arXiv:1809.10958v1 [math.SP] 28 Sep 2018 BAND FUNCTIONS OF IWATSUKA MODELS : POWER-LIKE AND FLAT MAGNETIC FIELDS PABLO MIRANDA AND NICOLAS POPOFF ABSTRACT. In this note we consider the Iwatsuka model with a postive increasing magnetic field having finite limits. The associated magnetic Laplacian is fibred through partial Fourier transform, and, for large frequencies, the band functions tend to the Landau levels, which are thresholds in the spectrum. The asymptotics of the band functions is already known when the magnetic field converge polynomially to its limits. We complete this analysis by giving the asymptotics for a regular magnetic field which is constant at infinity, showing that the band functions converge now exponentially fast toward the thresholds. As an application, we give a control on the current of quantum states localized in energy near a threshold. 1. THE I WATSUKA MODEL In this article we review and complete some results about the band function of the Iwtasuka model with an increasing positive magnetic field having finite limits. Assume that the magnetic field b : R 2 Ñp0, `8q depends only of one variable in the sense that bpx, y q“ bpxq. We assume moreover that b is C 0 , increasing, and have finite limits b ˘ as x Ñ ˘8, with 0 ă b ´ ă b ` . The model is gauge invariant, and we choose the magnetic potential Apx, y q :“p0,apxqq; with apxq :“ ż x 0 bptqdt. The magnetic Laplacian is then defined by H 0 :“ p´i∇ ´ Aq 2 “ ´B 2 x ` p´iB y ´ apxqq 2 acting in L 2 pR 2 q. Historically, this operator was introduced in order to provide an example of a magnetic Laplacian with absolutely continous spectrum see [12]. Then, this has been proved under various conditions on b, see [12, 13, 6, 20], but the fact that this is true as long as b is non-constant is still open, since [3]. With the years, this model has been widely studied, as a source of interesting questions linked to transport phenomena in the translationnally invariance direction. In this note, we describe and complete results on the asymptotics of the band functions, which is an important step when describing spectral properties at energy near the thresholds. A key tool for operators having a translation invariance is fibration through partial Fourier trans- form. In our case, denote by F y the partial Fourier transform in the y variable. Then there 1