  In this thesis we present three results about the ferromagnetic quantum XXZ
model: 1) Existence of a spectral gap above all infinite-volume ground states
in one dimension for any choice of spin S>1/2 (for S=1/2 this was already
known); 2) Characterization of one-dimensional "droplet" states as ground
states of the spin-1/2 model with up spin boundary fields; 3) An upper bound
for the spectral gap above the spin-1/2 interface ground states in dimension
two and higher. The upper bound vanishes as 1/R^2, where R is the linear size
of the system. This thesis contains research performed in collboration with
other authors: O. Bolina, P. Contucci, T. Koma, B. Nachtergaele.
