Efficient extraction of resonant states in systems with defects

TL;DR

This paper presents a novel numerical method for efficiently computing defect-induced resonances in crystals by solving an integral equation localized in the defect region, utilizing a new Green function computation technique called Brillouin Complex Deformation.

Contribution

The paper introduces a new integral equation approach combined with Brillouin Complex Deformation for efficient resonance computation in defective crystal systems.

Findings

Resonances can be expressed via a localized 'resonance source' within the defect.

The Green function of the perfect crystal can be computed efficiently using BCD.

The method enables accurate and efficient resonance calculations in defective crystals.

Abstract

We introduce a new numerical method to compute resonances induced by localized defects in crystals. This method solves an integral equation in the defect region to compute analytic continuations of resolvents. Such an approach enables one to express the resonance in terms of a "resonance source", a function that is strictly localized within the defect region. The kernel of the integral equation, to be applied on such a source term, is the Green function of the perfect crystal, which we show can be computed efficiently by a complex deformation of the Brillouin zone, named Brillouin Complex Deformation (BCD), thereby extending to reciprocal space the concept of complex coordinate transformations.