Fractional quantum dynamics in dissipative chemical systems: Memory-driven Schrödinger models for anomalous quantum transport

We present a unified fractional framework for quantum transport in dissipative chemical systems, incorporating temporal and spatial fractional operators to model memory-driven dynamics, anomalous diffusion, and long-range coherence. Starting from a generalized open-system Hamiltonian, we derive time- and space-fractional Schrödinger equations with dissipative potentials and nonlinear reactive couplings. Rigorous analysis establishes well-posedness, spectral properties, and algebraic energy decay, while linear stability and dispersion studies reveal memory-induced spectral shifts and fractional exceptional points. Numerical simulations using spectral and exponential time-differencing schemes illustrate subdiffusive spreading, anomalous decoherence, and quasi-stationary wave localization in reactive media. The framework provides a versatile mathematical and computational platform for predicting and interpreting fractional quantum transport phenomena in complex molecular and condensed-phase environments.