Edge states induced by Iwatsuka Hamiltonians with positive magnetic fields

We study purely magnetic Schr\"odinger operators in two-dimensions $(x,y)$ with magnetic fields $b(x)$ that depend only on the $x$-coordinate. The magnetic field $b(x)$ is assumed to be bounded, there are constants $0 < b_- < b_+ < \infty$ so that $b_- \leq b(x) \leq b_+$, and outside of a strip of small width $-\epsilon < x < \epsilon$, where $0 < \epsilon < b_-^{-1/2}$, we have $b(x) = b_\pm x$ for $\pm x > \epsilon$. The case of a jump in the magnetic field at $x=0$ corresponding to $\epsilon=0$ is also studied. We prove that the magnetic field creates an effective barrier near $x=0$ that causes edge currents to flow along it consistent with the classical interpretation. We prove lower bounds on edge currents carried by states with energy localized inside the energy bands of the Hamiltonian. We prove that these edge current-carrying states are well-localized in $x$ to a region of size $b_-^{-1/2}$, also consistent with the classical interpretation. We demonstrate that the edge currents are stable with respect to various magnetic and electric perturbations. For a family of perturbations compactly supported in the $y$-direction, we prove that the time asymptotic current exists and satisfies the same lower bound.