On fractional time quantum dynamics

TL;DR

This paper explores the application of fractional calculus to quantum dynamics, specifically through a fractional time Schrödinger equation, revealing connections to comb models and providing analytical Green functions.

Contribution

It introduces a fractional time Schrödinger equation and analyzes its properties, including spectral equivalence to comb models and derivation of Green functions.

Findings

Fractional time Schrödinger equation differs from standard by fractional derivative.

For α=1/2, the system is isospectral to a comb model.

Analytical Green functions are derived for the fractional system.

Abstract

Application of the fractional calculus to quantum processes is presented. In particular, the quantum dynamics is considered in the framework of the fractional time Schr\"odinger equation (SE), which differs from the standard SE by the fractional time derivative: $\frac{\prt}{\prt t}\to \frac{\prt^{\alpha}}{\prt t^{\alpha}}$. It is shown that for $\alpha =1/2$ the fractional SE is isospectral to a comb model. An analytical expression for the Green functions of the systems are obtained. The semiclassical limit is discussed.