2D Schrödinger operators with singular potentials concentrated near curves. Issue 13 (2nd September 2022) - Golovaty, Yuriy
ABSTRACT: We investigate the Schrödinger operators H ϵ = − Δ + W + V ϵ in R 2 with the short-range potentials V ϵ which are localized around a smooth closed curve γ . The operators H ϵ can be viewed as an approximation of the heuristic Hamiltonian H = − Δ + W + a ∂ ν δ γ + b δ γ, where δ γ is Dirac's δ -function supported on γ and ∂ ν δ γ is its normal derivative on γ . Assuming that the operator − Δ + W has only discrete spectrum, we analyse the asymptotic behaviour of eigenvalues and eigenfunctions of H ϵ . The transmission conditions on γ for the eigenfunctions u + = θ u −, θ ∂ ν u + − ∂ ν u − = β u − which arise in the limit as ϵ → 0 reveal a nontrivial connection between spectral properties of H ϵ and the geometry of γ .