Spectral analysis of Dirac operators for dislocated potentials with a purely imaginary jump

TL;DR

The paper delivers a comprehensive spectral and pseudospectral analysis of a non-Hermitian Dirac operator with a dislocated imaginary jump, including an explicit matrix Green function for the unperturbed case. By leveraging a Birman-Schwinger framework, it shows stability of the essential spectrum under $L^1$ perturbations and provides sharp pseudospectral enclosures inside the instability band, along with precise asymptotics for the resolvent norm. It also characterizes how the spectrum and pseudospectrum change under long-range perturbations, establishing eigenvalue localization for $m>0$, and presents detailed results for step potentials and a weak-coupling regime, including a complete description of the weakly coupled model. These findings advance the understanding of pseudospectra for non-selfadjoint Dirac operators, with implications for stability analyses in relativistic quantum systems and related mathematical physics models.

Abstract

In this paper we present a complete spectral analysis of Dirac operators with non-Hermitian matrix potentials of the form $i\operatorname{sgn}(x)+V(x)$ where $V\in L^1$. For $V=0$ we compute explicitly the matrix Green function. This allows us to determine the spectrum, which is purely essential, and its different types. It also allows us to find sharp enclosures for the pseudospectrum and its complement, in all parts of the complex plane. Notably, this includes the instability region, corresponding to the interior of the band that forms the numerical range. Then, with the help of a Birman-Schwinger principle, we establish in precise manner how the spectrum and pseudospectrum change when $V\not=0$, assuming the hypotheses $\|V\|_{L^1}<1$ or $V\in L^1\cap L^p$ where $p>1$. We show that the essential spectra remain unchanged and that the $\varepsilon$-pseudospectrum stays close to the instability region for small $\varepsilon$. We determine sharp asymptotic for the discrete spectrum, whenever $V$ satisfies further conditions of decay at infinity. Finally, in one of our main findings, we give a complete description of the weakly-coupled model.

Figures (5)

• Figure 1: The numerical range of $\mathscr{L}_{m}$.
• Figure 2: The red lines represent the spectrum of the operator $\mathscr{L}_{m}$ when $m>0$.
• Figure 3: The red lines show the continuous spectrum of $\mathscr{L}_{0}$. These lines coincide with the essential spectra $\mathrm{e}1,\ldots,\mathrm{e}4$. The inner part in light colour shows the point spectrum. The union of both these regions form the essential spectrum $\mathrm{e}5$ and also the full $\operatorname{Spec}(\mathscr{L}_{0})$.
• Figure 4: Regions $\Lambda^{0.1}_{+}(\varepsilon)$ and $\Lambda^{0.1}_{-}(\varepsilon)$ in the box as shown, for $m=1$, $\varepsilon=0.02$ (left) and $\varepsilon=0.01$ (right). The part in blue corresponds to $\Lambda^{0.1}_{+}(\varepsilon)\setminus \Lambda^{0.1}_{-}(\varepsilon)$ and $\Lambda^{0.1}_{-}(\varepsilon)$ is shows in brick.
• Figure 5: Partition of the resolvent set of $\mathscr{L}_{m}$ in subdomains for the proof of Theorem \ref{['Theo Location m>0']}.

Theorems & Definitions (37)

• Theorem 2.1
• Corollary 2.2
• proof
• Proposition 2.3
• Theorem 2.4
• Theorem 2.5
• Proposition 2.6
• Theorem 2.7
• Proposition 3.1
• Lemma 3.2