The authors calculate the second virial coefficient of the quantum Lorentz gas with point interaction. It can be evaluated in closed form for arbitrary strength of the interaction. Particular attention is given to the critical coupling which gives rise to a zero-energy resonance. Point interactions satisfy a modified Levinson's theorem; as a consequence they show that there is no cancellation between the bound state and the scattering contributions in the high-temperature expansion of the second virial coefficient, contrary to the regular potentials case. They also derive high- and low-energy sum rules for the point interaction. As an application of the sum rules they rederive a high-temperature expansion for the second virial coefficient.
Without any approximation, exact relations for the spin-density correlation functions in an interacting electron gas under an arbitrary nonuniform external magnetic field are obtained by making use of the nonperturbative canonical formulation of quantum many-body theory [T. Toyoda, Ann. Phys. (N.Y.) 173, 226 (1987)]. The obtained results contain
f-sum rules, a finite-temperature generalized Ward-Takahashi relation, as well as a finite-temperature version of the Nambu-Goldstone theorem with respect to the spontaneous symmetry breaking of the spin-rotational symmetry.