Transmission and Reflection coefficients for Schrödinger Operators with Truncated Periodic Potentials that support defect states

We consider scattering waves through truncated periodic potentials with perturbations that support localized gap eigenstates. In a small complex neighborhood around an assumed positive bound state of the model operator, we prove the existence of a distinct zero reflection state, or transmission resonance. We compare its location to a previously found scattering resonance and use the properties of solutions near these interesting points to analyze the behavior of transmission and reflection coefficients of scattering solutions near the assumed bound state. By example, we also discuss the truncated simple harmonic oscillator and compare the analysis to the crystalline case.