2D Schrödinger operators with singular potentials concentrated near curves. Issue 13 (2nd September 2022)

• Record Type: Journal Article
• Authors: Golovaty, Yuriy
• Is Part Of: Applicable analysis. Volume 101:Issue 13(2022)
• Page Start: 4512
• Page End: 4532
• Publication Date: 2022-09-02
• Subjects: Schrödinger operator -- singular interaction -- δ potential -- δ′-interaction -- interaction on curve -- asymptotics of eigenvalues; Primary 35P05 -- Secondary 81Q10 -- 81Q15; Mathematical analysis -- Periodicals; 515
• DOI: 10.1080/00036811.2020.1859496
• Languages: English
• ISSNs: 0003-6811

Abstract

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 .