Exactly Solvable Quantum Mechanics

TL;DR

This paper provides a comprehensive review of exactly solvable quantum mechanics, emphasizing recent advances in multi-indexed orthogonal polynomials and their applications in various quantum systems.

Contribution

It introduces the latest developments in multi-indexed orthogonal polynomials and their role in expanding the class of exactly solvable quantum models.

Findings

Introduction of multi-indexed orthogonal polynomials

Analysis of shape invariance and solution space structures

Development of deformation schemes and scattering problems

Abstract

A comprehensive review of exactly solvable quantum mechanics is presented with the emphasis of the recently discovered multi-indexed orthogonal polynomials. The main subjects to be discussed are the factorised Hamiltonians, the general structure of the solution spaces of the Schroedinger equation (Crum's theorem and its modifications), the shape invariance, the exact solvability in the Schroedinger picture as well as in the Heisenberg picture, the creation/annihilation operators and the dynamical symmetry algebras, coherent states, various deformation schemes (multiple Darboux transformations) and the infinite families of multi-indexed orthogonal polynomials, the exceptional orthogonal polynomials, and deformed exactly solvable scattering problems.