Spectral shift function for operators with crossed magnetic and electric fields

We obtain a representation formula for the derivative of the spectral shift function $\xi (\lambda ;B,ϵ)$ related to the operators $H_0(B,\epsilon) = (D_x - By)^2 + D_y^2 + \epsilon x$and$H(B, \epsilon) = H_0(B, \epsilon) + V(x,y), : B > 0, \epsilon > 0$. We establish a limiting absorption principle for $H(B,ϵ)$ and an estimate $\mathcal{O}({ϵ}^{n-2})$ for ${\xi }^{\mathrm{\prime }}(\lambda ;B,ϵ)$, provided $\lambda \mathrm{\notin }\sigma (Q)$, where $Q = (D_x - By)^2 + D_y^2 + V(x,y).$