Spectrum of the Iwatsuka Hamiltonian at thresholds

Abstract: We consider the bi-dimensional Schr\"odinger operator with unidirectionally constant magnetic field, $H_0$, 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 \emph{threshold} in the spectrum of $H_0$. In addition, for a non-negative electric perturbation $V$ we study the spectral density of $H_0\pm V$ by considering the Spectral Shift Function associated to the operator pair $(H_0\pm V,H_0)$. We describe the continuity and boundedness properties of the spectral shift function, and we compute the asymptotic behavior at the thresholds, which are the only points where it can grows to infinity.