Physics:Variable-order fractional Schrödinger equation

As a natural generalization of the fractional Schrödinger equation, the variable-order fractional Schrödinger equation has been exploited to study fractional quantum phenomena

[math]\displaystyle{ i\hbar \frac{\partial \psi^{\alpha(\mathbf{r})} (\mathbf{r},t)}{\partial t^{\alpha(\mathbf{r})} }= (-\hbar^2\Delta )^{\frac{\beta(t)}{2}}\psi (\mathbf{r},t)+V(\mathbf{r},t)\psi (\mathbf{r},t). }[/math]

where Δ = ∂²/∂r² is the Laplace operator and the operator (−ħ²Δ)^{β (t)/2} is the variable-order fractional quantum Riesz derivative.