# On a Schr\"odinger operator with a purely imaginary potential in the semiclassical limit

## TL;DR

This paper analyzes the spectral properties of a Schrödinger operator with a purely imaginary potential in the semiclassical limit, providing new resolvent estimates and extending previous results to more general boundary conditions and transmission problems.

## Contribution

It extends prior spectral analysis of the operator by removing restrictions on the potential and applies the techniques to Robin boundary conditions and transmission problems.

## Key findings

- Determined the left margin of the spectrum for the operator.
- Established resolvent estimates on the spectrum's left side.
- Extended analysis to Robin boundary conditions and transmission problems.

## Abstract

We consider the operator ${\mathcal A}_h=-\Delta+iV$ in the semi-classical $h\rightarrow 0$, where $V$ is a smooth real potential with no critical points. We obtain both the left margin of the spectrum, as well as resolvent estimates on the left side of this margin. We extend here previous results obtained for the Dirichlet realization of ${\mathcal A}_h$ by removing significant limitations that were formerly imposed on $V$. In addition, we apply our techniques to the more general Robin boundary condition and to a transmission problem which is of significant interest in physical applications.