  Recent advances in single atom manipulation have made it possible to create
"wires" in the form of atomic scale linear structures on a semiconductor
surface. If such structures are to be used in future electronic devices, it
will be necessary to know to what extent they behave as one-dimensional
conductors, in which the presence of electron-electron interactions in the wire
lead to properties quite different from those found in higher-dimensional
systems. In particular, it would be useful to know if these structures retain
any of the Luttinger liquid properties that are predicted for a pure
one-dimensional metal. However, experimental studies of such structures have so
far yielded unclear and contradictory results. We investigate this problem
theoretically by creating a highly simplified model of a wire on a surface--a
chain of atoms that includes electron-electron interactions to represent the
wire, coupled to a similar chain that excludes electron-electron interactions
to represent, albeit crudely, the surface. We present results for the
eigenstates and spectral functions for such systems that suggest that many
Luttinger liquid indicators are present. However the system as a whole retains
some residual Fermi liquid characteristics, and therefore merges properties of
both one and higher dimensions.
