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Generalised laplacian

  1. Nov 21, 2014 #1
    Hi, I was wondering if the following relation holds:

    $$ \frac{1}{r^{D-1}} \frac{\partial}{\partial r} \left( r^{D-1} \frac{\partial}{\partial r} \right) \psi = \frac{1}{r^{\frac{D-1}{2}}} \frac{\partial ^2}{\partial r^2} \left( r^{\frac{D-1}{2}} \right) \psi $$

    I've seen that the LHS evaluates to:

    $$\left( \frac{D-1}{r} \frac{\partial}{\partial r} + \frac{\partial ^2}{\partial r^2} \right) \psi $$

    while the RHS evaluates to:

    $$ \left( \frac{D-1}{r} \frac{\partial}{\partial r} + \frac{\partial ^2}{\partial r^2} + \left( \frac{D-1}{2} \right) \left( \frac{D-3}{2} \right) \frac{1}{r^2} \right) \psi $$

    Am I correct?
     
  2. jcsd
  3. Nov 21, 2014 #2

    ZetaOfThree

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    Gold Member

    That relation holds only if ##\left( \frac{D-1}{2} \right) \left( \frac{D-3}{2} \right) \frac{1}{r^2} \psi = 0##.
     
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