On topological effects in quantum mechanics: The harmonic oscillator in the pointed plane

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Abstract
Quantum mechanics of a (nonrelativistic) system S localized on a topologically nontrivial manifold M as its configuration space is based on a quantization method, which, in general, reflects global properties of M, i.e., some of the observables of S will ‘‘feel’’ the topology: There are topological effects and inequivalent quantizations on M. Some straightforward examples are given for such effects, using Borel quantization (BQ), the pointed plane as manifold M, and the energy operator with harmonic potential as observable. Two topological effects exist. There are unitarily inequivalent BQ on M, which are equivalent to the usual quantization on the plane with a topological potential, which has the form of a Bohm–Aharonov potential. There are different self‐adjoint extensions of the energy operator for a given BQ that in some cases are related to another kind of topological potential. These effects are discussed in detail, especially the self‐adjoint extensions of the energy operator. An experimental setup to verify some of the results is suggested.

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