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Eigenvalue equation

  1. Sep 18, 2007 #1
    Hello,

    I want to prove that the function [itex]\mathcal{A}[/itex] in the 1D case satisfy

    [itex]\mathcal{A}=\frac{48}{m}\sum_{j=1}^\infty \frac{\sin^2(qj/2)}{j^5}=\frac{12}{m}\left[2\zeta(5)-\text{Li}_5(e^{iq})-\text{Li}_5(e^{-iq})\right],[/itex]

    with [itex]\text{Li}_n(z)[/itex] the polylogarithm function, and the matrix [itex]\mathcal{A}[/itex] in the 2D case satisfy

    [itex]\mathcal{A}=\frac{3}{m}\sum_{j\neq 0}\left[1-\frac{5}{|\vec{r}_j^0|^2}\begin{pmatrix}(x_j^0)^2&x_jy_j\\x_jy_j&(y_j^0)^2\end{pmatrix}\right]\frac{\left(1-e^{i\vec{q}\vec{r}_j^0}\right)}{|\vec{r}_j^0|^5}[/itex]

    Can somebody help me, please, to do this? In 2D the equilibrium positions [itex]\vec{r}_i^0 [/itex] form a triangular lattice with basic lattice vectors [itex]a_1 = (1, 0) [/itex] and [itex]a_2 = (1,\sqrt{3})/2[/itex].

    Starting point:

    [itex]m\vec{\ddot x}_i=\frac{3}{2}\sum_{i\neq j}^{N}\left[\frac{5(\vec{r}_i^0 - \vec{r}_j^0)^2\left(\vec{x}_j-\vec{x}_i\right)}{|\vec{r}_i^0 - \vec{r}_j^0|^7}+\frac{\left(\vec{x}_i-\vec{x}_j\right)}{|\vec{r}_i^0 - \vec{r}_j^0|^5}\right][/itex]

    with the ansatz

    [itex]\vec{x}_i(t)=\epsilon_\lambda(\vec{q})e^{i\left(\vec{q}\vec{r}_i^0-\omega_\lambda(\vec{q})t\right)}[/itex]
    [itex]\vec{\ddot x}_i(t)=-\omega^2\epsilon_\lambda(\vec{q})e^{i\left(\vec{q}\vec{r}_i^0-\omega_\lambda(\vec{q})t\right)}[/itex]

    follows

    [itex]-\omega^2_\lambda(\vec{q})\epsilon_\lambda(\vec{q})=\mathcal{A}\epsilon_\lambda(\vec{q})[/itex]

    Thanks!
     
    Last edited: Sep 18, 2007
  2. jcsd
  3. Sep 18, 2007 #2

    mjsd

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    Homework Helper

    new to PF?
    what have you tried? is this a condense matter problem?
     
  4. Sep 18, 2007 #3
    Yes!

    with the ansatz in the equation of motion, i obtain

    [itex]\mathcal{A}=\frac{3}{2m}\sum_{i\neq j}\left[1-\frac{5}{|\vec{r}_i^0-\vec{r}_j^0|^2}\left(\vec{r}_i^0-\vec{r}_j^0\right)^2 \right]\frac{\left(1-e^{i\vec{q}(\vec{r}_i^0-\vec{r}_j^0)}\right)}{|\vec{r}_i^0-\vec{r}_j^0|^5}[/itex]

    What I have to do next? I do not make any differentiation about 1D and 2D.

    I consider a dipolar crystal.
     
  5. Sep 18, 2007 #4

    mjsd

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    your question is asked in some kind of a strange way... I don't even know what you are asking...or what are you trying to prove? just sub in the ansatz and check? what are you having problems with? it is difficult for anyone to help without knowing what you are stuck at. Besides in this forum you are meant to show us what you have tried
     
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