  We study the dynamic response of ultracold bosons trapped in one-dimensional
optical lattices using Quantum Monte Carlo simulations of the boson Hubbard
model with a confining potential. The dynamic structure factor reveals the
inhomogeneous nature of the low temperature state, which contains coexisting
Mott insulator and superfluid regions. We present new evidence for local
quantum criticality and shed new light on the experimental excitation spectrum
of 87Rb atoms confined in one dimension.
