Difference ladder operators for a harmonic Schrِdinger oscillator using unitary linear lattices

The formalism of raising and lowering operators is developed for the difference operator analogue of a quantum harmonic oscillator which acts on functions on a discrete support. The grid under consideration is a mixed version of an equidistant lattice and a q-linear grid. Several properties of the grid are described. The grids under consideration are referred to by the name unitary linear lattices. The ladder difference operators are derived and compared with the continuum situation. The arising spectral problems for these operators are dealt by using the theory of bilateral Jacobi operators in weighted l2(Z) spaces. Eventual applications to mathematical physics and numerical Schrِdinger theory are briefly discussed.