  We introduce a new method for analysing the Bose-Hubbard model for an array
of bosons with nearest neighbor interactions. It is based on a number-theoretic
implementation of the creation and annihilation operators that constitute the
model. One of the advantages of this approach is that it facilitates
computation with arbitrary accuracy, enabling nearly perfect numerical
experimentation. In particular, we provide a rigorous computer assisted proof
of quantum phase transitions in finite systems of this type.
  Furthermore, we investigate properties of the infinite array via harmonic
analysis on the multiplicative group of positive rationals. This furnishes an
isomorphism that recasts the underlying Fock space as an infinite tensor
product of Hecke spaces, i.e., spaces of square-integrable periodic functions
that are a superposition of non-negative frequency harmonics. Under this
isomorphism, the number-theoretic creation and annihilation operators are
mapped into the Kastrup model of the harmonic oscillator on the circle. It also
enables us to highlight a kinship of the model at hand with an array of spin
moments with a local anisotropy field. This identifies an interesting physical
system that can be mapped into the model at hand.
