Large-order behavior of lattice strong-coupling expansions

We examine the lattice strong-coupling expansion for the ground-state energy of the quantum-mechanical Hamiltonian $\frac{1}{2}{p}^2+g{\mathrm{}\left|x\right|\mathrm{}}^{\alpha }$, $\alpha >0\mathrm{}$. We are interested in the large-order behavior of this series for various values of $\alpha$. Treated as a quantum field theory, this simple model provides a laboratory for investigating the fascinating subtleties that arise when there are cancellations among graphs. The parameter $\alpha$ distinguishes between regions in which these cancellations occur with varying degrees of complexity. This paper considers various approaches to this very difficult problem and presents a partial solution.