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Propagating edge states for a magnetic Hamiltonian. (English) Zbl 0930.35144
The authors consider motion of a charged particle in a half plane subjected to a perpendicular constant magnetic field \(B\) and a weak impurity potential \(W_B\), with \(\|W_B\|_\infty< \delta<1/2\). It is known that circular trajectories close to the edge, i.e. at a distance smaller than \((E/B)^{1/2}\) will cause the particle to speed along the edge with velocity of order \(E^{1/2}\), where \(E\) is the energy of the particle, while further away from the edge in the so-called bulk states, it moves much slower. However not much is known when impurities are presented. The magnetic Hamiltonian considered here is \[ H_0= 1/2 p^2_x+ 1/2(p_y- Bx)^2, \] with a Dirichlet boundary condition at \(x= 0\). Adding the weak impurity modifies the Hamiltonian to \(\widehat H= H_0+ W_B\). The magnetic length scale is given by \(B^{-1/2}\), indicating that the edge effects are felt only at the \(B^{-1/2}\) distance from the edge. The authors introduce for each Landau band the \(H_0\)-invariant band and edge spaces.
The authors comment that most of their results are well established for the classical case. However in the quantized setting these results are new. For example, no matter how fast the impurity potential \(W_B\) fluctuates, some states propagate with velocity close to \(B^{1/2}\). They also give a rigorous proof of existence of a phenomenon related to the quantum Hall effect, which was pointed out in 1982 by B. I. Halperin [Phys. Rev. B 38, 285-290 (1982)].

MSC:
35Q40 PDEs in connection with quantum mechanics
81V70 Many-body theory; quantum Hall effect
81Q05 Closed and approximate solutions to the Schrödinger, Dirac, Klein-Gordon and other equations of quantum mechanics
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