Abstract
We consider the “weighted” operator Pk= − ∂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 Pk. If the coefficient is periodic on a finite interval (locally periodic) with k identical cells, then the resonance spectrum of Pk 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.
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.