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Edge currents and eigenvalue estimates for magnetic barrier Schrödinger operators. (English) Zbl 1303.35080

Summary: 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.

MSC:

35Q40 PDEs in connection with quantum mechanics
35J10 Schrödinger operator, Schrödinger equation
35P15 Estimates of eigenvalues in context of PDEs
35P30 Nonlinear eigenvalue problems and nonlinear spectral theory for PDEs
81Q10 Selfadjoint operator theory in quantum theory, including spectral analysis
35B40 Asymptotic behavior of solutions to PDEs
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