Distribution of eigenvalues for a large circular domain wall

Determine the distribution of the eigenvalues of the two-dimensional Dirac operator with circular domain wall mass profile \(\kappa=1-2\mathbf{1}_{B_R(0)}\) in the spectral interval \((-1,1)\) as the radius \(R\) tends to infinity.

Background

The paper considers a Dirac operator whose mass changes sign across the boundary of a large disk, so that one topological phase occupies a bounded region and the other occupies its exterior. For every fixed radius, the operator is a compact perturbation of the gapped constant-mass Dirac operator, and its spectrum in $(-1,1)$ therefore consists only of eigenvalues.

The authors state that they expect the spectrum in $(-1,1)$ to become dense as $R\to\infty$, but they do not establish how those eigenvalues are distributed. Understanding this distribution would describe how the finite circular interface approximates the continuous edge spectrum of an unbounded domain wall.