Stability and Simulation of Fractional Dynamics in Nonhomogeneous Media via Generalized Quantum-Hadamard Operators

Abstract

In this study, we extend the concept of the quantum Gamma function (q-Gamma) by introducing a new (q,\tau)-deformed Gamma function. This generalization allows us to construct an enriched family of Hadamard-type fractional operators, which we then apply to the analysis of memory and decoherence in open quantum systems. The inclusion of the deformation parameter q together with the delay-like scaling parameter \tau makes the proposed (q,\tau)-Hadamard framework particularly suited to capture nonlocal and non-Markovian effects, thereby offering a flexible tool for describing structured reservoirs and anomalous dissipation. To investigate the analytical consequences, we employ a (q,\tau)-Mittag-Leffler function, through which explicit solutions are obtained for a class of fractional differential equations, with particular emphasis on population dynamics models. These solutions reveal a variety of memory-driven features, including sub-exponential decay of coherence and the occurrence of revival phenomena. Both the fractional order alpha and the deformation parameters play a decisive role in shaping the temporal behavior. Beyond population models, the framework also provides insights into fractional quantum master equations, quantum walks with fractional memory, and noise effects in quantum information processing.