On the calculation of the discrete spectra of one‐dimensional Dirac operators.

In this work, we consider the one‐dimensional Dirac operator DQ+Qsu(x)=1iσ2ddx+Q(x)+Qs(x)u(x),$$ {\mathfrak{D}}_{\mathrm{Q}+{\mathrm{Q}}_s}u(x)=\left(\frac{1}{\mathrm{i}}{\sigma}_2\frac{d}{dx}+\mathrm{Q}(x)+{\mathrm{Q}}_s(x)\right)u(x), $$where σ2$$ {\sigma}_2 $$ is Pauli's matrix, Q$$ \mathrm{Q} $$ is a 2×2$$ 2\times 2 $$‐matrix representing a regular potential that includes the electrostatic and scalar interactions as well as the anomalous magnetic momentum, Qs$$ {\mathrm{Q}}_s $$ is a singular potential consisting of N$$ N $$ delta distributions Qs(x)=∑j=1NAjδx−xj,$$ {\mathrm{Q}}_s(x)=\sum \limits_{j=1}^N{\mathrm{A}}_j\delta \left(x-{x}_j\right), $$Aj$$ {\mathrm{A}}_j $$ (j=1,⋯,N$$ j=1,\cdots, N $$) are 2×2$$ 2\times 2 $$‐matrices representing the strengths of Dirac deltas, and u=u1u2⊤$$ u={\left({u}^1\kern0.5em {u}^2\right)}^{\top } $$ is a two‐spinor. We associate to the operator DQ+Qs$$ {\mathfrak{D}}_{\mathrm{Q}+{\mathrm{Q}}_s} $$ an unbounded in L2ℝ,ℂ2$$ {L}^2\left(\mathbb{R},{\mathbb{C}}^2\right) $$ symmetric operator denoted by DQ,Ω$$ {\mathcal{D}}_{\mathrm{Q},\Omega} $$, where Ω=xjj=1N$$ \Omega ={\left\{{x}_j\right\}}_{j=1}^N $$ is the support of singular potential Qs$$ {\mathrm{Q}}_s $$. The operator DQ,Ω$$ {\mathcal{D}}_{\mathrm{Q},\Omega} $$ includes only the regular potential Q$$ \mathrm{Q} $$ together with certain interaction conditions at each point xj∈Ω$$ {x}_j\in \Omega $$. The paper presents a method for determining the discrete spectrum of the operator DQ,Ω$$ {\mathcal{D}}_{\mathrm{Q},\Omega} $$ for arbitrary potential Q$$ \mathrm{Q} $$ whose entries are given by L∞(ℝ)$$ {L}^{\infty}\left(\mathbb{R}\right) $$‐functions. The eigenvalues λn$$ {\lambda}_n $$ of the operator DQ,Ω$$ {\mathcal{D}}_{\mathrm{Q},\Omega} $$ are the zeros of a dispersion equation ηΩ(λ)=0$$ {\eta}_{\Omega}\left(\lambda \right)=0 $$, where the characteristic function ηΩ$$ {\eta}_{\Omega} $$ is determined explicitly in terms of power series involving the spectral parameter λ$$ \lambda $$. The construction of the characteristic function ηΩ$$ {\eta}_{\Omega} $$ from a set of monodromy matrices and the interaction conditions is presented in the paper. Moreover, its power series representation leads to an efficient numerical method for calculating the eigenvalues of the Dirac operator DQ,Ω$$ {\mathcal{D}}_{\mathrm{Q},\Omega} $$ from the zeros of certain approximate function η˜Ω$$ {\tilde{\eta}}_{\Omega} $$ which is obtained by truncating the series up to a finite number of terms. Several examples show the applicability and accuracy of the numerical method. [ABSTRACT FROM AUTHOR]