Spectral problems for operators with crossed magnetic and electric fields

We obtain a representation formula for the derivative of the spectral shift function $ξ(λ; B, ε)$ related to the operators $H_0(B,ε) = (D_x - By)^2 + D_y^2 + εx$ and $H(B, ε) = H_0(B, ε) + V(x,y), \: B > 0, ε> 0$. We prove that the operator $H(B, ε)$ has at most a finite number of embedded eigenvalues on $\R$ which is a step to the proof of the conjecture of absence of embedded eigenvalues of $H$ in $\R.$ Applying the formula for $ξ'(λ, B, ε)$, we obtain a semiclassical asymptotics of the spectral shift function related to the operators $H_0(h) = (hD_x - By)^2 + h^2D_y^2 + εx$ and $H(h) = H_0(h) + V(x,y).$

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