Non-commutativity as a measure of inequivalent quantization

We show that the strength of non-commutativity could play a role in determining the boundary condition of a physical problem. As a toy model, we consider the inverse-square problem in non-commutative space. The scale invariance of the system is explicitly broken by the scale of non-commutativity Theta. The effective problem in non-commutative space is analyzed. It is shown that despite the presence of a higher singular potential coming from the leading term of the expansion of the potential to first order in Theta, it can have a self-adjoint extension. The boundary conditions are obtained, which belong to a 1-parameter family and are related to the strength of non-commutativity.