Remarks on the quasi-position representation in models of generalized uncertainty principle

This note aims to elucidate certain aspects of the quasi-position representation frequently used in the investigation of one-dimensional models based on the generalized uncertainty principle (GUP). We specifically focus on two key points: (i) contrary to recent claims, the quasi-position operator can possess physical significance even though it is non-Hermitian, and (ii) in the quasi-position representation, operators associated with the position, such as the potential energy, also behave as a derivative operator on the quasi-position coordinate, unless the method of computing expectation values is modified. The development of both points revolves around the observation that the position and quasi-position operators share the same set of eigenvalues and are connected through a non-unitary canonical transformation. This outcome may have implications for widely referenced constraints on GUP parameters.