Claudia Kalhori: Examples of Quasi-Exactly Solvability in Quantum Mechanics

Abstract: A linear quantum mechanical system (Hamiltonian) is called exactly solvable if all its eigenfunctions and eigenvalues can be found explicitly. This means that for all its energy levels and corresponding wave functions, an explicit expression can be obtained and will give us the entire spectrum. Such systems include the standard quadratic oscillator and several other examples, but very few.

In the late 1980s a new type of quantum mechanical systems has been discovered which are called quasi-exactly solvable. In such systems, it is possible to explicitly find a limited part of the spectrum, but not all of it, which means that there is a finite part of the energy spectrum, and related eigenfunctions that can be found exactly. Examples of such systems are quasi-exactly solvable potentials which usually depend on an integer parameter M. Usually for a given positive integer M, one can explicitly find the first M eigenfunctions and eigenvalues.

In this paper we give a short introduction to exactly solvable systems and quasi-exactly solvable systems. We give some important examples and an analysis of a particular quasi-exactly solvable system appearing in recent research. We introduce the framework of Schrödinger equations in quantum mechanics and study in details the case of the quantum harmonic oscillator. We also give a short introduction to the WKB approximation, which is a method one can use to obtain approximate information about quasi-exactly solvable systems.