Exploring Quantum Information Entropy in Double Hyperbolic Potentials Under the Fractional Schrödinger Framework

In this work, we investigate quantum information entropy for double hyperbolic well potentials within the framework of the fractional Schrödinger equation (FSE). Specifically, we analyze the position and momentum Shannon entropies for two hyperbolic potentials, $U_1$ and $U_2$, as a function of the fractional derivative order $\mu$. Our findings reveal that decreasing $\mu$ enhances wave function localization in position space, thereby reducing spatial uncertainty while simultaneously increasing momentum uncertainty. We confirm the validity of the Beckner–Bialynicki-Birula–Mycielski inequality for both potentials, demonstrating its robustness across different degrees of nonlocality. Furthermore, we explore the behavior of Fisher information, observing that it increases in position space while decreases in momentum space as the well depths grow.