  Quantum mechanics in the presence of $\delta$-function potentials is known to
be plagued by UV divergencies which result from the singular nature of the
potentials in question. The standard method for dealing with these divergencies
is by constructing self-adjoint extensions of the corresponding Hamiltonians.
Two particularly interesting examples of this kind are nonrelativistic spin
zero particles in $\delta$-function potential and Dirac particles in
Aharonov-Bohm magnetic background. In this paper we show that by extending the
corresponding Schr\"odinger and Dirac equations onto the flat noncommutative
space a well-defined quantum theory can be obtained. Using a star product and
Fock space formalisms we construct the complete sets of eigenfunctions and
eigenvalues in both cases which turn out to be finite.
