Dirac density matrix and the Legendre transform of the kinetic energy generated by one-dimensional model potentials

In density functional theory, the single-particle kinetic energy is still not known in an orbital-free form. The natural tool to calculate such kinetic energy for a given one-body potential V is the Dirac density matrix gamma. Here, by first taking solvable one-dimensional examples, and in particular the sech(2)(x) potential, forms for the Dirac density matrix are exhibited in terms of the potential.This in turn generates the Legendre transform of the single-particle kinetic energy. Finally, one-dimensional perturbation theory for the kinetic energy density, summed to all orders in V, is presented in the appendix.