Prepotential approach to exact and quasi-exact solvabilities of Hermitian and non-Hermitian Hamiltonians

Abstract

In this talk I present a simple and unified approach to both exact and quasi-exact solvabilities of the one-dimensional Schr\"odinger equation. It is based on the prepotential together with Bethe ansatz equations. This approach gives the potential as well as the eigenfunctions and eigenvalues simultaneously. In this approach the system is completely defined by the choice of the change of variables, and the so-called zero-th order prepotential. We illustrate the approach by several examples of Hermitian and non-Hermitian Hamiltonians with real energies. The method can be easily extended to the constructions of exactly and quasi-exactly solvable Dirac, Pauli, and Fokker-Planck equations, and to quasinormal modes.