Quasi-Classical Asymptotics for the Pauli Operator

Abstract:

We study the behaviour of the sums of the eigenvalues of the Pauli operator in , in a magnetic field and electric field V(x) as the Planck constant ħ tends to zero and the magnetic field strength μ tends to infinity. We show that for the sum obeys the natural Weyl type formula

where σ = (d- 2)/2 + γ, with an explicit constant C _{γ, d} . If the field B has a constant direction, then this formula is uniform in μ≥ 0. The method is based on Colin de Verdiere's approach proposed in his work on “magnetic bottles” (Commun. Math Phys, 105 , 327—335 (1986)).