The formalism of Ursell operators provides a self-consistent integral equation for the one-particle reduced operator. In three dimensions this technique yields values of the shift in the Bose-Einstein condensation (BEC) transition temperature, as a function of the scattering length, that are in good agreement with those of Green's function and quantum Monte Carlo methods. We have applied the same equations to a uniform two-dimensional system and find that, as we alter the chemical potential, an instability develops so that the self-consistent equations no longer have a solution. This instability, which seems to indicate that interactions restore a transition, occurs at a non-zero value of an effective chemical potential. The non-linear equations are limited to temperatures greater than or equal to Tc, so that they do not indicate the nature of the new stable state, but we speculate concerning whether it is a Kosterlitz-Thouless state or a ``smeared'' BEC, which might avoid any violation of the Hohenberg theorem, as described in an accompanying paper.
We extend our theory of slow magnons in a two-component Bose-Einstein condensate to the case of two spatial dimensions (2D). We provide a detailed discussion of polaronic corrections to the magnon branch of the elementary excitation spectrum in weakly and strongly interacting regimes. In a dilute system, the latter may be achieved by adjusting interspecies attraction so as to obtain a resonance in the
p-wave scattering channel. Resonantly enhanced
p-wave attraction results in the self-localization of magnons and the formation of a magnetoroton. The nature of a ground state beyond the magnetoroton instability remains to be explored. We suggest a dilute
p-wave crystal of alternating polarization as a possible candidate. In contrast to 3D, the required strength of the attractive potential in the
s-wave channel here corresponds to tight binding, and we suggest a potential realization of our model with excitons in 2D semiconductors.