  In this work we explore the self-adjointness of the GUP-modified momentum and
Hamiltonian operators over different domains. In particular, we utilize the
theorem by von-Newmann for symmetric operators in order to determine whether
the momentum and Hamiltonian operators are self-adjoint or not, or they have
self-adjoint extensions over the given domain. In addition, a simple example of
the Hamiltonian operator describing a particle in a box is given. The solutions
of the boundary conditions that describe the self-adjoint extensions of the
specific Hamiltonian operator are obtained.
