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. In this paper we study the perturbed quadratic Hamiltonian in two-dimensional case, P (b, ω) = (Dx − by) + D y + ωx − √ b2 + ω2 + V (x, y). Here, b is the strong constant magnetic field, ω 6= 0 is a fixed constant, and the potential V vanishes at infinity. For f ∈ C 0 ((−∞, 0);R) and b large enough, we give a full asymptotic expansion in powers of b of the trace of f(P (b, ω)). Moreover, we also obtain a Weyl formula with optimal remainder estimate of the counting function of eigenvalues of P (b, ω) as b → ∞.

Spectral asymptotics for two-dimensional Schrödinger operators with strong magnetic fields