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We show how to establish a correspondence between the recently introduced formulation of quantum mechanics based on orthogonal polynomials in the energy and physical parameters (without reference to a potential function) and the convention formulation. In this work, we obtain the potential function for the newly introduced quantum system associated with the Wilson and Racah polynomials [A. D. Alhaidari and T. J. Taiwo, J. Math. Phys. 58, 022101 (2017)] using any of the proposed procedures in the work of Alhaidari [Commun. Theor. Phys. 68, 711 (2017)]. To achieve that, we use the matrix elements of the potential function and the basis in which these were calculated. [ABSTRACT FROM AUTHOR]

We develop a new operator algebraic formulation of the Nakajima-Mori-Zwanzig (NMZ) method of projections. The new theory is built upon rigorous mathematical foundations, and it can be applied to both classical and quantum systems. We show that a duality principle between the NMZ formulation in the space of observables and in the state space can be established, analogous to the Heisenberg and Schrödinger pictures in quantum mechanics. Based on this duality we prove that under natural assumptions, the projection operators appearing in the NMZ equation must be conditional expectations. The proposed formulation is illustrated in various examples. [ABSTRACT FROM AUTHOR]

Consequently, the correct energy spectrum is, in fact, as written by us in Eq. (13). In a comment by Fernández[1] on our article,[2] it was stated that the expression for the energy spectrum of the Kratzer-Fues potential obtained by us using the tridiagonal representation approach (TRA) does not appear to be correct. The parameter I i depends on the energy, which may give the impression that potential (1) is energy dependent due to the first term. [Extracted from the article]