The time-dependent Schrödinger equation in non-integer dimensions for constrained quantum motion

Abstract

We propose a theoretical model, based on a generalized Schrödinger equation, to study the behavior of a constrained quantum system in non-integer, lower than two-dimensional space. The non-integer dimensional space is formed as a product space X×Y, comprising x-coordinate with a Hausdorff measure of dimension α1=D−1 (1<D<2) and y-coordinate with the Lebesgue measure of dimension of length (α2=1). Geometric constraints are set at y=0. Two different approaches to find the Green's function are employed, both giving the same form in terms of the Fox H-function. For D=2, the solution for two-dimensional quantum motion on a comb is recovered.