Trace asymptotics formula for the Schrödinger operators with constant magnetic fields

In this paper, we consider the 2D  -Schrödinger operator with constant magnetic field H(V)=(Dx−By)2+Dy2+Vh(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 Vh(x,y)=V(hx,hy)Vh(x,y)=V(hx,hy), and the large coupling constant limit case Vh(x,y)=h−δV(x,y)Vh(x,y)=h−δV(x,y). We obtain a complete asymptotic expansion in powers of h2h2 of tr(Φ(H(V),h))tr(Φ(H(V),h)), where Φ(⋅,h)∈C0∞(R;R). We also give a Weyl type asymptotics formula with optimal remainder estimate of the counting function of eigenvalues of H(V)H(V).