Abstract:The spectral properties of two-dimensional Schrödinger operators with δ'-potentials supported on star graphs are discussed. We describe the essential spectrum and give a complete description of situations in which the discrete spectrum is non-trivial but finite. A more detailed study is presented for the case of a star graph with two branches, in particular, the small angle asymptotics for the eigenvalues is obtained.
We consider the one-parametric family of self-adjoint realizations of the two-dimensional massive Dirac operator with a Lorentz scalar δ-shell interaction of strength τ∈R\{−2,0,2} supported on a broken line of opening angle 2ω with ω∈(0,π2). The essential spectrum of any such self-adjoint realization is symmetric with respect to the origin with a gap around zero whose size depends on the mass and, for τ < 0, also on the strength of the interaction, but does not depend on ω. As the main result, we prove that for any N∈N and strength τ ∈ (−∞, 0)\{−2} the discrete spectrum of any such self-adjoint realization has at least N discrete eigenvalues, with multiplicities taken into account, in the gap of the essential spectrum provided that ω is sufficiently small. Moreover, we obtain an explicit estimate on ω sufficient for this property to hold. For τ ∈ (0, ∞)\{2}, the discrete spectrum consists of at most one simple eigenvalue.
We consider the one-parametric family of self-adjoint realizations of the two-dimensional massive Dirac operator with a Lorentz scalar δ-shell interaction of strength τ∈R\{−2,0,2} supported on a broken line of opening angle 2ω with ω∈(0,π2). The essential spectrum of any such self-adjoint realization is symmetric with respect to the origin with a gap around zero whose size depends on the mass and, for τ < 0, also on the strength of the interaction, but does not depend on ω. As the main result, we prove that for any N∈N and strength τ ∈ (−∞, 0)\{−2} the discrete spectrum of any such self-adjoint realization has at least N discrete eigenvalues, with multiplicities taken into account, in the gap of the essential spectrum provided that ω is sufficiently small. Moreover, we obtain an explicit estimate on ω sufficient for this property to hold. For τ ∈ (0, ∞)\{2}, the discrete spectrum consists of at most one simple eigenvalue.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.