Mathematical calculation for exact solutions of infinitely strongly interacting Fermi gases in tight waveguides

Exact analytical solutions of the fundamental systems of N quasi-one-dimensional spin-1/2 fermions with infinite delta repulsion in an arbitrary confining potential were presented in our previous letter (Guan et al 2009 Phys. Rev. Lett. 102 160402). The solutions are the simultaneous eigenstates of the Hamiltonian H and the total spin operators S2 and Sz, which fulfil Girardeau's hard-core contacting boundary condition and are antisymmetric under odd permutations among fermions. They are approximate solutions when the coefficient in the delta repulsion is large but finite, and according to the Lieb-Mattis theorem, the solution with the lowest S is the ground state for the system. A detailed mathematical calculation for the exact solutions is presented. The property of the spin-dependent reduced one-body density is discussed.