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

We investigate the approximate solutions of the Dirac equation with the position-dependent mass particle in the Eckart potential field including the Coulomb tensor interaction by using the parametric Nikiforov-Uvarov method. Taking an appropriate approximation to deal with the centrifugal term, the Dirac energy states and the corresponding normalized two-spinor components of the wave function are obtained in closed form. Some special cases of our solution are investigated. Furthermore, we present the correct solutions obtained via the asymptotic iteration method which are in agreement with the parametric Nikiforov-Uvarov method results.

Bound states of spatially dependent mass Dirac equation with the Eckart potential including Coulomb tensor interaction

Dirac and Klein Gordon equations with equal scalar and vector potentials

Analytical approximations to the solutions of the Manning-Rosen potential with centrifugal term

Bound states of the Klein–Gordon equation with vector and scalar Rosen–Morse-type potentials☆

Bound states of the five-parameter exponential-type potential model☆

We propose a unified treatment of generating exactly solvable trigonometric potentials with PT symmetry. Our symmetric approach reproduces a number of earlier results. We show that the energy spectra for the exactly solvable PT-symmetric trigonometric potentials can be obtained from a general energy spectrum formula.

Systematic study on exactly solvable trigonometric potentials with pt symmetry

We construct three new solvable pseudo-Hermitian potential models with real spectra starting from the five-parameter exponential-type potential model.

Jian-Yi Liu

Papers

Bound states of the Klein–Gordon equation with vector and scalar Rosen–Morse-type potentials☆

Bound states of the five-parameter exponential-type potential model☆

Systematic study on exactly solvable trigonometric potentials with pt symmetry

New solvable pseudo-hermitian potential models with real spectra