There is no pointwise consistent quasicontinuum energy

Much work has gone into the construction of quasicontinuum energies that reduce the coupling error along the interface between atomistic and continuum regions. The largest consistency errors are typically pointwise $O(\frac{1}{\text{eps}})$ errors, and in some cases this has been reduced to pointwise O(1) errors. In this paper we show that one cannot create a coupling method using a finite-range coupling interface that has o(1)-consistency in the interface, and we use this to give an upper bound on the order of convergence in discrete $w^{1,p}$-norms in 1D.