SPECTRAL ASYMPTOTICS FOR MAGNETIC SCHRODINGER OPERATOR WITH SLOWLY VARYING POTENTIAL

Consider the Schrodinger operator with constant magnetic field and smooth potential V : H(epsilon) = H + V(epsilon x, epsilon y), H = D-x(2) + (D-y + mu x)(2), (x, y) is an element of Omega(d), with Dirichlet boundary conditions. Here Omega(d) = Pi(d)(j=1)] - a(j), a(j)[xR(y)(d). The spectral properties of two operators H and H(epsilon) are investigated. For epsilon small enough, we study the effect of the slowly varying potential V(epsilon x, epsilon y). In particular, we derive asymptotic trace formula and we give an asymptotic expansion in powers of epsilon of the spectral shift function corresponding to (H(epsilon), H).