Dirichlet and Neumann Eigenvalues for Half-Plane Magnetic Hamiltonians

2012/12/07 by Vincent Bruneau, Bruneau, Vincent, Pablo Miranda +3
Mathematics · Physics and Astronomy · #35J10 #35P20 #47F05 #81Q10 #Analysis of PDEs (math.AP) #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Spectral Theory (math.SP) #math-ph #math.AP #math.MP #math.SP #msc:35J10 #msc:35P20 #msc:47F05 #msc:81Q10

paper · pdf · doi:10.48550/arxiv.1212.1727

23 pages

arxiv created 2012/12/07 · arxiv updated 2012/12/11

Abstract

Let H_{0, D} (resp., H_0,N) be the Schroedinger operator in constant magnetic field on the half-plane with Dirichlet (resp., Neumann) boundary conditions, and let H_ℓ : = H_{0, 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 H_D and H_N below the respective infima of the essential spectra. To this end, we construct effective Hamiltonians which govern the asymptotic behaviour of the discrete spectrum of H_ℓ near inf σ_ess(H_ℓ) = inf σ(H_0,ℓ), ℓ = D,N. Applying these Hamiltonians, we show that σ_disc(H_D) is infinite even if V has a compact support, while σ_disc(H_N) could be finite or infinite depending on the decay rate of V.

Related

• Eigenvalue Asymptotics for a Schrödinger Operator with Non-Constant Magnetic Field Along One Direction
• Threshold Singularities of the Spectral Shift Function for a Half-Plane\n Magnetic Hamiltonian
• On the asymptotic number of edge states for magnetic Schrödinger operators
• Asymptotics for the magnetic Dirichlet-to-Neumann eigenvalues in general domains
• Spectral Properties of a Magnetic Quantum Hamiltonian on a Strip