Resonances for magnetic Stark Hamiltonians in two-dimensional case

We study the resonances of the two-dimensional Schrödinger operator $P_1{(B;\beta)}={(D_x-By)}^2+D_y^2+\beta x+V{(x,y)}$ , B > 0, β > 0, with constant magnetic and electric fields. We define the resonances of P1(B; β) and the spectral shift function ξ(λ) related to P1(B; β) and P0(B; β) = P1(B; β) − V(x, y) without any restriction on B and β. For strong magnetic fields (B → ∞) we obtain a representation of the derivative of ξ(λ), a trace formula for $\text{tr}{(f{(P_1{(B;\beta)})}-f{(P_0{(B;\beta)})})}$ and an upper bound for the number of the resonances lying in $\{z\in{3/8}\text{:}{\vert\Re_z-{(2n-1)}B\vert}\leq\alpha B$ , $\operatorname{Im}z\geq\mu\operatorname{Im}\theta\}$ , 0 < α < 1, 0 < μ < 1, Im θ < 0. Moreover, for B → ∞ we examine the resonance-free domains and show that the resonances are included in the neighborhoods $\{z\in 3/8\text{:}{\vert\Re_z-{(2n-1)}B\vert}\leq C_0\}$ , where (2n − 1)B are the Landau levelsand C0 > 0 is a constant independent of B and n ∈ ℕ* = ℕ \ {0}.