Spectral problems for operators with crossed magnetic and electric fields

We obtain a representation formula for the derivative of the spectral shift function xi(lambda; B, epsilon) related to the operators H(0)(B, epsilon) = (D(x) - B(y))(2) + D(y)(2) + epsilon x and H(B, epsilon) = H(0)(B, epsilon) + V (x, y), B > 0, epsilon > 0. We prove that the operator H(B, epsilon) has at most a finite number of embedded eigenvalues on R which is a step to the proof of the conjecture of the absence of embedded eigenvalues of H in R. Applying the formula for xi'(lambda, B, epsilon), we obtain a semiclassical asymptotics of the spectral shift function related to the operators H(0)(h) = (hD(x) - B(y))(2) + h(2)D(y)(2) + epsilon x and H(h) = H(0)(h) + V (x, y).