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 κ>0 interacting with an exciton. We study the existence of discrete eigenvalues as κ is varied. On one hand, we show that for sufficiently small κ there exists a unique bound state whose binding energy behaves like κ4, and we explicitly compute its leading coefficient. On the other hand, if κ is larger than some critical value, then the system has no bound states.