Affiliations: Department of Mathematics, University of Helsinki, Helsinki, Finland. E-mail: [email protected] | Department of Mathematics, University of Kentucky, Lexington, KY, USA. E-mail: [email protected] | CPT, CNRS UMR 7332, Aix Marseille Université, Marseille, France and Université du Sud Toulon-Var, La Garde, France. E-mail: [email protected]
Note: [] Corresponding author: Peter D. Hislop, Department of Mathematics, University of Kentucky, Lexington, KY 40506-0027, USA. E-mail: [email protected]
Abstract: We study two-dimensional magnetic Schrödinger operators with a magnetic field that is equal to b>0 for x>0 and −b for x<0. This magnetic Schrödinger operator exhibits a magnetic barrier at x=0. The unperturbed system is invariant with respect to translations in the y-direction. As a result, the Schrödinger operator admits a direct integral decomposition. We analyze the band functions of the fiber operators as functions of the wave number and establish their asymptotic behavior. Because the fiber operators are reflection symmetric, the band functions may be classified as odd or even. The odd band functions have a unique absolute minimum. We calculate the effective mass at the minimum and prove that it is positive. The even band functions are monotone decreasing. We prove that the eigenvalues of an Airy operator, respectively, harmonic oscillator operator, describe the asymptotic behavior of the band functions for large negative, respectively positive, wave numbers. We prove a Mourre estimate for perturbations of the magnetic Schrödinger operator and establish the existence of absolutely continuous spectrum in certain energy intervals. We prove lower bounds on magnetic edge currents for states with energies in the same intervals. We also prove that these lower bounds imply stable lower bounds for the asymptotic currents. Because of the unique, non-degenerate minimum of the first band function, we prove that a perturbation by a slowly decaying negative potential creates an infinite number of eigenvalues accumulating at the bottom of the essential spectrum from below. We establish the asymptotic behavior of the eigenvalue counting function for these infinitely-many eigenvalues below the bottom of the essential spectrum.
Keywords: magnetic Schrödinger operators, snake orbits, magnetic field, magnetic edge states, edge conductance
