  We study a nonrelativistic quantum mechanical particle on an interval of
finite length with a Hamiltonian that has a $p^3$ correction term, modelling
potential low energy quantum gravity effects. We describe explicitly the $U(3)$
family of the self-adjoint extensions of the Hamiltonian and discuss several
subfamilies of interest. As the main result, we find a family of self-adjoint
Hamiltonians, indexed by four continuous parameters and one binary parameter,
whose spectrum and eigenfunctions are perturbatively close to those of the
uncorrected particle with Dirichlet boundary conditions, even though the
Dirichlet condition as such is not in the $U(3)$ family. Our boundary
conditions do not single out distinguished discrete values for the length of
the interval in terms of the underlying quantum gravity scale.
