Discrete spectrum asymptotics for the Schrodinger operator with a singular potential and a magnetic field

Object of the study is the operator H = H-0(h,mu)+ V in L(2)(R(d)), d greater than or equal to 2, where H-0(h,mu) is the Schrodinger operator with a magnetic field of intensity mu greater than or equal to 0 and the Planck constant h epsilon (0, h(0)). The electric (real-valued) potential V = V(x) is assumed to be asymptotically homogeneous of order -beta, beta greater than or equal to 0 as x --> 0. One obtains asymptotic formulae with remainder estimates as h --> 0, mu h less than or equal to C for the trace M(s) = tr{phi g(s)(H)} where phi epsilon C-0(infinity)(R(d)), g(lambda) = lambda(-)(s), s epsilon [0, 1]. Due to the condition mu h less than or equal to C the leading term of M(s) does not depend on mu. It depends on the relation between the parameters d, s and beta. There are five regions, in which either leading terms or remainder estimates have different form. In one of these regions M(s) admits a two-term asymptotics. In this case, for an asymptotically Coulomb potential the second term coincides with the well-known Scott correction term.