  We develop an algebraic formulation for the discrete quantum harmonic
oscillator (DQHO) with a finite, equally-spaced energy spectrum and energy
eigenfunctions defined on a discrete domain, which is known as the su(2) or
Kravchuk oscillator. Unlike previous approaches, ours does not depend on the
discretization of the Schr\"odinger equation and recurrence relations of
special functions. This algebraic formulation is endowed with a natural su(2)
algebra, each finite dimensional irreducible representation of which defines a
distinct DQHO labeled by its resolution. In addition to energy ladder
operators, the formulation allows for resolution ladder operators connecting
all DQHOs with different resolutions. The resolution ladder operators thus
enable the dynamic scaling of the resolution of finite degree-of-freedom
quantum simulations. Using the algebraic DQHO formalism, we are able to
rigorously derive the energy eigenstate wave functions of the QHO in a purely
algebraic manner without using differential equations or differential
operators, which is impossible in the continuous or infinite discrete setting.
The coherent state of the DQHO is constructed, and its expected position is
proven to oscillate as a classical harmonic oscillator. The DQHO coherent state
recovers that of the quantum harmonic oscillator at large resolution. The
algebraic formulation also predicts the existence of an inverse DQHO that has
no known continuous counterpart.
