Spectrum Of The Iwatsuka Hamiltonian At Thresholds

We consider the bi-dimensional Schrödinger operator with unidirectionally constant magnetic field, H₀, sometimes known as the “Iwatsuka Hamiltonian”. This operator is analytically fibered, with band functions converging to finite limits at infinity. We first obtain the asymptotic behavior of the band functions and its derivatives. Using this results we give estimates on the current and on the localization of states whose energy value is close to a given threshold in the spectrum of H₀. In addition, for non-negative electric perturbations V we study the spectral properties of H₀±V, by considering the Spectral Shift Function associated to the operator pair (H₀±V,H₀). We compute the asymptotic behavior of the Spectral Shift Function at the thresholds, which are the only points where it can grow to infinity.