content_image/438712.jpg content_image/438718.jpg content_image/438711.jpg content_image/438710.jpg content_image/438709.jpg content_image/438708.jpg content_image/438707.jpg
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To solve the eigenvalue problem (F10) we employ the following identities for Hermite polynomials 43 

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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 = √$_{3) are}$ 

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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 

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Finally, the displacement eigenmodes normalized by the condition 

can be written as follows 

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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. 

