Comb-like geometric constraints leading to emergence of the time-fractional Schrödinger equation

Abstract

This paper presents an overview over several examples, where the comb-like geometric constraints lead to emergence of the time-fractional Schrödinger equation. Motion of a quantum object on a comb structure is modeled by a suitable modification of the kinetic energy operator, obtained by insertion of the Dirac delta function in the Laplacian. First, we consider motion of a free particle on two- and three-dimensional comb structures, and then we extend the study to the interacting cases. A general form of a nonlocal term, which describes the interactions of the particle with the medium, is included in the Hamiltonian, and later on, the cases of constant and Dirac delta potentials are analyzed. At the end, we discuss the case of non-integer dimensions, considering separately the case of fractal dimension between one and two, and the case of fractal dimension between two and three. All these examples show that even though we are starting with the standard time-dependent Schrödinger equation on a comb, the time-fractional equation for the Green’s functions appears, due to these specific geometric constraints.