Transmission and Reflection coefficients for Schr\"odinger Operators with Truncated Periodic Potentials that support defect states

TL;DR

This paper analyzes how truncated periodic potentials with defect states influence scattering, revealing conditions for zero reflection states and comparing them to scattering resonances, with applications to models like the harmonic oscillator.

Contribution

It introduces a detailed analysis of transmission and reflection coefficients near bound states in truncated periodic potentials supporting defect states, including new insights into zero reflection states.

Findings

Existence of zero reflection states near bound states

Comparison between transmission resonances and scattering resonances

Application to the truncated simple harmonic oscillator

Abstract

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.