Quantum Version of Norton's Dome

Question

I was recently revisiting Norton's dome and was reading about its connect with Bohmian mechanics. I completely understand its role in classical physics, but a more fundamental question would be: Is a quantum version of the Norton dome possible, where the wave function is bound to the surface. What would be the energy Eigen state which would have exactly the same amount of energy necessary for a classical particle to rest on the dome, assuming the angular momentum of this state to be zero? How would this change with a non-zero angular momentum?

Does the dome point out any anomalies in quantum mechanics as it did in classical physics?

Comments

"How time takes an electron between 2 bouncing potential barriers to out through and what way he takes" might be the quantum analogous if you are not focusing on gravity. The dome top is also made by 2 weak barriers ( and others higher to keep the ball following the dome ). In both cases, you may say nothing without contextual informations on possible perturbations. Is the quantum analysis accurate at the macroscropic scale ? there is the scale common answer somehow related to the amount of the relative uncertainty ( cf Compton analysis of the quantum scale ).

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Do you know any papers where they propose this idea?

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Don't cheat through Rudolf-Ortvay!!!

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This was actually an assignment given to me by my prof. I have my semester exams going on, so I'm not in a position even close to participating in the contest. If I really am taking part in the contest why would I ask the question in an open forum like this one, for the world to see?