Unbound States and Mixed Bound--Unbound Phases in Near-Infinitely Deep Potentials

We investigate the robustness of unbound states in one-dimensional quasiperiodic models with near-infinitely deep potentials. By constructing a deeper extension of the Liu-Xia model and combining inverse participation ratio (IPR) calculations with Lyapunov-exponent analysis based on Avila's global theory, we show that increasing the potential depth does not eliminate unbound states. Instead, it shifts and narrows their energy window to $-2t-V