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Generalized Green function of harmonic oscillator

  1. Oct 23, 2013 #1
    1. The problem statement, all variables and given/known data
    The generalized Green function is
    $$
    G_g(x, x') = \sum_{n\neq m}\frac{u_n(x)u_n(x')}{k_m^2 - k_n^2}.
    $$
    Show [itex]G_g[/itex] satisfies the equation
    $$
    (\mathcal{L} + k_m^2)G_g(x, x') = \delta(x - x') - u_m(x)u_m(x')
    $$
    where [itex]\delta(x - x') = \frac{2}{\ell}\sum_{n = 1}^{\infty}\sin(k_nx)\sin(k_nx')[/itex]
    and the condition that
    $$
    \int_0^{\ell}u_m(x)G_g(x, x')dx = 0.
    $$


    2. Relevant equations



    3. The attempt at a solution
    From a previous problem, I found
    $$
    u_n(x) = \sqrt{\frac{2}{\ell}}\sin(k_nx).
    $$
    I then end up with
    \begin{gather}
    (\mathcal{L} + k_m^2)G_g(x, x') = \frac{2}{\ell}\sum_{n = 1}^{\infty}\sin(k_nx)\sin(k_nx') - u_m(x)u_m(x') \\
    \sum_{n\neq m}^{\infty}\sin(k_nx)\sin(k_nx') = \left(\sum_{n = 1}^{\infty}\sin(k_nx)\sin(k_nx')\right) - \sin(k_mx)\sin(k_mx')
    \end{gather}
    What is going wrong?
    1. The problem statement, all variables and given/known data



    2. Relevant equations



    3. The attempt at a solution
    1. The problem statement, all variables and given/known data



    2. Relevant equations



    3. The attempt at a solution
     
  2. jcsd
  3. Oct 25, 2013 #2

    vela

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    Why do you think anything is going wrong? Please explain what you did. I can't tell if you just plugged in the expression for ##\delta(x-x')## to get the first line or if you evaluated the lefthand side and got the righthand side or did something else.
     
  4. Oct 25, 2013 #3

    vanhees71

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    2016 Award

    The Fourier method to solve differential equations is always to make an ansatz of the unknown function in terms of a Fourier series. In your case
    [tex]G(x,x')=\sum_{n=1}^{\infty} A_n(x') u_n(x).[/tex]
    Then you apply the operator, [itex]\mathcal{L}+k_m^2[/itex] to this ansatz, expand the inhomogeneity of the equation also in terms of a Fourier series and then compare the coefficients on both sides, which lets you solve for [itex]A_n(x')[/itex].

    Note that in this case, where [itex]\lambda=-k_m^2[/itex] is an eigenvalue of [itex]\mathcal{L}[/itex], the inhomoeneity must be perpendicular to the eigenfunction [itex]u_m[/itex] for consistency. That's why you have to subtract the part parallel to this eigenmode. Note also that any solution of the inhomogeneous equation is only determined up to a function proportional to this eigenmode!
     
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