On the spectral theory of Gesztesy–Šeba realizations of 1-D Dirac operators with point interactions on a discrete set

Abstract

We investigate spectral properties of Gesztesy-Šeba realizations DX,α and D X,β of the 1-D Dirac differential expression D with point interactions on a discrete set X = {xn} ∞ n=1 ⊂ R. Here α := {αn} ∞ n=1 and β := {βn} ∞ n=1 ⊂ R. The Gesztesy-Šeba realizations DX,α and D X,β are the relativistic counterparts of the corresponding Schrödinger operators HX,α and H X,β with δ-and δ ′ -interactions, respectively. We define the minimal operator DX as the direct sum of the minimal Dirac operators on the intervals (xn−1, xn). Then using the regularization procedure for direct sum of boundary triplets we construct an appropriate boundary triplet for the maximal operator D * X in the case d * (X) := inf{|xi − xj| , i = j} = 0. It turns out that the boundary operators BX,α and B X,β parameterizing the realizations DX,α and D X,β are Jacobi matrices. These matrices substantially differ from the ones appearing in spectral theory of Schrödinger operators with point interactions. We show that certain spectral properties of the operators DX,α and D X,β correlate with the corresponding spectral properties of the Jacobi matrices BX,α and B X,β , respectively. Using this connection we investigate spectral properties (self-adjointness, discreteness, absolutely continuous and singular spectra) of Gesztesy-Šeba realizations. Moreover, we investigate the non-relativistic limit as the velocity of light c → ∞. Most of our results are new even in the case d * (X) > 0. are naturally associated with (1.1). Namely, assuming that I = (a, +∞) and I = N one defines the operators H 0

Boundary pairs associated with quadratic forms

“…One of the main examples in is the Laplacian on with infinitely many delta interactions on it (in other words, Robin‐type perturbations of the Neumann Laplacian on , see Remark (ii)), a case also treated in detail in (see Remark 6.1 and also , , for other recent applications of boundary triple methods to delta‐type Schrödinger operators). Brasche et al.…”

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

“…Recall that, according to (22) we can decompose the form domain Y as the orthogonal sum of the positive and negative spectral subspaces for the operator D, i.e.…”

“…and thus the resolvent of the operator D can be regarded as a perturbation of the resolvent of the operator D 0 . In the above formula γ(·) and M (·) are the gamma-field and the Weyl function, respectively, associated with D (see [22]). It turns out that the operator appearing in the righthand side of (83) is of finite rank.…”

Limiting absorption principle and scattering matrix for Dirac operators with δ-shell interactions

“…The two δ-perturbation terms with strengths η, τ ∈ R define the electrostatic shell interaction ηI 4 δ Γ and the Lorentz scalar shell interaction τ βδ Γ , respectively. Singular perturbations of the Dirac operator have been introduced first in [25], where the one dimensional Dirac operator with point interactions is considered, see also [3,19,21,36,41] for more results on Dirac operators with point interactions in R. Shell interactions supported on a sphere in R 3 were then introduced in [22] by using the one-dimensional results and a decomposition to spherical harmonics. This problem has been recently reconsidered in [4,5,6], where in the case of a C 2 -surface the self-adjointness and several properties of Dirac operators with electrostatic δ-perturbations are derived.…”