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 $-2t-V<E<2t-V$. We further extend the analysis to non-Hermitian quasiperiodic potentials with gain and loss. In these systems, unbound states survive within analytically determined real-energy intervals, but they no longer occupy the whole interval uniformly; rather, they coexist with bound states and form a mixed bound-unbound phase. The corresponding boundaries between the mixed region and the pure bound-state regions are obtained exactly from the Lyapunov exponent. These results demonstrate that unbound states in extreme quasiperiodic potentials are controlled not only by the potential depth but also by the spectral and localization structures induced by non-Hermiticity.