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We present an analytical solution for the full spectrum of Kitaev's one-dimensional p-wave superconductor with arbitrary hopping, pairing amplitude and chemical potential in the case of an open chain. We also discuss the structure of the zero-modes in the presence of both phase gradients and next nearest neighbor hopping and pairing terms. As observed by Sticlet et al., one feature of such models is that in a part of the phase diagram, zero-modes are present at one end of the system, whiledoi:10.1088/2399-6528/aab7e5 fatcat:ivoom2muijcu3isoub4ooaxivm