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Fractional-time Schrödinger equation: fractional dynamics on a comb. (English) Zbl 1225.81053
Summary: The physical relevance of the fractional time derivative in quantum mechanics is discussed. It is shown that the introduction of the fractional time Schrödinger equation (FTSE) in quantum mechanics by analogy with the fractional diffusion $\frac{\partial }{\partial t}\to \frac{{\partial }^{\alpha }}{\partial {t}^{\alpha }}$ can lead to an essential deficiency in the quantum mechanical description, and needs special care. To shed light on this situation, a quantum comb model is introduced. It is shown that for $\alpha =1/2$, the FTSE is a particular case of the quantum comb model. This exact example shows that the FTSE is insufficient to describe a quantum process, and the appearance of the fractional time derivative by a simple change $\frac{\partial }{\partial t}\to \frac{{\partial }^{\alpha }}{\partial {t}^{\alpha }}$ in the Schrödinger equation leads to the loss of most of the information about quantum dynamics.
##### MSC:
 81Q05 Closed and approximate solutions to quantum-mechanical equations 35R11 Fractional partial differential equations 26A33 Fractional derivatives and integrals (real functions)