Effective Potential from the Generalized Time-Dependent Schrodinger Equation

We analyze the generalized time-dependent Schrodinger equation for the force free case, as a generalization, for example, of the standard time-dependent Schrodinger equation, time fractional Schrodinger equation, distributed order time fractional Schrodinger equation, and tempered in time Schrodinger equation. We relate it to the corresponding standard Schrodinger equation with effective potential. The general form of the effective potential that leads to a standard time-dependent Schrodinger equation with the same solution as the generalized one is derived explicitly. Further, effective potentials for several special cases, such as Dirac delta, power-law, Mittag-Leffler and truncated power-law memory kernels, are expressed in terms of the Mittag-Leffler functions. Such complex potentials have been used in the transport simulations in quantum dots, and in simulation of resonant tunneling diode