Symmetries and Their Role in Rationally Extended Real and Complex Potentials

This book is divided into seven chapters. In the first chapter, we discuss the development of supersymmetry (SUSY) in quantum mechanics, PT symmetry and construction of exactly solvable real as well as complex systems in detail. The basics about SUSY in quantum mechanics, the PT symmetry in non-hermitan quantum systems, the newly discovered exceptional orthogonal polynomials (EOPs) and the important methodologies, which have been used to obtain the important results in the other chapter of the book are described in the second chapter. Some important properties of SUSY and PT symmetry are also discussed in the same chapter. In chapter 3, we study the different rationally extended SIPs associated with Xm EOPs using Co-ordinate transformation approach. These are rationally extended radial oscillator, rationally extended trigonometric Scarf (also known as Scarf-I) potential, rationally extended generalized Poschl-Teller (GPT) and extended Poschl-Teller-II potentials. The bound state solutions of all these real extended potentials are also obtained in terms of exceptional Laguerre (in the case of extended radial oscillator potential) and exceptional Jacobi (in the case of extended Scarf-I and GPT potentials) orthogonal polynomials. A well known complex and PT symmetric rationally extended potential, the extended Scarf-II potential is also considered and solutions of this potential are also obtained. The scattering state solutions for some of the these extended potentials are also obtained in Chapter 4. In chapter 5 and 6, we construct some of these potentials using group theoretic method and obtain their bound as well as scattering state solutions.