Trace asymptotics formula for the Schr\"odinger operators with constant magnetic fields

In this paper, we consider the 2D- Schr"odinger 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 $\mathrm{t}\mathrm{r}(\mathrm{\Phi }(H(V),h))$, where $\mathrm{\Phi }(\cdot ,h)\in C_0^{\mathrm{\infty }}(\mathbb{R};\mathbb{R})$. We also give a Weyl type asymptotics formula with optimal remainder estimate of the counting function of eigenvalues of $H(V)$.