Q.M. on S^6, add potential gives Q.M. on S^3?

[Spinnor]

Say we have a free quantum mechanical particle constrained to move on the surface of S^6;

x1^2 + x2^2 + x3^2 + x4^2 + x5^2 + x6^2 + x7^2 = R^2

Can we add a simple potential such that for low energies the particle is "constrained" to move in some sub space of S^6, say S^3, but for higher energies the particle has "full access" of the space S^6.

What might be such a simple potential?

Does this potential have a topology?

Is their a group represented by the quantum mechanics of a free particle constrained to move on the surface S^3?


Thanks for your help.

 

[Spinnor]

Would V = a*a*(x1^2 + x2^2 + x3^2) work? Here a is a real constant large enough so that for a given R it works. Such a potential tends to keep the particle near x1 = x2 = x3 = 0 while it is free to roam about in the space x4^2 + x5^2 + x6^2 + x7^2 = R^2 which defines the space S^3? So for low energies the particle seems to live in one space and for higher energies it lives in another space?

Thanks for your thoughts.