Spectrum of the Iwatsuka Hamiltonian at thresholds

Abstract

ABSTRACT. We consider the bi-dimensional Schrödinger 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 threshold in the spectrum of H 0 . In addition, for non-negative electric perturbations V we study the spectral density of H 0˘V , by considering the Spectral Shift Function associated to the operator pair pH 0V , H 0 q. We compute the asymptotic behavior of the Spectral Shift Function at the thresholds, which are the only points where it can grows to infinity.

Absence of embedded eigenvalues for translationally invariant magnetic Laplacians

“…Note that this issue is closely connected to the existence of edge currents (quantified by Mourre estimates), as explained for instance in [4], where positive magnetic fields are considered. The reader might also want to consider -the physical considerations in [11], -the paper [3] considering the dispersion curves associated with non-smooth magnetic fields, -the contribution [13] generalizing Iwatsuka's result by adding a translationnaly invariant electric potential, -the paper [14] devoted to dimension three and fields having cylindrical and longitudinal symmetries, -or [7] where various estimates of the band functions are established for increasing, positive, and bounded magnetic fields, and applied to the estimate of quantum currents. Then L has no eigenvalue.…”

Magnetic fields and boundary conditions in spectral and asymptotic analysis

“…On the opposite side, as we explain in Section 5.3, the quantum states localized near these thresholds have a component with very small velocity along the invariance direction, moreover they are localized at infinity, both in space and in frequency. This analysis can be found in [77,103].…”

“…It is expected that this function may be singular at the thresholds. In Section 6.1, we present the results from [103]: we consider the Iwatsuka model submitted to an electric perturbation, and we provide the a priori and the precise behavior of this function near thresholds, depending on the hypotheses on the decay of the magnetic field and the electric perturbation.…”

“…There exist a slightly more general version of this theorem, for a magnetic field which satisfies (5.2) only asymptotically, see [103,Theorem 2.2]. This theorem includes a two-term asymptotic, but we have preferred to present the above version, easier to read.…”

“…Here, E j " E j " 2j ´1 in model A and E j " B `Ej in model B. Then, using the asymptotics of Section 5.2, we get in [77] and [103]:…”