On the other hand, the first moment sum rule (E1) can be rewritten as which is manifestly satisfied by χ el µν by virtue of the com- pleteness relation (E5) for ˜ u$_{λ}$ ( r ). When applied to the exact response function the f -sum rule implies that Comparing the last two equations we conclude that Then we see that i.e. the sum rule is satisfied in the elastic approximation. More pointedly, making use of Eq. (E3) the sum rule can be written in the form from which it follows that where the “oscillator strengths” f λ n are positive quantities defined as with As a final step we prove that for all λ . This is of course nothing but the f -sum rule Here Ψ$_{0}$( x, x ′ ) is the ground state two-particle wave function, which in this case coincides with the square root of the two-particle density matrix, n$_{0}$ ( x ) is the ground state density, and T$_{0}$ ( x ) is the ground state kinetic stress tensor defined by Eq. (59). Equation (F2) possesses an analytic solution in the limit of strong Coulomb interaction e $^{2}$m$^{2}$ω$_{0}$ ≫ 1 when the ground state wave function reduces to the following asymptotic form At the exact many-body theory level the system is de- scribed by the Hamiltonian of Eq. (100). Within our con- tinuum mechanics the excitation energies are obtained from the solution of the following “elastic” eigenvalue problem for the displacement u ( x, t ) = u ( x ) e − iωt (see Eq. (61) in the main text) In this appendix we present an asymptotically exact solution of the 1D continuum mechanics eigenvalue prob- lem for two particles confined by the harmonic potential V$_{0}$ = 1 $_{2}$mω 2 $_{0}$x 2 and interacting with a soft-Coulomb po- tential QED. − 1 π ∫ ∞ 0 dωω ℑ mχ$_{µν}$ ( r , r ′ , ω ) = 1 2 m √ n$_{0}$ ( r ) ˜ M$_{µν}$ ( r , r ′ ) √ n$_{0}$ ( r ′ ) . (E12) − 1 π ∫ ∞ 0 dωω ℑ mχ$_{ik}$ ( r , r ′$_{, ω}$$_{) =}$ − 1 π ∞ dωω ℑ mχ el $_{ik}$( r , r ′ , ω ) , (E13) ∑ λ ω 2 $_{λ}$[˜ u$_{λ}$ ( r )]$_{µ}$[˜ u$_{λ}$ ( r ′$_{)]}$$_{ν}$ = 2 m ∑ n ω$_{n}$$_{0}$ [ j$_{µ}$ ( r )]$_{0}$$_{n}$ √ n$_{0}$ ( r ) [ j$_{ν}$ ( r ′ )]$_{n}$$_{0}$ √ n$_{0}$ ( r ′ ) , (E14) ω 2 λ = ∑ n ω 2 n $_{0}$f λ n , (E15) f λ n = 2 m | F λ n | 2 ω$_{n}$$_{0}$ (E16) F λ n ≡ ∫ d r ˜ u$_{λ}$ ( r ) · [ j ( r )]$_{0}$$_{n}$ √ n$_{0}$ ( r ) (E17) ∑ l f λ n = 1 (E18) − 1 π ∫ ∞ 0 dω ℑ mχ$_{µν}$ ( r , r ′ , ω ) ω = n$_{0}$ ( r ) 2 m δ$_{µν}$δ ( r − r ′ ) , (E19) mω $^{2}$n$_{0}$ u ( x ) = 1 4 m ∂ 2 x [ n$_{0}$∂ 2 $_{x}$u ( x )] − 3 ∂$_{x}$ [ T$_{0}$∂$_{x}$u ( x )] + mω 2 $_{0}$n$_{0}$ u ( x ) + 2 ∫ $_{dx}$′$_{[}$$_{∂}$ 2 $_{x}$w ( x − $_{x}$′$_{)]Ψ}$ 2 $_{0}$( $_{x, x}$′$_{)[}$$_{u}$$_{(}$$_{x}$$_{)}$ − u ( $_{x}$′$_{)]}$$_{.}$ (F2) Ψ$_{0}$( x$_{1}$, x$_{2}$ ) = 1 ℓ $_{∞}$√$_{π}$ $_{e}$− ( x 1 + x 2 ) 2 $_{2}$√$_{3}$$_{ℓ}$ ∞ [ $_{e}$− ( x 1 − x 2 − x 0 ) 2 2 ℓ ∞ + $_{e}$− ( x 1 − x 2 + x 0 ) 2 2 ℓ ∞ ] , (F3) w ( x − x ′ ) = e 2 √ ( x − x ′ ) 2 + a 2 . (F1) ∑ n 2 m | F λ n | 2 ω$_{n}$$_{0}$ = 2 m ∫ d r ∫ d r ′ ∑ n [ j$_{µ}$ ( r )]$_{0}$$_{n}$[ j$_{ν}$ ( r ′ )]$_{n}$$_{0}$ ω$_{n}$ [˜ u$_{λ}$ ( r )]$_{µ}$ √ n$_{0}$ ( r ) [˜ u$_{λ}$ ( r ′ )]$_{ν}$ √ n$_{0}$ ( r ′ ) = ∫ d r ∫ d r ′$_{n}$$_{0}$$_{(}$ r ) δ$_{µν}$δ ( r − r ′$_{)}$ [˜ u$_{λ}$ ( r )]$_{µ}$ √ n$_{0}$ ( r ) [˜ u$_{λ}$ ( r ′ )]$_{ν}$ √ n$_{0}$ ( r ′ ) = 1 . (E21) ∑ n [ j$_{µ}$ ( r )]$_{0}$$_{n}$[ j$_{ν}$ ( r ′ )]$_{n}$$_{0}$ ω$_{n}$$_{0}$ = n$_{0}$ ( r ) 2 m δ$_{µν}$δ ( r − r ′$_{)}$$_{.}$ (E20) 21 Appendix F: Analytic solution of the elastic eigenvalue problem in the strong-correlation regime