Exactly solvable Schrödinger equations play important roles in
theoretical physics. Most of them are known to enjoy some remarkable
algebraic structures, such as supersymmetry, shape invariance, or
existence of annihilation/creation operators. In this talk I shall
present yet another (possibly new) Lie-algebraic approach to exactly
solvable bound-state problems in one-dimensional quantum mechanics. I
shall revisit five specific examples: Coulomb, Rosen-Morse,
Manning-Rosen, Eckart, and Liouville potentials, all of whose
bound-state problems are exactly solvable. I shall discuss that these
bound-state problems can be solved by making use of unitary
representations of the Lie algebras so(2,1), so(3), or iso(2) in a
little bit strange way.