  In one-dimensional quantum mechanics, or the Sturm-Liouville theory, Crum's
theorem describes the relationship between the original and the associated
Hamiltonian systems, which are iso-spectral except for the lowest energy state.
Its counterpart in `discrete' quantum mechanics is formulated algebraically,
elucidating the basic structure of the discrete quantum mechanics, whose
Schr\"odinger equation is a difference equation.
