  0  0.5  1  1.5  2  2.5 r s =1 r s =5 r s =3  3  0.5  1  1.5  2  2.5  3 FIG. 1: (Color online) Longitudinal (a) and transverse (b) modes for a homogeneous electron gas at r$_{s}$ = 1 , 3 , 5. Wave vector q is in units of k$_{F}$ and frequency is in units of 2 E$_{F}$ . The curves labelled ω + ( q ) and ω − ( q ) are the boundaries of the electron-hole continuum. $^{35}$. Single particle excitations exist for ω − ( q ) < ω < ω + ( q ), whereas multiparticle excitations are distributed all over the plane. The elastic approximation replaces the exact spectrum by the two branches ω$_{L}$ ( q ) and ω$_{T}$ ( q ), which carry the entire spectral weight. Notice that the r$_{s}$ -dependence is barely discernible on a large q scale, but becomes clearly visible at smaller q as shown in the insets. excitation energies ω$_{n}$ = ( Z $^{2}$/ 2)(1 − 1 /n $^{2}$) ( n = 1 , 2 , .. ). The corresponding eigenfunctions are given by Laguerre polynomials: As a final example let us consider the case of two electrons repelling each other with the “soft” Coulomb potential e 2 √ ( $^{x$_{1}$}$−x$_{2}$ ) $^{2}$+ a 2 in a one-dimensional parabolic trap of natural frequency ω$_{0}$ (the cutoff a > 0 serves to eliminate the pathological behavior of the interaction at x$_{1}$ = x$_{2}$ ). This is a model that can be solved nu- merically thanks to the separation of center of mass and where ℓ$_{0}$ = √ 2 ℏ /mω$_{0}$ . The ground-state density is a gaussian centered at the origin. The excitation energies, expressed in units of ω$_{0}$ , are sums of the excitation ener- gies of two identical harmonic oscillators: In the non interacting limit ( ω$_{0}$ → ∞ , or e 2 → 0) the relative wave function is can be either symmetric or antisymmetric, depending on the parity of n , and has n nodes. Here ℓ$_{cm}$ = √ ℏ / 2 mω$_{0}$ . The ground-state has n = 0 , m = 0 and all the other states are characterized by positive values of the integers n and m . The ground-state of the system is a spin singlet ( S = 0) and for this reason in the following we consider only singlet states, which are connected to the ground-state by the current density operator. The relative wave function for such states is symmetric: φ$_{m}$ ( − x ) = φ$_{m}$ ( x ). This wave function has 2 m nodes, of which m with x > 0 and m (symmetrically placed) with x < 0. The center of mass wave function, where ψ$_{n}$ ( X ) is the wave function of the n-th excited state of the center of mass hamiltonian and φ$_{m}$ ( x ) is the wave function of the m -th excited state of the relative hamiltonian. where ψ$_{0}$ ( X ) and φ$_{0}$ ( x ) are, respectively, the ground- state wave functions of the center of mass and of the relative hamiltonian. The general excited state is The ground-state wave function is where X = x$_{1}$ + x$_{2}$ 2 and P = p$_{1}$ + p$_{2}$ are, respectively, the coordinate and the momentum of the center of mass and x = x$_{1}$ − x$_{2}$ , p = $^{p$_{1}$}$−p$_{2}$ 2 are the coordinate and the momentum in the relative channel. Notice that for a fixed strength e 2 of the interaction we can go from the weakly correlated to the strongly correlated regime by varying the value of ω$_{0}$ : ω$_{0}$ → ∞ corresponds to the non- interacting limit, and ω$_{0}$ → 0 to the strongly correlated limit. relative variable, and analytically in the limit of strong correlation. The hamiltonian is 13 ˆ H$_{0}$ = P 2 4 m + mω 2 $_{0}$X$^{2}$+ p 2 m + m 4 ω 2 $_{0}$x$^{2}$+ e 2 √$_{x}$ 2 + a 2 , (100) Ψ$_{0}$( x$_{1}$, x$_{2}$ ) = ψ$_{0}$ ( X ) φ$_{0}$ ( x ) , (101) Ψ$_{nm}$( x$_{1}$, x$_{2}$ ) = ψ$_{n}$ ( X ) φ$_{m}$ ( x ) (102) ψ$_{n}$ ( X ) = H$_{n}$ ( X ℓ$_{cm}$ ) e − X $^{2}$/ 2 ℓ 2 cm , (103) φ$_{m}$ ( x ) = H$_{2}$$_{m}$ ( x ℓ$_{0}$ ) e − x $^{2}$/ 2 ℓ 2 r , (104) u$_{n}$ ( r ) ∝ L 2 $^{n}$− 2 ( 2 r n ) . (99) C. Two-electron systems lim $^{ω$_{0}$}$→∞ E$_{nm}$ ω$_{0}$ = n + 2 m (105)