  We present a simple algebraic procedure that can be applied to solve a range
of quantum eigenvalue problems without the need to know the solution of the
Schr\"odinger equation. The procedure, presented with a pedagogical purpose, is
based on algebraic consistency conditions that must be satisfied by the
eigenvalues of a couple of operators proper of the problem. These operators can
be either bilinear forms of the raising and lowering operators appropriate to
the problem, or else auxiliary operators constructed by resorting to the
factorization of the Hamiltonian. Different examples of important
quantum-mechanical textbook problems are worked out to exhibit the clarity and
simplicity of the algebraic procedure for determining the spectrum of
eigenvalues without knowing the eigenfunctions. For this reason the material
presented may be particularly useful for undergraduate students or young
physicists.
