Two-term asymptotics for the sum of eigenvalues of the Schrödinger operator with Coulomb singularities in a homogeneous magnetic field

Abstract

We study the sum of negative eigenvalues M(h,μ) of the Schrödinger operator H_V(h,μ) in L²(R³) with a decaying at infinity potential V and a homogeneous magnetic field of intensity μ. The parameter h denotes the Planck constant. Assuming that V has a finite number of Coulomb singularities and is smooth outside them, we establish the two-term asymptotics of M(h,μ) as h→0 and μ→∞. The second term is determined only by the singularities of V and it turns out to be the same as in the case μ=0.