To solve the eigenvalue problem (F10) we employ the following identities for Hermite polynomials 43 where k = 0 , 1 , 2 , . . . is the quantum number labeling the eigenmodes (for every n there are two modes of opposite parity). Note that the second, third, and fourth terms on the right hand side of Eq. (F13) are, respectively, the eigenvalues of the first, second, and the third terms on the right hand side in Eq. (F10). With a little algebra we simplify the eigenvalues of Eq. (F13) as follows In the physical units of length the eigenfunctions in the whole space take the form Finally, the displacement eigenmodes normalized by the condition can be written as follows Equations (F14) and (F17) give the asymptotically exact solutions of the elastic eigenvalue problem in the limit of strong correlations. In Sec. IV C we have used this solutions to control the accuracy of our numerical results. $_{u}$± k ( x ) = 1 √$_{2}$ k $^{+1}$k ! $_{e}$− ( $^{x}$−x 0 / 2) 2 λ 2 ∞ H$_{k}$ ( $^{x}$−x$_{0}$/ 2 λ ∞ ) ± ( − 1) k $_{e}$− ( x + x 0 / 2) 2 λ 2 ∞ H$_{k}$ ( x + x$_{0}$/ 2 λ ∞ ) $_{e}$− ( $^{x}$−x 0 / 2) 2 λ 2 ∞ + $_{e}$− ( x + x 0 / 2) 2 λ 2 ∞ . (F17) ∫ dxn$_{0}$ ( x ) $_{u}$± k ( x ) $_{u}$± l ( x ) = δ$_{kl}$ (F16) n$_{0}$ ( x ) $_{u}$± k ( x ) ∼ $_{e}$− ( $^{x}$−x 0 / 2) 2 λ 2 ∞ H$_{k}$ ( x − x$_{0}$/ 2 $^{λ}$∞ ) ± ( − 1) k $_{e}$− ( x + x 0 / 2) 2 λ 2 ∞ H$_{k}$ ( x + x$_{0}$/ 2 $^{λ}$∞ ) . (F15) $_{ω}$± k = ω$_{0}$ [ $_{2 + 3}$√$_{3}$$_{k}$ + 6 k ( k − 1)(2 − √$_{3)}$ ∓ ( − 1) $^{k}$(2 − √$_{3)}$ k ] 1 / 2 . (F14) ( $_{ω}$± k ω$_{0}$ ) 2 = 2 + 3 4 k ( k − 1) $_{(}$√$_{3 + 1)}$ 2 $_{+ 3}$√$_{3}$$_{k}$ ∓ ( − 1) k ( √$_{3}$ − 1 √$_{3 + 1}$ ) k , (F13) ( ω 2 ω 2 0 − 2 ) u ± = 3 e ξ $^{2}$∂ $^{2}$[ e − ξ $^{2}$∂ $^{2}$u ± ] $_{(}$√$_{3 + 1)}$ 2 − $_{3}$√$_{3}$ 2 e ξ $^{2}$∂ [ e − ξ $^{2}$∂u ± ] ∓ √$_{3 + 1}$ $_{2}$√$_{π}$$_{3}$ 1 / 4 ∞ ∫ −∞ dξ ′ $_{e}$− $_{(}$√$_{3+1)}$ 2 $_{4}$√$_{3}$ ( ξ ′ + √$_{3}$ − 1 √$_{3+1}$ ξ ) 2 u ± ( ξ ′ ) . (F10) 23 1 W. M. C. Foulkes, L. Mitas, R. J. Needs, and G. Rajagopal, Rev. Mod. Phys. 73 , 33 (2001). 2 Time-Dependent Density Functional Theory , Lecture Notes in Physics, Vol. 706, edited by M.A.L. Marques, C.A. Ullrich, F. Nogueira, A. Rubio, K. Burke, and E.K.U. Gross (Springer, Berlin, 2006). 3 E.K.U. Gross, J.F. Dobson and M. Petersilka, in Density Functional Theory , v. 181 of Topics in current chemistry , ed. R.F. Nalewajski (Springer-Verlag, Berlin 1996). 4 M.E. Casida, in Recent Advances in Density Functional Methods , edited by D.P. Chong (World Scientific, Singa- pore, 1995), p. 155. e x $^{2}$∂$_{x}$ ( e − x $^{2}$∂$_{x}$ H$_{k}$ ) = − 2 kH$_{k}$ , e x $^{2}$∂ 2 x ( e − x $^{2}$∂ 2 $_{x}$H$_{k}$ ) = 4 k ( k − 1) H$_{k}$ , (F11) a + 1 $_{2}$√$_{aπ}$ ∞ ∫ −∞ dx ′ $_{e}$− ( a +1) 2 4 a ( x ′ + $^{a}$− 1 a $_{+1}$x ) $^{2}$H$_{k}$ ( x ′ ) = ( − 1) k ( a − 1 a + 1 ) k H$_{k}$ ( x ) . (F12) From the identities (F11)and (F12) we see that Hermite polymomials are the eigenfunctions of each of three terms in the integro-differential operator (both for odd and for even modes) on the right hand side in (F10). The corresponding eigenvalues (one should apply the identity (F12) with a = √$_{3) are}$