---
abstract: 'We study statistics of resonances in a one-dimensional disordered chain coupled to an outer world simulated by a perfect lead. We consider a limiting case for weak disorder and derive some results which are new in these studies. The main focus of the present study is to describe statistics of the scattered complex energies. We derive compact analytic statistical results for long chains. A comparison of these results has been found to be in good agreement with numerical simulations.'
address:
- '$^1$Instituto de Ciencias F'' isicas, Universidad Nacional Aut'' onoma de M'' exico, Cuernavaca, M'' exico'
- '$^2$Department of Physics, Technion-Israel, Institute of Technology, Haifa 32000, Israel'
author:
- 'Vinayak$^{1,2}$'
title: 'Statistics of resonances in a one-dimensional chain: a weak disorder limit'
---

Introduction
============

Resonant phenomena have received much attention in atomic and nuclear physics and more recently in chaotic and disordered systems [@disordered; @GG:00; @Casati:99; @KS:06; @KS:08; @JF:09; @chaos]. Complex energies, $\tilde{{E}}_{\alpha}=E_{\alpha}-\frac{i}{2}\,\Gamma_{\alpha}$, which correspond to poles of the scattering matrix on the unphysical sheet, characterize resonances [@LandauL]. Resonances correspond to the long-lived quasi-stationary states which eventually decay to continuum while distribution of resonance widths, $P(\Gamma)$, determines decay of the corresponding survival probability with time.

In recent 