x - b \right) } + {\delta \left( x + b \right) } \right), \quad \tau, \beta > 0,$$ i.e., $L^0$ is perturbed by a pair of attractive point interactions of equal strength whose centers are situatied at the same distance from the origin. In this case the operator $L = L^0 + W$ is self-adjoint; the techniques used are based on Green’s function analysis.

D. Haag, H. Cartarius, and G. Wunner [@HCW], motivated by analysis of Bose-Einstein condensates with $\mathcal{PT}$-symmetric loss and gain, focused on the case of non-Hermitian perturbations $$W = i \gamma \left[ {\delta \left( x - b \right) } - {\delta \left( x + b \right) } \right].$$ Their numerical estimates showed that for small $\gamma$ the spectrum of $L = L^0 + W$ is on the real line ${\mathbb{R}}$, and they gave some predictions on the state of decay of the disk radii where the eigenvalues of the operator $L$ are located. Now we provide a rigorous mathematical analysis of the asymptotics of eigenvalues $\lambda_n = \lambda_n(L^0 + W)$.

We follow the techniques used in [@DM06B], [@DM13Diff], [@AdMipub], [@MiSi], [@Elton03], [@Elton04] and based on careful estimates related to the resolvent representation

$$\begin{gathered}
\label{eq:rudefintro}
R  = 