ABSTRACT
The number of exactly solvable eigenvalue problems has been rapidly increasing since the advent of quantum mechanics and there are many efficient ways to treat them. As a general rule, it is possible to write exactly solvable abstract eigenvalue equations in terms of the generators of finite-dimensional Lie algebras and solve them by means of purely algebraic methods. Matrix elements are commonly necessary for the application of perturbation theory and variational principles, among other approximate methods. The chapter discusses examples of three-dimensional Lie algebras with well-known quantum-mechanical realizations. Typically, exactly solvable quantum-mechanical problems are either onedimensional models or lead to separable equations in many dimensions. In the coordinate representation both cases reduce to ordinary differential equations of second order that determine the eigenvalues and separation constants compatible with the physical boundary conditions. In its purest form, a Lie algebraic method is representation-independent. Extensive application of algebraic methods to selected physical problems is already available elsewhere.
