Defect Resonances of Truncated Crystal Structures

Defects in the atomic structure of crystalline materials may spawn electronic bound states, known as \emph{defect states}, which decay rapidly away from defect. Simplified models defect states typically assume is surrounded on all sides by an infinite perfectly material. In reality surrounding must be finite, and certain contexts can small enough that edge effects are significant. this work we investigate these prove following result. Suppose a one-dimensional material hosting positive energy state truncated distance $M$ Then, for sufficiently large $M$, there exists resonance \emph{exponentially close} (in $M$) to eigenvalue. It follows hosts metastable with exponentially long lifetime. Our methods allow both frequency associated resonant computed orders $e^{-M}$. We expect result particular interest context photonic crystals, where used wave-guiding structures relatively small. Finally, under mild additional assumption if has negative then exponentially-close energy.