On two-dimensional Dirac operators with critical delta-shell interactions

TL;DR

The paper analyzes the spectral effects of critical delta-shell interactions for a two-dimensional Dirac operator when supported on a straight line versus a circle. Using self-adjoint realizations and fiber decompositions, it shows that in the critical regime the spectral point $z_*=-\frac{\tau}{\eta}m$ behaves markedly differently by geometry: an infinitely degenerate eigenvalue on the line, and a non-eigenvalue accumulation point of a double sequence of eigenvalues on the circle. It provides explicit eigenfunctions and a detailed asymptotic description, including the role of Bessel functions in the circle case and the translation-invariant line case formulas. The results lead to conjectures about generic curves and universal features of critical singular interactions, suggesting a broader geometric/topological influence on spectral accumulation phenomena.

Abstract

We study two-dimensional Dirac operators with singular interactions of electrostatic and Lorentzscalar type, supported either on a straight line or a circle. For certain critical values of the interaction strengths, the essential spectrum of such operators comprises an isolated point lying within the mass gap. We clarify the nature of this point in both geometries. For the straight line model, this point is known to be an eigenvalue of infinite multiplicity, and we provide a detailed analysis of the corresponding eigenfunctions. By contrast, in the case of a circle, we show that the said point is not itself an eigenvalue, but rather an accumulation point of a double sequence of simple eigenvalues. In view of the high degree of symmetry of the configurations under analysis, this behavior is unexpected and our findings lead us to formulate some conjectures concerning critical singular interactions supported on generic smooth curves.

Figures (7)

• Figure 1: Plot of $\vert \psi_{\Xi}(x,y)\vert^2$ for $\eta=2$, $\tau=0$ and $\Xi(k) = b_0(k) = (\tfrac{2}{\pi})^{1/4} e^{-k^2}$.
• Figure 2: Plot of $\vert \psi_{\Xi}(x,y)\vert^2$ for $\eta=\sqrt{13}$, $\tau=-3$ and $\Xi(k)=\frac{2}{\sqrt 5}\!\left(\frac{2}{\pi}\right)^{1/4}(k+1)\,e^{-k^2}$.
• Figure 3: The discrete eigenvalues $z_k$ of $H_{\eta,\tau}$, for $\eta = +\sqrt{4+\tau^2}$ and $\tau=-5,0,5$, in yellow, blue and red, respectively.
• Figure 4: Plots of the square-modulus $\vert \psi_k\vert^2$ of the normalized eigenfunctions \ref{['eq:formk']}, as a function of the radial coordinate $r > 0$, for $k=5$ (blue), $k=10$ (red) and $k=20$ (yellow), for $m = R = 1$ and $\tau = 0$.
• Figure 5: Plot of $\vert \psi_k\vert^2$ for $k=0$ and $\eta=2\sqrt 2$, $\tau=-2$.

Theorems & Definitions (33)

• Proposition 1.1
• Theorem 1.2
• Remark 1.3
• Remark 1.4
• Remark 1.5
• Remark 1.6
• Remark 1.7
• Corollary 1.8
• Remark 1.9
• Proposition 1.10