Generalized time-dependent Schrödinger equation in two dimensions under constraints

Abstract

We investigate a generalized two-dimensional time-dependent Schrödinger equation on a comb with a memory kernel. A Dirac delta term is introduced in the Schrödinger equation so that the quantum motion along the x-direction is constrained at y = 0. The wave function is analyzed by using Green’s function approach for several forms of the memory kernel, which are of particular interest. Closed form solutions for the cases of Dirac delta and power-law memory kernels in terms of Fox H-function, as well as for a distributed order memory kernel, are obtained. Further, a nonlocal term is also introduced and investigated analytically. It is shown that the solution for such a case can be represented in terms of infinite series in Fox H-functions. Green’s functions for each of the considered cases are analyzed and plotted for the most representative ones. Anomalous diffusion signatures are evident from the presence of the power-law tails. The normalized Green’s functions obtained in this work are of broader interest, as they are an important ingredient for further calculations and analyses of some interesting effects in the transport properties in low-dimensional heterogeneous media.

Anomalous diffusion and random search in xyz-comb: exact results

“…The inverse Laplace transform yields the time fractional diffusion equation on a twodimensional comb [29]:…”

Anomalous Relaxation and Three-Level System: A Fractional Schrödinger Equation Approach

“…These fractional operators, and the processes connected to them, have been widely analyzed in several fields, particularly in quantum mechanics, as a fundamental physics theory that describes the physical properties of nature in lighting on a subatomic and atomic level. The pioneer works of N. Laskin [27][28][29] (based on extensions of the path integral formulation [30]), lead us to a fractional Schrödinger equation and have been followed by other extensions incorporating fractional differential operators in time and space [31], non-local terms [32,33], constraints among the different spatial coordinates (comb-model) [34][35][36][37][38], and others [39,40]. Space and time fractional derivatives in the Schrödinger equation were considered to obtain soliton-type solutions in the optical field [41], while numerical solutions for the fractional Schrödinger equations were obtained in Ref [42].…”