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We present calculations of the tunneling density of states in ananisotropically paired superconductor for two different sample geometries: asemi-infinite system with a single specular wall, and a slab of finitethickness and infinite lateral extent. In both cases we are interested in theeffects of surface pair breaking on the tunneling spectrum. We take the stablebulk phase to be of $d_{x^2-y^2}$ symmetry. Our calculations are performedwithin two different band structure environments: an isotropic cylindricalFermi surface with a bulk order parameter of the form $\Delta\sim k_x^2-k_y^2$,and a nontrivial tight-binding Fermi surface with the order parameter structurecoming from an anti-ferromagnetic spin-fluctuation model. In each case we findadditional structures in the energy spectrum coming from the surface layer.These structures are sensitive to the orientation of the surface with respectto the crystal lattice, and have their origins in the detailed form of themomentum and spatial dependence of the order parameter. By means of tunnelingspectroscopy, one can obtain information on both the anisotropy of the energygap, $|\Delta(\p)|$, as well as on the phase of the order parameter,$\Delta(\p) = |\Delta(\p)|e^{i\varphi(\p)}$.Comment: 14 pages of revtex text with 11 compressed and encoded figures. To  appear in J. Low Temp. Phys., December, 199

The effect of surfaces on the tunneling density of states of an anisotropically paired superconductor