There is, I think, something ethereal about i —the square root of minus one. I remember first hearing about it at school. It seemed an odd beast at that time—an intruder hovering on the edge of reality.

Usually familiarity dulls this sense of the bizarre, but in the case of i it was the reverse: over the years the sense of its surreal nature intensified. It seemed that it was impossible to write mathematics that described the real world in …

I and i

An asymptotic interation method for solving second-order homogeneous linear differential equations of the form y'' = lambda(x) y' + s(x) y is introduced, where lambda(x) \neq 0 and s(x) are C-infinity functions. Applications to Schroedinger type problems, including some with highly singular potentials, are presented.

https://arxiv.org/pdf/math-ph/0309066.pdf

Asymptotic iteration method for eigenvalue problems

This paper reviews a recently discovered link between integrable quantum field theories and certain ordinary differential equations in the complex domain. Along the way, aspects of PT-symmetric quantum mechanics are discussed, and some elementary features of the six-vertex model and the Bethe ansatz are explained.

The ODE/IM correspondence

The Schrodinger equation is solved exactly for some well known potentials. Solutions are obtained reducing the Schrodinger equation into a second order differential equation by using an appropriate coordinate transformation. The Nikiforov-Uvarov method is used in the calculations to get energy eigenvalues and the corresponding wave functions.

A general approach for the exact solution of the schrodinger equation

Hidden pseudospin and spin symmetries and their origins in atomic nuclei

Exact solution of Schrodinger equation for the pseudoharmonic potential is obtained for an arbitrary angular momentum. The energy eigenvalues and corresponding eigenfunctions are calculated by Nikiforov–Uvarov method. Wavefunctions are expressed in terms of Jacobi polynomials. The energy eigenvalues are calculated numerically for some values of l and n with n ≤  5 for some diatomic molecules.

Exact solution of Schrödinger equation for Pseudoharmonic potential

Effective mass Schrodinger equation is solved exactly for a given potential. Nikiforov- Uvarov method is used to obtain energy eigenvalues and the corresponding wave functions. A free parameter is used in the transformation of the wave function. The effective mass Schrodinger equation is also solved for the Morse potential transforming to the constant mass Schrodinger equation for a potential. One can also get solution of the effective mass Schrodinger equation starting from the constant mass Schrequation. PACS numbers: 03.65.-w; 03.65.Ge; 12.39.Fd

A new approach to the exact solutions of the effective mass Schrodinger equation

Using the NU method [A F Nikiforov and V B Uvarov 1988 "Special Functions of Mathematical Physics" (Birkhauser, Basel)], we investigated the real eigenvalues of the complex and/or PT-symmetric, non-Hermitian and the exponential type systems, such as Poschl–Teller and Morse potentials.

Exponential Type Complex and Non-Hermitian Potentials in PT-Symmetric Quantum Mechanics

The Klein–Gordon equation is solved approximately for the Hulthen potential for any angular momentum quantum number l with the position-dependent mass. Solutions are obtained by reducing the Klein–Gordon equation into a Schrodinger-like differential equation using an appropriate coordinate transformation. The Nikiforov–Uvarov method is used in the calculations to get energy eigenvalues and the wavefunctions. It is found that the results in the case of constant mass are in good agreement with the ones obtained in the literature.

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Papers

A general approach for the exact solution of the schrodinger equation

Exact solution of Schrödinger equation for Pseudoharmonic potential

A new approach to the exact solutions of the effective mass Schrodinger equation

Exponential Type Complex and Non-Hermitian Potentials in PT-Symmetric Quantum Mechanics

Approximate analytical solutions of the Klein–Gordon equation for the Hulthén potential with the position-dependent mass