DISCRETE SPECTRUM ASYMPTOTICS FOR THE SCHRÖDINGER OPERATOR WITH A SINGULAR POTENTIAL AND A MAGNETIC FIELD

Object of the study is the operator H=H0(h, µ)+V in L (Rd), d≥2, where H0(h, μ) is the Schrödinger operator with a magnetic field of intensity μ≥0 and the Planck constant h∈(0, h0]. The electric (real-valued) potential V=V(x) is assumed to be asymptotically homogeneous of order −β, β≥0 as x→0. One obtains asymptotic formulae with remainder estimates as h→0, μh≤C for the trace Ms=tr{ɸgs(H)} where [Formula: see text], s∈[0, 1]. Due to the condition μh≤C the leading term of Ms does not depend on μ. It depends on the relation between the parameters d, s and β. There are five regions, in which either leading terms or remainder estimates have different form. In one of these regions Ms admits a two-term asymptotics. In this case, for an asymptotically Coulomb potential the second term coincides with the well-known Scott correction term.