Particle in a cavity in one-dimensional bandlimited quantum mechanics

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].