Dirichlet and Neumann eigenvalues for half-plane magnetic Hamiltonians

Abstract

Let H0,D (respectively, H0,N) be the Schrödinger operator in constant magnetic field on the half-plane with Dirichlet (respectively, Neumann) boundary conditions, and let Hℓ := H0,ℓ - V, ℓ = D, N, where the scalar potential V is non-negative, bounded, does not vanish identically, and decays at infinity. We compare the distribution of the eigenvalues of HD and HN below the respective infima of the essential spectra. To this end, we construct effective Hamiltonians which govern the asymptotic behavior of the discrete spectrum of Hℓ near inf σ ess (Hℓ) = inf σ(H0,ℓ), ℓ = D, N. Applying these Hamiltonians, we show that σ disc (HD) is infinite even if V has a compact support, while σ disc (HN) could be finite or infinite depending on the decay rate of V.