Rigorous semiclassical results for the magnetic response of an electron gas

Consider a free electron gas in a confining potential and a magnetic field in arbitrary dimensions. If this gas is in thermal equilibrium with a reservoir at temperature T > 0, one can study its orbital magnetic response (omitting the spin). One defines a conveniently smeared out magnetization M, and the corresponding magnetic susceptibility chi, which will be analyzed from a semiclassical point of view, namely when h (the Planck constant) is small compared to classical actions characterizing the system. Then various regimes of temperature T are studied where M and chi can be obtained in the form of suitable asymptotic h-expansions. In particular when T is of the order of h, oscillations a la de Haas-van Alphen appear, that can be linked to the classical periodic orbits of the electronic motion.