Thermodynamic limit of the free electron gas on a circle

We show that for the ground state of a one dimensional free electron gas on a circle the analytic expression for the canonical ensemble partition function can be easily derived from the density matrix by assuming that the thermodynamic limit coincides with the limit of the eigenfunction expansion of the kinetic energy. This approximation fails to give the finite temperature partition function because those two limits cannot be chosen as coincident.
Comment: 3 pages