Does the Generalised Laplacian satisfy a certain relation?

[spaghetti3451]

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?

 

[ZetaOfThree]

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