  The effects of the generalized uncertainty principle (GUP) on the low-energy
stationary states of a particle moving in a cavity with no sharp boundaries are
determined by means of the perturbation expansion in the framework of
one-dimensional bandlimited quantum mechanics. A realization of GUP resulting
in the existence of a finite ultraviolet (UV) wave-vector cutoff $K\sim
1/\ell_P$ (with the Planck length $\ell_P$) is considered. The cavity of the
size $\ell \gg \ell_P$ is represented by an infinitely deep trapezoid-well
potential with boundaries smeared out in a range $R$ satisfying the
inequalities $\ell\gg R\gtrsim \ell_P$. In order to determine the energy shifts
of the low-lying stationary states, the usual perturbation expansion is
reformulated in a manner that enables one to treat consistently order-by-order
the direct and indirect GUP effects, i.e., those due to the modification of the
Hamiltonian and the lack of the UV modes, respectively. It is shown that the
leading terms of the indirect and the direct GUP effects are of the first and
second order, respectively, in the small parameter $\ell_P/\ell$ in agreement
with our previous finding in a more naive approach [1].
