Landau quantization and particle-particle ladder sums in a magnetic field

We evaluate the sum of particle-particle ladder diagrams for a two-dimensional (2D) electron gas in the presence of an external magnetic field. The evaluation depends on the observation that the quantized center-of-mass kinetic energy is conserved in electron-electron scattering events. For a delta-function interaction scattering channels with different center-of-mass kinetic energy separate and the integral equation reduces to an algebraic equation which can be solved explicitly. Divergences associated with superconducting instabilities are identified. In the weak-field limit a connection is established between the different scattering channels and different solutions of the linearized Ginzburg-Landau equations.