We found universal laws for the spectrum of two-dimensional three-body systems, composed by two identical particles and a distinct one (AAB). These universal laws appear when the potential range (r_0) is much smaller than the size of the system. In two dimensions this condition is formulated as (E2 is the two-body energy and ? is the reduced mass). The zero range model, which is very appropriated to establish the universal laws, is introduced through the ?-Dirac potential. We derive the corresponding two-dimensional Faddeev equations for the three-body system and solve them numerically in momentum space. Our results showed that the three-body binding energy monotonically increases with the two-body binding energy, and such dependence is more pronounced than the mass variations. We found that the three-body energy depends logarithmic on the two-body energy for large values. Furthermore, been m=mB/mA the ratio between the masses of the B and A particles, the three-body energy is mass-independent for m ? ? and increase without bounds for m ?0. The limit of two non-interacting identical particles is also studied in the AAB system. We found that the two-dimensional three-body system always support at least two bound states and more bound states appear for m<0.22. Finally, we analyze the particular limit of m ?0 using the adiabatic approximation. This approximation can be used to study the three-body system in two-dimensions with an accuracy better than 10% compared to the solutions of the Faddeev equations, for m ? 0.01.

Macro properties of cold atomic gases are driven by few-body correlations, even if the gas has thousands of particles. Quantum systems composed of two and three particles with attractive zero-range pairwise interactions are considered for general masses and interaction strengths in two and three dimensions (2D and 3D). The Faddeev decomposition is used to derive the equations for the bound state, which is the starting point for the investigation of universal properties of few-body systems, i.e. those that all potentials with the same physics at low energy are able to describe in a model-independent form. In 2D, the number of bound states in a three-body system increases without bound as the mass of one particle becomes much lighter than the other two. The analytic form of an effective potential between the heavy particles explains the mass-dependence on the number of bound energy levels. An exact analytic expression for the large-momentum asymptotic behavior of the spectator function in the Faddeev equation is presented. The spectator function and its asymptotic form define the two- and three-body contact parameters. The two-body parameter is found to be independent of the quantum state in some specific 2D systems. The 2D and 3D momentum distributions have a distinct sub-leading form whereas the 3D term depends on the mass of the particles. A model that interpolates between 2D and 3D is proposed and a sharp transition in the energy spectrum of three-body systems is found.