Resonance spectrum for one-dimensional layered media

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.