Analysis of the one-dimensional motion in the two-parameter potential (proposed in [Phys. Rev. A 112, 013319 (2025)]), which can be considered as a discrete generalization of the inverse-square potential, is presented. For correct formulation of the problem, the self-adjointness of the corresponding Hamiltonian is discussed. The structure of the eigenvalue spectrum, which follows from the scale invariance, is determined for different potential’s parameters. In particular, the Efimov-like spectra of this and the inverse-square potential are compared.