Resonance spectrum for one-dimensional layered media

TL;DR

This paper analyzes the resonance spectrum of a one-dimensional layered medium operator, showing how the spectrum evolves as the medium transitions from finite to semi-infinite, revealing band structures and convergence properties.

Contribution

It provides a detailed study of the resonance spectrum transition from finite to infinite layered media, establishing convergence to the real axis and describing band structures.

Findings

Resonance spectrum forms band structures in finite layered media.

Resonances are localized below the periodic spectrum in the complex plane.

As the medium becomes semi-infinite, the resonance spectrum converges to the real axis.

Abstract

We consider the "weighted" operator $P_k=-\partial_x a(x)\partial_x$ on the line with a step-like coefficient which appears when propagation of waves thorough a finite slab of a periodic medium is studied. The medium is transparent at certain resonant frequencies which are related to the complex resonance spectrum of $P_k.$ If the coefficient is periodic on a finite interval (locally periodic) with $k$ identical cells then the resonance spectrum of $P_k$ has band structure. In the present paper we study a transition to semi-infinite medium by taking the limit $k\to \infty.$ The bands of resonances in the complex lower half plane are localized below the band spectrum of the corresponding periodic problem ($k=\infty$) with $k-1$ or $k$ resonances in each band. We prove that as $k\to \infty$ the resonance spectrum converges to the real axis.