Dirac–Krein Systems on Star Graphs

Abstract

Abstract. We study the spectrum of a self-adjoint Dirac-Krein operator with potential on a compact star graph G with a finite number n of edges. This operator is defined by a Dirac-Krein differential expression with summable matrix potentials on each edge, by self-adjoint boundary conditions at the outer vertices, and by a self-adjoint matching condition at the common central vertex of G. Special attention is paid to Robin matching conditions with parameter τ ∈ R ∪ {∞}. Choosing the decoupled operator with Dirichlet condition at the central vertex as a reference operator, we derive Krein's resolvent formula, introduce corresponding Weyl-Titchmarsh functions, study the multiplicities, dependence on τ , and interlacing properties of the eigenvalues, and prove a trace formula. Moreover, we show that, asymptotically for R → ∞, the difference of the number of eigenvalues in the intervals [0, R) and [−R, 0) deviates from some integer κ0, which we call dislocation index, at most by n + 2.Mathematics Subject Classification (2010). 81Q10, 81Q35, 47A10, 47A75.

Nonlinear Dirac Equation on Graphs with Localized Nonlinearities: Bound States and Nonrelativistic Limit

“…Lemma 4.3. Let p ∈ (2,6). Under the assumptions (18), (19) and (20), the sequence (ψ n ) is bounded in H 1 (G, C 2 ) (uniformly with respect to n).…”

Completeness Property of One-Dimensional Perturbations of Normal and Spectral Operators Generated by First Order Systems

“…(ii) Note also that an important role in proving Riesz basis property in [24,27] and [39] is playing the following asymptotic formula λ n = λ 0 n + o(1), as n → ∞, n ∈ Z, (3.36) for the eigenvalues {λ n } n∈Z of the operator L C,D (B, Q) with regular BC (and summable potential matrix Q), where {λ 0 n } n∈Z is the sequence of eigenvalues of the unperturbed operator L C,D (B, 0). Note also that formula (3.36) has recently been applied to investigation of spectral properties of Dirac systems on star graphs [1].…”

“…Since Q 12 (x) = 0 for x ∈ [a,1], the solution of the second equation in (4.12) on the interval [a, 1] is f 2 (x) = C 2 e ib 2 λx . The second boundary condition in (4.11) implies that C 2 = 0, hencef 2 (x) = 0 for x ∈ [a, 1].…”

A note on the Dirac operator with Kirchoff-type vertex conditions on noncompact metric graphs

“…The study of the Dirac operator (and other kind of operators) on metric graphs has has been carried on in several works in the last years (see, e.g., [7,12,18,43,37,39]). On the contrary, concerning the existence of standing waves for the NLDE on metric graphs, to our knowledge, the first rigorous mathematical work on the subject is [16], where a nonlinearity concentrated on the compactcore of the graph is considered.…”