where $^{ℓ}$∞ = (2 ℏ $_{/}$√$_{3}$$_{mω}$$_{0}$$_{)}$ 1 2 , and x$_{0}$ = (2 e $^{2}$/mω2 $_{0}$) 1 3 is the classical distance between particles, i.e., the distance that minimizes the classical energy of two charged particles in the 1D harmonic potential. The corresponding ground state density takes the form of two well separated “blobs” The kinetic stress tensor Eq. (59) for the ground state wave function (F3) becomes simply proportional to the density: where the size $^{λ}$∞ of the density blobs located at x = ± x$_{0}$/ 2 is Another technical observation which simplifies calcu- lations in the strong interaction limit is that the cross- products of the two exponentials in the square brack- ets in (F3) are irrelevant for the expressions of the type Ψ$_{0}$( x, x$_{2}$ )Ψ$_{0}$( x ′ , x$_{2}$ ) in the limit of x$_{0}$ √ mω$_{0}$ ≫ 1. For the two-particle density matrix entering the nonlocal term in (F2) this implies the following result Simplification of the integral term in the equation of motion (F2) comes from the fact that in the limit of x$_{0}$ √ mω$_{0}$ ≫ 1 the pair correlation function (F7) is peaked at | x − x $^{′}$|  ∼ x$_{0}$ (this keeps the particles at a distance close to the classical value). Hence in the integral kernel the interaction factor can be approximated as Substituting (F6) and (F8) into the (F2) and using the obvious identity we reduce the equation of motion to the following form Now the following observations are in order: (i) The integro-differential operator in (F9) (in fact in the original Eq. (F2)) is symmetric under inversion of x . Hence the solutions can be classified by parity u ± ( x ) = ± u ± ( − x ). Therefore it is sufficient to consider Eq. (F9) only in the region of positive x ; (ii) In (F9) for x > 0 all local terms contain n$_{0}$ ( x ) which is a narrow Gaussian located at x ∼ x$_{0}$/ 2. There- fore these terms are nonzero only around x$_{0}$/ 2; (iii) The integral kernel Ψ 2 $_{0}$( x, x ′ ) for x > 0 is a product of two Gaussian peaks, one at x + x ′ ∼ 0, and another at x − x ′ ∼ x$_{0}$ , which confines x to the region of the right density blob, x ∼ x$_{0}$/ 2, and x ′ to the region of the left blob, x ′ ∼ − x$_{0}$/ 2. Therefore for positive x all terms in (F9) are nonzero only in the region x ∼ x$_{0}$/ 2, while the integration re- gion in the nonlocal (interaction) term is confined by the Gaussian factors to x ′ ∼ − x$_{0}$/ 2. Note that for this reason the integral term will contribute with opposite signs to the equations of motion for the modes of opposite parity. To further simplify the eigenvalue problem in the strong coupling limit we proceed as follows. (i) Con- sidering Eq. (F9) in the region x > 0 we make a shift of coordinates x → x + x$_{0}$/ 2, and x ′ → x ′ − x$_{0}$/ 2. After that because of the Gaussian factors the integration can be extended to the whole axis. This completely elimi- nates x$_{0}$ (i.e. the coupling constant) from the problem, as it should be in the strong coupling limit; (ii) Go to di- mensionless coordinates ξ = $^{x/λ}$∞ , and ξ ′ = x ′ $^{/λ}$∞ ; (iii) Divide everything by the ground state density n$_{0}$ (which is simply a Gaussian located at the origin after the above shift, n$_{0}$ ( ξ ) = e − ξ 2 $_{/}$√$_{π}$$_{).}$ The result of these three steps is the following dimen- sionless equation of motion 22 n$_{0}$ ( x ) = 1 λ $_{∞}$√$_{π}$ [ $_{e}$− ( $^{x}$−x 0 / 2) 2 λ 2 ∞ + $_{e}$− ( x + x 0 / 2) 2 λ 2 ∞ ] , (F4) T$_{0}$ ( x ) = √$_{3 + 1}$ 4 ω$_{0}$n$_{0}$ ( x ) . (F6) $^{λ}$∞ = ( ℏ $_{(}$√$_{3 + 1)}$ $_{2}$√$_{3}$$_{mω}$$_{0}$ ) 1 2 . (F5) 2Ψ 2 $_{0}$( x, x ′ ) = 2 ℓ 2 ∞ π $_{e}$− ( x + x ′ ) 2 √$_{3}$$_{ℓ}$ 2 ∞ [ $_{e}$− ( $^{x}$−x ′ − x 0 ) 2 ℓ 2 ∞ + $_{e}$− ( $^{x}$−x ′ + x 0 ) 2 ℓ 2 ∞ ] . (F7) ∂ 2 $_{x}$w ( x − x ′ ) ≈ 2 e 2 | x − x $^{′}$| 3 ≈ 2 e 2 x 3 0 = mω 2 0 . (F8) m [ ω 2 − 2 ω 2 0 ] n$_{0}$ ( x ) u ( x ) = − mω 2 0 ∫ dx ′ 2Ψ 2 $_{0}$( x, x ′ ) u ( x ′ ) + 1 4 m ∂ 2 $_{x}$[ n$_{0}$ ( x ) ∂ 2 $_{x}$u ( x )] − 3 $_{ω$_{0}$}$√$_{3 + 1}$ 4 ∂$_{x}$ [ n$_{0}$ ( x ) ∂$_{x}$u ( x )] . (F9) n$_{0}$ ( x ) = ∫ $_{dx}$′$_{2Ψ}$ 2 $_{0}$( $_{x, x}$′$_{)}$$_{,}$