Bounds for Schr\"odinger operators on the half-line perturbed by dissipative barriers

We consider Schr\"odinger operators of the form $H_R = - d^2/ d x^2 + q + i \gamma \chi_{[0,R]}$ for large $R>0$, where $q \in L^1(0,\infty)$ and $\gamma > 0$. Bounds for the maximum magnitude of an eigenvalue and for the number of eigenvalues are proved. These bounds complement existing general bounds applied to this system, for sufficiently large $R$.