Accumulation of complex eigenvalues of an indefinite Sturm--Liouville operator with a shifted Coulomb potential

For a particular family of long-range potentials $V$, we prove that the eigenvalues of the indefinite Sturm–Liouville operator $A = $\mathrm{s}\mathrm{i}\mathrm{g}\mathrm{n}$(x)(-\Delta + V(x))$ accumulate to zero asymptotically along specific curves in the complex plane. Additionally, we relate the asymptotics of complex eigenvalues to the two-term asymptotics of the eigenvalues of associated self-adjoint operators.