On the existence of impurity bound excitons in one-dimensional systems with zero range interactions

We consider a three-body one-dimensional Schrodinger operator with zero range potentials, which models a positive impurity with charge $kappa > 0$ interacting with an exciton. We study the existence of discrete eigenvalues as $kappa$ is varied. On one hand, we show that for sufficiently small $kappa$ there exists a unique bound state whose binding energy behaves like $kappa^4$, and we explicitly compute its leading coefficient. On the other hand, if $kappa$ is larger than some critical value then the system has no bound states.