New numerical analysis of Riemann-Liouville time-fractional Schrödinger with power, exponential decay, and Mittag-Leffler laws

The mathematical equation that describes how the quantum state of a quantum system changes during time was considered within the framework of fractional differentiation with three different derivatives in Riemann-Liouville sense. The fractional derivatives used in this work are constructed based on power, exponential decay, and Mittag-Leffler law. A new numerical scheme for fractional derivative in Riemann-Liouville sense is presented and used to solve numerically the Schrödinger equation. The stability analysis of each model is presented in detail.