Trace asymptotics formula for the Schrِdinger operators with constant magnetic fields

In this paper, we consider the 2D-Schrödinger operator with constant magnetic field H ( V ) = ( D x − B y ) 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 ( h x , h y ) , and the large coupling constant limit case V h ( x , y ) = h − δ V ( x , y ) . We obtain a complete asymptotic expansion in powers of h 2 of tr ( Φ ( H ( V ) , h ) ) , where Φ ( ⋅ , h ) ∈ C 0 ∞ ( R ; R ) . We also give a Weyl type asymptotics formula with optimal remainder estimate of the counting function of eigenvalues of H ( V ) .