Resonance spectrum for one-dimensional layered media

Abstract

We consider the “weighted” operator P_k= − ∂_x a(x)∂ _x on the real line with a step-like coefficient which appears when propagation of waves through 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 article, we study a transition to semi-infinite medium by taking the limit k→ ∞ . The bands of resonances in the complex lower half plane are localized below the band spectrum of the corresponding periodic problem (k=∞) with k − 1 or k resonances in each band. We prove that as k→ ∞ , the resonance spectrum converges to the real axis.

Keywords:

• One-dimensional
• Layered
• Truncated periodic
• Scattering resonances

Keywords:

• 2000 Mathematics Subject Classifications:
• 47A10
• 47A40
• 81Q10

Acknowledgements

The author would like to thank Maciej Zworski for suggesting to look at the problem considered in the present paper and for helpful discussions.