Trace asymptotics formula for the Schrodinger operators with constant magnetic fields

In this paper, we consider the 2D-Schrodinger operator with constant magnetic field H(V) = (D-x - By)(2) + D-y(2) + V-h(x,y), where V tends to zero at infinity and h is a small positive parameter. We will be concerned with two cases: the semi-classical limit regime V-h(x,y) = V(hx,hy), and the large coupling constant limit case V-h(x,y) = h(-delta)V(x,y). We obtain a complete asymptotic expansion in powers of h(2) of tr(Phi(H(V), h)), where Phi (., h) is an element of C-0(infinity) (R;R). We also give a Weyl type asymptotics formula with optimal remainder estimate of the counting function of eigenvalues of H(V). (C) 2014 The Authors. Published by Elsevier Inc. This is an open access article under the CC BY-NC-ND license (http://creativecoranions.org/licenses/by-nc-nd/3.0/).