Schrödinger equation for a noninteracting Bose gas on a spatiotemporal lattice

We propose a spatiotemporal discretization of the Schrödinger equation for field operators of noninteracting bosons. The scheme, being based on the associated space of the creation and annihilation operators and on a discretization of the Schrödinger equation, preserves equal time commutation relations, possesses Hermitian action, unitary evolution operator, and all relevant integrals which are counterparts of the respective conservation quantities in the continuum limit. Although the scheme is implicit, it allows a reduction to a nonlocal but explicit scheme. In the limit of small time steps the scheme is reduced to a model studied earlier. Some particular examples of the discrete-time Schrödinger equation in the presence of external potentials are given.