2010/08/30 by Vincent Bruneau, Bruneau, Vincent, Pablo Miranda +3
Mathematics · Physics and Astronomy · #35J10 #35P20 #47F05 #81Q10 #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Spectral Theory (math.SP) #math-ph #math.MP #math.SP #msc:35J10 #msc:35P20 #msc:47F05 #msc:81Q10
paper · pdf · doi:10.48550/arxiv.1008.5182
32 pages
arxiv created 2010/08/30 · arxiv updated 2010/09/01
We consider the unperturbed operator H0 : = (-i ∇ - A)2 + W, self-adjoint in L2(\R2). Here A is a magnetic potential which generates a constant magnetic field b>0, and the edge potential W is a non-decreasing non constant bounded function depending only on the first coordinate x ∈ \R of (x,y) ∈ \R2. Then the spectrum of H0 has a band structure and is absolutely continuous; moreover, the assumption limx → ∞(W(x) - W(-x)) < 2b implies the existence of infinitely many spectral gaps for H0. We consider the perturbed operators H± = H0 ± V where the electric potential V ∈ L∞(\R2) is non-negative and decays at infinity. We investigate the asymptotic distribution of the discrete spectrum of H_± in the spectral gaps of H0. We introduce an effective Hamiltonian which governs the main asymptotic term; this Hamiltonian involves a pseudo-differential operator with generalized anti-Wick symbol equal to V. Further, we restrict our attention on perturbations V of compact support and constant sign. We establish a geometric condition on the support of V which guarantees the finiteness of the eigenvalues of H± in any spectral gap of H0. In the case where this condition is violated, we show that, generically, the convergence of the infinite series of eigenvalues of H+ (resp. H-) to the left (resp. right) edge of a given spectral gap, is Gaussian.