Graduate 3D oscillator trapped away from centre

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The discussion centers on the spherical oscillator problem, specifically addressing a scenario where a particle oscillates around a shifted position due to a small barrier. The potential is proposed to be in the form of 1/2 m w^2 (r-r0)^2, indicating that it is centered around the new position r0. Participants suggest using a change of variables to solve for the energy levels in this modified potential. The conversation emphasizes the need for a clear formulation of the potential to proceed with the solution. Overall, the thread seeks to explore the implications of this shifted potential on the energy levels of the oscillator.
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The spherical oscillator problem is for a potential centered at the origin.. but what if the particle is confined to oscillate at a small shift by a small (not infinite) barrier? The potential is now centered about the new position.. Could someone please suggest a way to solve the problem for the energy levels?
 
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Can you write down the potential?
 
I assume the general form of the potential is
1/2 m w^2 (r-r0)^2
 
Then it can be solved using a change of variables,
 

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