A solution for the mathematical problem of functional calculus with the Laplace-Beltrami operator on surfaces with axial symmetry is found. A quantitative analysis of the spectrum is presented.

We consider a one dimensional fermionic gas confined on a circle of finite length. At finite temperatures, in the absence of any background potential, one can show that a constant density of particles is a solution to the Hartree equation, regardless of the type of interparticle interaction. Moreover, at finite temperatures and small coupling constants, our previous analysis shows that this is the only solution of the Hartree equations. We show in

We investigate the thermodynamic limit of the Hartree approximation for periodic background potentials and short range two-body interactions. We prove that, for any finite volume, the Hartree problem has a unique solution among the periodic densities of particles provided the coupling constant is smaller than a certain value. This value is independent of volume. We also prove that these solutions converge as the thermodynamic limit is considered.

We consider the Hartree approximation at finite temprature. We give a justification of this approximation by using the methods of functional integration. For finite temperatures, a fixed point approach to solving the Hartree problem is proposed. For a class of two-body interactions and background potentials that include the Coulomb interaction, we prove that the Hartree equation has a unique solution provided the coupling constant is small. We also