  In the present paper we deal with the issue of finding the self-adjoint
extensions of a $p^4$-corrected Hamiltonian. The importance of this subject
lies on the application of the concepts of quantum mechanics to the
minimal-length scale scenario which describes an effective theory of quantum
gravity. We work in a finite one dimensional interval and we give the explicit
$U(4)$ parametrization that leads to the self-adjoint extensions. Once the
parametrization is known, we can choose appropriate $U(4)$ matrices to model
physical problems. As examples, we discuss the infinite square-well, periodic
conditions, anti-periodic conditions and periodic conditions up to a prescribed
phase. We hope that the parametrization we found will contribute to model other
interesting physical situations in further works.
