Generalized Schrödinger operators of the graph type

Pavel Exner Czech Academy of Sciences

We discuss generalized Schrödinger operators of the form −Δ − αδ(x − Γ) in L^{2}(ℝ^{2}), where Γ is a planar graph. Various spectral properties of these operators can be derived by combining minimax estimates with the generalized Birman-Schwinger principle. In particular we give conditions under which the geometry gives rise to a non-void discrete spectrum for a noncompact Γ. We also discuss the relations between the asymptotic properties of such operators as α → ∞ and the spectral behavior of a suitable operator on L^{2}(Γ) in the situation when Γ an infinite curve or a loop, as well as emergence of persistent currents in such an asymptotics.