  We study an impurity in a one-dimensional potential consisting of a harmonic
and a periodic part using both the time-evolving block decimation (TEBD)
algorithm and a variational ansatz. Attractive and repulsive contact
interactions with a sea of fermions are considered. We find excellent agreement
between TEBD and variational results and use the variational ansatz to
investigate higher lattice bands. We conclude that the lowest band
approximation fails at sufficiently strong interactions and develop a new
method for computing the Tan contact parameter.
