  We implement the so-called Weyl-Heisenberg covariant integral quantization in
the case of a classical system constrained by a bounded or semi-bounded
geometry. The procedure, which is free of the ordering problem of operators, is
illustrated with the basic example of the one-dimensional motion of a free
particle in an interval, and yields a fuzzy boundary, a position-dependent mass
(PDM), and an extra potential on the quantum level. The consistency of our
quantization is discussed by analyzing the semi-classical phase space portrait
of the derived quantum dynamics, which is obtained as a regularization of its
original classical counterpart.
