First time anharmonic potential V (r) = ar 2 + br − c r , (a > 0) is examined for N -dimensional Schrödinger equation via Laplace transformation method. In transformed space, the behavior of the Laplace transform at the singular point of the differential equation is used to study the eigenfunctions and the energy eigenvalues.The results are easy to derive and identical with those obtained by other methods.

In this paper we obtain approximate bound state solutions of N -dimensional fractional time independent Schrödinger equation for generalised Mie-type potential, namely V (r α ) = A r 2α + B r α +C. Here α(0 < α < 1) acts like a fractional parameter for the space variable r. When α = 1 the potential converts into the original form of Mie-type of potential that is generally studied in molecular and chemical physics. The entire study is composed with Jumarie type fractional derivative approach. The solution is expressed via Mittag-Leffler function and fractionally defined confluent hypergeometric function. To ensure the validity of the present work, obtained results are verified with the previous works for different potential parameter configurations, specially for α = 1.At the end, few numerical calculations for energy eigenvalue and bound states eigenfunctions are furnished for a typical diatomic molecule.

The bound state solutions of the D-dimensional Schrödinger equation for new mixed class of potential, V (r) = V 1 r 2 + V 2 e −αr r + V 3 cothαr + V 4 , are studied within the framework of the Pekeris approximation for any arbitrary ℓ-state. Asymptotic iteration method (AIM) is used for the work.The energy spectrum are obtained as well as their corresponding normalized eigenfunctions are derived in terms of generalized hypergeometric functions 2 F 1 (a, b, c; z). It is shown that using the Pekeris approximation, present potential model is very much capable of deriving other well known potentials quite easily and corresponding solutions are in excellent agreement with the previous work carried out in literature.

The second order N -dimensional Schrödinger equation with Mie-type potentials is reduced to a first order differential equation by using the Laplace transformation. Exact bound state solutions are obtained using convolution or Faltungs theorem. The Ladder operators are also constructed for the Mie-type potentials in N -dimensions. Lie algebra associated with these operators are studied and it is found that they satisfy the commutation relations for the SU(1,1) group.

In this paper we obtain approximate bound state solutions of N -dimensional time independent fractional Schrödinger equation for generalised pseudoharmonic potential which has the form V (r α ) = a 1 r 2α + a 2 r 2α + a 3 . Here α(0 < α < 1) acts like a fractional parameter for the space variable r. The entire study is composed with the Jumarie type derivative and the elegance of Laplace transform. As a result we successfully able to express the approximate bound state solution in terms of Mittag-Leffler function and fractionally defined confluent hypergeometric function. Our study may be treated as a generalization of all previous works carried out on this topic when α = 1 and N arbitrary. We provide numerical result of energy eigenvalues and eigenfunctions for a typical diatomic molecule for different α close to unity. Finally, we try to correlate our work with Cornell potential model which corresponds to α = 1 2 with a 3 = 0 and predict the approximate mass spectra of quarkonia.