We deal with the Hamiltonian hierarchy problem of the Hulthén potential within the frame of the supersymmetric quantum mechanics and find that the associated superymmetric partner potentials simulate the effect of the centrifugal barrier. Incorporating the supersymmetric solutions and using the first-order perturbation theory we obtain an expression for the energy levels of theHulthén potential which gives satisfactory values for the non-zero angular momentum states.

The eigenvalues of the potentials V 1 (r) = r 6 , and of the special cases of these potentials such as the Kratzer and Goldman-Krivchenkov potentials, are obtained in Ndimensional space. The explicit dependence of these potentials in higherdimensional space is discussed, which have not been previously covered.

An application of the recently introduced method [M. Çapak et al., J. Math. Phys. 52, 102102 (2011)] to the bound-state eigenvalue problem in the elementary quarkonium potential V(r) = -a/r + br + cr2 is described, proved and illustrated for [Formula: see text] and [Formula: see text] systems. The quasi- and conditionally-exactly solvable spin-averaged mass spectra of heavy quarkonia are obtained in compact forms. The comparison of the present predictions with those of other theories in the related literature, together with the available data, has shown the success of the model used in this work and also revealed that the use of different confinings in the perturbed Coulomb potential descriptions has no considerable effect on the mass spectra of such systems.

The concept of the elegant work introduced by Lévai in Ref. [5] is extended for the solutions of the Schrödinger equation with more realistic other potentials used in different disciplines of physics. The connection between the present model and the other alternative algebraic technique in the literature is discussed.

The recently introduced scheme [20,21] is extended to propose an algebraic non-perturbative approach for the analytical treatment of Schrödinger equations with non-solvable potentials involving an exactly solvable potential form together with an additional piece. As an illustration the procedure is successfully applied to the Cornell potential by means of very simple algebraic manipulations. However, instead of providing numerical eigenvalues for the only consideration of the small strength of the related linear potential as in the previous reports, the present model puts forward a clean route to interpret related experimental or precise numerical results involving wide range of the linear potential strengths. We hope this new technique will shed some light on the questions concerning with the limitations of the traditional perturbation techniques.