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

## TL;DR

This paper investigates the existence of impurity bound excitons in a one-dimensional quantum system with zero-range interactions, revealing conditions for bound state formation and their energy dependence on impurity charge.

## Contribution

It provides a rigorous analysis of bound state existence in a 1D Schrödinger operator with zero-range potentials, including explicit energy asymptotics for small impurity charges.

## Key findings

- Existence of a unique bound state for small impurity charge with energy proportional to ppa^4
- No bound states exist when impurity charge exceeds a critical value
- Explicit calculation of the leading coefficient in the energy expansion

## Abstract

We consider a three-body one-dimensional Schr\"odinger 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.