# Resonances - lost and found

## TL;DR

This paper studies the behavior of resonances in large one-dimensional Schrödinger operators with shifted or delta potentials, revealing their convergence patterns and influence on quantum dynamics.

## Contribution

It characterizes the asymptotic distribution of resonances for large $L$ and shows their impact on the time evolution despite not converging to the resonances of the limiting operator.

## Key findings

- Resonances crowd onto a horizontal line as $L o

- Resonances below the line converge to reflectionless points and other resonances
- Resonances influence long-time quantum dynamics even when not close to the limiting resonances

## Abstract

We consider the large $L$ limit of one dimensional Schr\"odinger operators $H_L=-d^2/dx^2 + V_1(x) + V_{2,L}(x)$ in two cases: when $V_{2,L}(x)=V_2(x-L)$ and when $V_{2,L}(x)=e^{-cL}\delta(x-L)$. This is motivated by some recent work of Herbst and Mavi where $V_{2,L}$ is replaced by a Dirichlet boundary condition at $L$. The Hamiltonian $H_L$ converges to $H = -d^2/dx^2 + V_1(x)$ as $L\to \infty$ in the strong resolvent sense (and even in the norm resolvent sense for our second case). However, most of the resonances of $H_L$ do not converge to those of $H$. Instead, they crowd together and converge onto a horizontal line: the real axis in our first case and the line $\Im(k)=-c/2$ in our second case. In the region below the horizontal line resonances of $H_L$ converge to the reflectionless points of $H$ and to those of $-d^2/dx^2 + V_2(x)$. It is only in the region between the real axis and the horizontal line (empty in our first case) that resonances of $H_L$ converge to resonances of $H$. Although the resonances of $H$ may not be close to any resonance of $H_L$ we show that they still influence the time evolution under $H_L$ for a long time when $L$ is large.