  We study numerically the phase structure and the gluon propagator of the
SU(2) gauge-Higgs model in two dimensions. First, we calculate gauge-invariant
quantities, in particular the static potential from Wilson Loop, the W
propagator, and the plaquette expectation value. Our results suggest that a
confinement-like region and a Higgs-like region appear even in two dimensions.
In the confinement-like region, the static potential rises linearly, with
string breaking at large distances, while in the Higgs-like region, it is of
Yukawa type, consistent with a Higgs-type mechanism. The correlation length
obtained from the W propagator has a finite maximum between these regions. The
plaquette expectation value shows a smooth cross-over consistent with the
Fradkin-Shenker-Osterwalder-Seiler theorem. From these results, we suggest that
there is no phase transition in two dimensions. We also calculate a
gauge-dependent order parameter in Landau gauge. Unlike gauge invariant
quantities, the gauge non-invariant order parameter has a line of discontinuity
separating these two regions. Finally we calculate the gluon propagtor. We
infer from its infrared behavior that the gluon propagator would vanish at zero
momentum in the infinite-volume limit, consistent with an analytical study.
