Exactly solvable models in supersymmetric quantum mechanics and connection with spectrum-generating algebras

Analytic expressions for the eigenvalues and eigenfunctions of nonrelativistic shape-invariant Hamiltonians can be derived using the well-known method of supersymmetric quantum mechanics. Most of these Hamiltonians also possess spectrum-generating algebras and are hence solvable by an independent group theory method. We demonstrate the equivalence of the two solution methods by developing an algebraic framework for shape-invariant Hamiltonians with a general parameter change involving nonlinear extensions of Lie algebras.