# Resonances in the one dimensional Stark effect in the limit of small field

## TL;DR

This paper investigates the behavior of resonances in a one-dimensional quantum system under a small electric field, analyzing their asymptotic properties and relation to scattering coefficients.

## Contribution

It provides a detailed asymptotic analysis of resonances in the 1D Stark effect at small fields, linking them to zeros of the analytic continuation of reflection coefficients.

## Key findings

- Resonances occur near the real axis and specific complex lines.
- Asymptotic formulas for resonance positions are derived.
- Remarks on extending results to higher dimensions are included.

## Abstract

We discuss the resonances of Hamiltonians with constant electric field in one dimension in the limit of small field. These resonances occur near the real axis, near zeros of the analytic continuation of a reflection coefficient for potential scattering, and near the line arg z = -2\pi/3. We calculate their asymptotics. In conclusion we make some remarks about the higher dimensional problem.