Solutions to the Schrödinger Equation: Nonlocal Terms and Geometric Constraints

Abstract

Here, we investigate a three-dimensional Schrödinger equation that generalizes the standard framework by incorporating geometric constraints. Specifically, the equation is adapted to account for a backbone structure exhibiting memory effects dependent on both time and spatial position. For this, we incorporate an additional term in the Schrödinger equation with a nonlocal dependence governed by short- or long-tailed distributions characterized by power laws associated with Lévy distributions. This modification also introduces a backbone structure within the system. We derive solutions that reveal various behaviors using Green’s function approach expressed in terms of Fox H-functions.

1. Introduction

Some common features of the complex media leading to anomalous dynamics in classical and quantum systems are heterogeneity, anisotropy, and nonlocality. In the classical limit, it was shown that these complexities lead to non-Gaussian behavior of the diffusion processes, which can be sufficiently modeled by adequate generalizations of the diffusion equation [1,2,3,4,5,6,7]. Such generalizations often include time and space-fractional operators [2,7,8,9,10], providing a powerful tool for modeling heterogeneous and anisotropic media with memory [7,11,12,13,14,15]. On the other hand, an explicit incorporation of the anisotropy by imposing geometric constraints, e.g., comb-like constraints in the conventional partial differential equations of integer order, also leads to anomalous behavior and it can also be connected to fractional models [16,17,18,19]. Another ingredient that affects the anomalous dynamics in complex media is nonlocality. In general, nonlocality can be spatial and temporal. The spatial nonlocality is associated with long-range correlations and it cannot be neglected when the dynamics of the observed particle are substantially influenced not only by the nearest neighbours, but also by all other particles of the system. In mathematical models, this effect can be represented through appropriately formulated potential energy terms that depend on the interparticle distances. Temporal nonlocality indicates the connection between events, which results in the memory effects and causes the non-Markovianity. The inclusion of temporal nonlocality in mathematical diffusion models can be achieved using the time-dependent memory kernel [20,21,22,23,24].

The present work deals with the temporal and spatial nonlocalities in quantum systems. Comprehending the quantum dynamics in complex realistic environments often requires an extension of the models based on the standard Schrödinger equation [25,26,27,28]. Suitable adaptations, comprising heterogeneity, anisotropy, and nonlocality in a quantum context, have found applications in modeling of the anomalous dynamics in disordered materials [29], anomalous diffusion and transport in low-dimensional systems [29,30,31], and cold atomic gases in optical lattices [32,33,34]. In this paper, we introduce a model based on the time-dependent three-dimensional Schrödinger equation with a nonlocal term, given by

(1)$\begin{array}{c}\hfill \begin{array}{cc}\hfill i\mathit{ћ}{\displaystyle \frac{\partial }{\partial t}}\psi (x,y,z,t)& =-{\displaystyle \frac{{\mathit{ћ}}^2}{2m}}{\nabla }^2\psi (x,y,z,t)\hfill \\ & -\delta \left(y\right){\int }_0^{t}d{t}^{\prime }{\int }_{-\infty }^{\infty }d{x}^{\prime }{\int }_{-\infty }^{\infty }d{z}^{\prime }\mathcal{K}\left(x-x^{\prime },z-z^{\prime },t-t^{\prime }\right)\psi (x^{\prime },y,z^{\prime },t^{\prime }),\hfill \end{array}\end{array}$

where $\mathcal{K}(x,z,t)$ contains both spatial and temporal nonlocal correlations. The Dirac delta function in the nonlocal potential energy operator enables long-range interactions to be restricted only at $y=0$, which means that the motion of the particle is only affected by these correlations in the $xOz$- plane. At the same time, for $y\ne 0$, the model describes a free particle. This model grasps the features of two-dimensional materials and provides an elegant way to introduce coupling between the directions, which distinguishes it from other nonlocal terms that were studied previously [22,35,36,37,38,39,40]. It should be emphasized that in the comb-like models, the Dirac delta function is introduced in the kinetic energy operator, achieving geometric constraints suitable for modeling the anisotropic branched structures without any potential energy terms, which means in the absence of potential. generalizations of the Schrödinger equation in two and three dimensions, involving comb-like geometric constraints, were considered in [17,41]. Hence, the model investigated in the present paper widens the previous research by including anisotropy in the interactions.

The paper is organized as follows. In Section 2, we provide detailed solutions for the considered Schrödinger Equation (1) using the Green’s function methods. In addition, in Section 2, we analyze how different nonlocal kernels impact the system’s behavior and induce patterns suggestive of anomalous spreading. In Section 3, we discuss the reduced Green function to show the effect on the wave package spreading. Our findings and their implications for nonlocal interactions in quantum systems are discussed in Section 4.

2. Schrödinger Equation

Let us start our discussion on the solutions to Equation (1), i.e.,

(2)$\begin{array}{c}\hfill \begin{array}{cc}\hfill i\mathit{ћ}{\displaystyle \frac{\partial }{\partial t}}\psi (x,y,z,t)=& -{\displaystyle \frac{{\mathit{ћ}}^2}{2m}}\left({\displaystyle \frac{{\partial }^2}{\partial x^2}}+{\displaystyle \frac{{\partial }^2}{\partial y^2}}+{\displaystyle \frac{{\partial }^2}{\partial z^2}}\right)\psi (x,y,z,t)\hfill \\ & -\delta \left(y\right){\int }_0^{t}d{t}^{\prime }{\int }_{-\infty }^{\infty }d{x}^{\prime }{\int }_{-\infty }^{\infty }d{z}^{\prime }\mathcal{K}\left(x-x^{\prime },z-z^{\prime },t-t^{\prime }\right)\psi (x^{\prime },y,z^{\prime },t^{\prime }).\hfill \end{array}\end{array}$

As we mentioned in the introduction, the nonlocal term influences the spreading of the initial wave package, mainly at $y=0$, which means within the $xOz$-plane. Equation (2) is subjected to the boundary condition ${lim}_{\left|\mathbf{r}\right|\to \infty }\psi (\mathbf{r},t)=0$ and the initial condition $\psi (\mathbf{r},0)=\phi \left(\mathbf{r}\right)$, where $\phi \left(\mathbf{r}\right)$ is a normalized function and $\mathbf{r}=(x,y,z)$. To find the solutions for this equation and analyze the effect of the nonlocal term on the propagation of the initial condition, we use integral transforms (Fourier and Laplace) and the Green function approach [42].

By applying the Fourier transform (x and z variables), which is defined by $f\left(k_{\xi }\right)=\mathfrak{F}\left\{f\left(\xi \right)\right\}={\int }_{-\infty }^{\infty }f\left(\xi \right)e^{-i{k}_{\xi }\xi }d\xi$, where $\xi =\{x,y,z\}$ (thus, the inverse Fourier transform reads $f\left(\xi \right)={\mathfrak{F}}^{-1}\left\{f\left(k_{\xi }\right)\right\}={\displaystyle \frac{1}{2\pi }}{\int }_{-\infty }^{\infty }f\left(k_{\xi }\right)e^{i{k}_{\xi }\xi }d{k}_{\xi }$), Equation (2) can be simplified to

(3)$\begin{array}{c}\hfill \begin{array}{cc}\hfill i\mathit{ћ}{\displaystyle \frac{\partial }{\partial t}}\psi (k_x,y,k_z,t)=& -{\displaystyle \frac{{\mathit{ћ}}^2}{2m}}{\displaystyle \frac{{\partial }^2}{\partial y^2}}\psi (k_x,y,k_{z}t)+{\displaystyle \frac{{\mathit{ћ}}^2}{2m}}\left(k_x^2+k_z^2\right)\psi (k_x,y,k_z,t)\hfill \\ & -\delta \left(y\right){\int }_0^{t}d{t}^{\prime }\mathcal{K}\left(k_x,k_z,t-t^{\prime }\right)\psi (k_x,y,k_z,t^{\prime }).\hfill \end{array}\end{array}$

By using the Green function approach, the solution for this equation can be found and written as follows:

(4)$\begin{array}{c}\hfill \psi (k_x,y,k_z,t)={\int }_{\infty }^{\infty }d{y}^{\prime }\mathcal{G}(k_x,y,y^{\prime },k_z,t)\phi (k_x,k_z,y^{\prime }),\end{array}$

with the Green function obtained from the following partial differential equation

(5)$\begin{array}{c}\hfill \begin{array}{cc}\hfill i\mathit{ћ}{\displaystyle \frac{\partial }{\partial t}}\mathcal{G}(k_x,y,y^{\prime },k_z,t)& +{\displaystyle \frac{{\mathit{ћ}}^2}{2m}}{\displaystyle \frac{{\partial }^2}{\partial y^2}}\mathcal{G}(k_x,y,y^{\prime },k_z,t)-{\displaystyle \frac{{\mathit{ћ}}^2}{2m}}\left(k_x^2+k_z^2\right)\mathcal{G}(k_x,y,y^{\prime },k_z,t)\hfill \\ & =-\delta \left(y\right){\int }_0^{t}d{t}^{\prime }\mathcal{K}\left(k_x,k_z,t-t^{\prime }\right)\mathcal{G}(k_x,y,y^{\prime },k_z,t^{\prime })+i\mathit{ћ}\delta (y-y^{\prime })\delta \left(t\right),\hfill \end{array}\end{array}$

subjected to the conditions ${lim}_{\left|y\right|\to \infty }\mathcal{G}(k_x,y,y^{\prime },k_z,t)=0$ and $\mathcal{G}(k_x,y,y^{\prime },k_z,t)=0$ for $t<0$. By applying the Fourier transform (y variable), we may simplify Equation (2) and obtain

(6)$\begin{array}{c}\hfill \begin{array}{cc}\hfill i\mathit{ћ}{\displaystyle \frac{\partial }{\partial t}}\mathcal{G}(k_x,k_y,y^{\prime },k_z,t)& -{\displaystyle \frac{{\mathit{ћ}}^2}{2m}}\left(k_y^2+k_x^2+k_z^2\right)\mathcal{G}(k_x,k_y,y^{\prime },k_z,t)\hfill \\ & =-{\int }_0^{t}d{t}^{\prime }\mathcal{K}(k_x,k_z,t-t^{\prime })\mathcal{G}(k_x,y=0,y^{\prime },k_z,t^{\prime })+i\mathit{ћ}{e}^{-i{k}_y{y}^{\prime }}\delta \left(t\right).\hfill \end{array}\end{array}$

In the Laplace domain (the Laplace transform is defined by $f\left(s\right)=\mathcal{L}\left\{f\left(t\right)\right\}={\int }_0^{\infty }f\left(t\right)e^{-st}dt$ and the inverse Laplace transform by $f\left(t\right)={\mathcal{L}}^{-1}\left\{f\left(s\right)\right\}={\displaystyle \frac{1}{2\pi i}}{\int }_{-i\infty +c}^{i\infty +c}f\left(s\right)e^{st}ds$), the solution for Equation (6) is provided by

(7)$\begin{array}{c}\hfill \begin{array}{cc}\hfill \mathcal{G}(k_x,k_y,y^{\prime },k_z,s)=& {\mathcal{G}}_f(k_x,k_y,k_z,s)e^{-i{k}_y{y}^{\prime }}\hfill \\ & +{\displaystyle \frac{i}{\mathit{ћ}}}\mathcal{K}(k_x,k_z,s){\mathcal{G}}_f(k_x,k_y,k_z,s)\mathcal{G}(k_x,0,y^{\prime },k_z,s),\hfill \end{array}\end{array}$

where ${\mathcal{G}}_f(k_x,k_y,k_z,s)$ corresponds to the Green function for the free case in the absence of the nonlocal term, i.e.,

(8)$\begin{array}{c}\hfill {\mathcal{G}}_f(k_x,k_y,k_z,s)={\displaystyle \frac{1}{s+\frac{i\mathit{ћ}}{2m}\left(k_x^2+k_y^2+k_z^2\right)}}.\end{array}$

From the previous results, after performing some calculations, it is possible to show that

(9)$\begin{array}{c}\hfill \mathcal{G}(k_x,y=0,y^{\prime },k_z,s)={\displaystyle \frac{{\mathcal{G}}_f(k_x,y^{\prime },k_z,s)}{1-(i/\mathit{ћ})\mathcal{K}(k_x,k_z,s){\mathcal{G}}_f(k_x,y=0,k_z,s)}}.\end{array}$

Thus, the solution for Equation (7), in the Fourier and Laplace domain, can be written as

(10)$\begin{array}{c}\hfill \begin{array}{cc}\hfill \mathcal{G}(k_x,y,y^{\prime },& k_z,s)={\mathcal{G}}_f(k_x,y-y^{\prime },k_z,s)\hfill \\ & +{\displaystyle \frac{(i/\mathit{ћ})\mathcal{K}(k_x,k_z,s)}{1-(i/\mathit{ћ})\mathcal{K}(k_x,k_z,s){\mathcal{G}}_f(k_x,y=0,k_z,s)}}{\mathcal{G}}_f(k_x,y,k_z,s){\mathcal{G}}_f(k_x,y^{\prime },k_z,s),\hfill \end{array}\end{array}$

and can be rewritten as

(11)$\begin{array}{c}\hfill \begin{array}{cc}\hfill \mathcal{G}(k_x,y,y^{\prime },k_z,s)=& {\mathcal{G}}_f(k_x,y-y^{\prime },k_z,s)\hfill \\ & +{\displaystyle \frac{(i/\mathit{ћ})\mathcal{K}(k_x,k_z,s){\mathcal{G}}_f(k_x,y=0,k_z,s)}{1-(i/\mathit{ћ})\mathcal{K}(k_x,k_z,s){\mathcal{G}}_f(k_x,y=0,k_z,s)}}{\mathcal{G}}_f(k_x,\left|y\right|+|y^{\prime }|,k_z,s),\hfill \end{array}\end{array}$

where

(12)$\begin{array}{c}\hfill {\mathcal{G}}_f(k_x,y,k_z,s)=\sqrt{{\displaystyle \frac{m}{2i\mathit{ћ}}}}{\displaystyle \frac{e^{-\sqrt{s+\frac{i\mathit{ћ}{k}_{xz}^2}{2m}}\sqrt{\frac{2m}{i\mathit{ћ}}}\left|y\right|}}{\sqrt{s+\frac{i\mathit{ћ}{k}_{xz}^2}{2m}}}},\end{array}$

and $k_{xz}^2=k_x^2+k_z^2$.

Now, we analyze some cases taking into account two cases for the kernel $\mathcal{K}(k_x,k_z,s)$. The first one is $\mathcal{K}(k_x,k_z,s)=$ − ${\mathfrak{K}}_x{\left|k_x\right|}^{{\mu }_x}$ − ${\mathfrak{K}}_z{\left|k_z\right|}^{{\mu }_z}$, which implies that $\mathcal{K}(x,y,t)\propto \delta \left(t\right)\left[\delta \left(z\right){\mathfrak{K}}_x/{\left|x\right|}^{{\mu }_x+1}+\delta \left(x\right){\mathfrak{K}}_x/{\left|z\right|}^{{\mu }_z+1}\right]$ and, consequently,

(13)$\begin{array}{c}\hfill \begin{array}{cc}\hfill {\mathfrak{F}}^{-1}\left\{{\mathcal{L}}^{-1}\right\{& \mathcal{K}(k_x,k_z,s)\psi (k_x,y,k_z,s);t\};x,z\}\hfill \\ & =-{\mathfrak{F}}^{-1}\left\{\left({\mathfrak{K}}_x|k_x{|}^{{\mu }_x}+{\mathfrak{K}}_z{\left|k_z\right|}^{{\mu }_z}\right)\psi (k_x,y,k_z,t);x,z\right\}\hfill \\ & ={\int }_{-\infty }^{\infty }d{x}^{\prime }{\int }_{-\infty }^{\infty }d{z}^{\prime }\left({\mathcal{K}}_{{\mu }_z}{\displaystyle \frac{\delta (z-z^{\prime })}{|x-x^{\prime }{|}^{{\mu }_x+1}}}+{\mathcal{K}}_{{\mu }_x}{\displaystyle \frac{\delta (x-x^{\prime })}{|z-z^{\prime }{|}^{{\mu }_z+1}}}\right)\psi (x^{\prime },y,z^{\prime },t),\hfill \end{array}\end{array}$

where ${\mathcal{K}}_{{\mu }_{x\left(z\right)}}=(2{\mathfrak{K}}_{x\left(z\right)}/\pi )\left[2\mathsf{\Gamma }(1+{\mu }_{x\left(z\right)})sin({\mu }_{x\left(z\right)}\pi /2)\right]$. This choice for the kernel results in long-range nonlocal effects geometrically constrained in the $xOz$-plane at $y=0$, which depends on the exponents ${\mu }_x$ and ${\mu }_z$, as is evident from Equation (13), while there is no nonlocality in time. As we will see later, this nonlocality in space is responsible for the emergence of the anomalous dynamics in the system.

The other choice for the kernel is $\mathcal{K}(k_x,k_z,s)=-\mathfrak{K}{\left[s+i\mathit{ћ}{k}_{xz}^2/\left(2m\right)\right]}^{\gamma }{k}_{xz}^2$, which results in the following form for the nonlocal term

(14)$\begin{array}{c}\hfill \begin{array}{cc}\hfill {\mathfrak{F}}^{-1}& \left\{{\mathcal{L}}^{-1}\left\{\mathcal{K}(k_x,k_z,s)\psi (k_x,y,k_z,s);t\right\};x,z\right\}\hfill \\ & =-{\mathfrak{F}}^{-1}\left\{k_{xz}^2{e}^{-{\displaystyle \frac{i\mathit{ћ}{k}_{xz}^2}{2m}}t}{\displaystyle \frac{\mathfrak{K}}{\mathsf{\Gamma }(1-\gamma )}}{\displaystyle \frac{\partial }{\partial t}}{\int }_0^t{\displaystyle \frac{d{t}^{\prime }}{{(t-t^{\prime })}^{\gamma }}}\psi (k_x,y,k_z{t}^{\prime });x,z\right\}\hfill \\ & ={\int }_{-\infty }^{\infty }d{z}^{\prime }{\int }_{-\infty }^{\infty }d{x}^{\prime }{\displaystyle \frac{\partial }{\partial t}}{\mathcal{G}}_{f,xz}(x-x^{\prime },z-z^{\prime },t){\displaystyle \frac{2m\mathfrak{K}}{i\mathit{ћ}\mathsf{\Gamma }(1-\gamma )}}{\displaystyle \frac{\partial }{\partial t}}{\int }_0^t{\displaystyle \frac{d{t}^{\prime }}{{(t-t^{\prime })}^{\gamma }}}\psi (x^{\prime },y,z^{\prime },t^{\prime }),\hfill \end{array}\end{array}$

where

(15)$\begin{array}{c}\hfill {\mathcal{G}}_{f,xz}(x,z,t)={\displaystyle \frac{m}{2\pi i\mathit{ћ}t}}{e}^{-{\displaystyle \frac{m}{2i\mathit{ћ}t}}\left(x^2+z^2\right)}\end{array}$

corresponds to the free particle propagator in two dimensions for the x and z directions. As we see from Equation (14), this choice for the kernel also results in a nonlocality in time, which is controlled by the parameter $\gamma$. As we will see later, such a nonlocal term in combination with the geometrical constraint of the nonlocal effects in the $xOz$-plane will induce anomalous dynamics in the system.

For the first case with the nonlocal term of form (13), after substituting the kernel $\mathcal{K}(k_x,k_z,s)$ in Equation (11), we have

(16)$\begin{array}{c}\hfill \begin{array}{cc}\hfill \mathcal{G}(k_x,y,y^{\prime },k_z,s)& ={\mathcal{G}}_f(k_x,y-y^{\prime },k_z,s)\hfill \\ & -{\displaystyle \frac{(i/\mathit{ћ})\left({\mathfrak{K}}_x|k_x{|}^{{\mu }_x}+{\mathfrak{K}}_z{\left|k_z\right|}^{{\mu }_z}\right){\mathcal{G}}_f(k_x,0,k_z,s)}{1+(i/\mathit{ћ})\left({\mathfrak{K}}_x|k_x{|}^{{\mu }_x}+{\mathfrak{K}}_z{\left|k_z\right|}^{{\mu }_z}\right){\mathcal{G}}_f(k_x,0,k_z,s)}}{\mathcal{G}}_f(k_x,\left|y\right|+|y^{\prime }|,k_z,s).\hfill \end{array}\end{array}$

By using Equation (12), the previous equation can be written as follows

(17)$\begin{array}{c}\hfill \begin{array}{cc}\hfill \mathcal{G}(k_x,y,y^{\prime },k_z,s)& ={\mathcal{G}}_f(k_x,y-y^{\prime },k_z,s)-{\mathcal{G}}_f(k_x,\left|y\right|+|y^{\prime }|,k_z,s)\hfill \\ & +\sqrt{{\displaystyle \frac{m}{2i\mathit{ћ}}}}{\displaystyle \frac{e^{-\sqrt{s+\frac{i\mathit{ћ}{k}_{xz}^2}{2m}}\sqrt{\frac{2m}{i\mathit{ћ}}}\left(\right|y|+|y^{\prime }\left|\right)}}{\sqrt{s+\frac{i\mathit{ћ}{k}_{xz}^2}{2m}}+\frac{i}{\mathit{ћ}}\sqrt{\frac{m}{2i\mathit{ћ}}}\left({\mathfrak{K}}_x|k_x{|}^{{\mu }_x}+{\mathfrak{K}}_z{\left|k_z\right|}^{{\mu }_z}\right)}}.\hfill \end{array}\end{array}$

By performing the inverse Laplace transform, we can obtain that

(18)$\begin{array}{c}\hfill \begin{array}{cc}\hfill \mathcal{G}(x,& y,y^{\prime },z,t)={\mathcal{G}}_f(x,y-y^{\prime },z,t)-{\mathcal{G}}_f(x,|y|+|y^{\prime }|,z,t)\hfill \\ & +\sqrt{{\displaystyle \frac{m}{8\pi i\mathit{ћ}{t}^3}}}{\int }_0^{t}d\overline{t}\left[\overline{t}+\sqrt{{\displaystyle \frac{2m}{i\mathit{ћ}}}}\left(\left|y\right|+|y^{\prime }|\right)\right]e^{-{\displaystyle \frac{1}{4t}}{\left(\overline{t}+\sqrt{{\displaystyle \frac{2m}{i\mathit{ћ}}}}\left(\left|y\right|+|y^{\prime }|\right)\right)}^2}{\mathcal{G}}_{f,xz}^{({\mu }_x,{\mu }_z)}(x,z,\overline{t},t),\hfill \end{array}\end{array}$

where

(19)$\begin{array}{c}\hfill \begin{array}{cc}\hfill {\mathcal{G}}_{f,xz}^{({\mu }_x,{\mu }_z)}(x,z,\overline{t},t)& ={\mathcal{G}}_{f,x,{\mu }_x}(x,\overline{t},t){\mathcal{G}}_{f,z,{\mu }_z}(z,\overline{t},t).\hfill \end{array}\end{array}$

with

(20)$\begin{array}{c}\hfill \begin{array}{cc}\hfill {\mathcal{G}}_{f,x,{\mu }_x}(x,\overline{t},t)& =\sqrt{{\displaystyle \frac{m}{2\pi i\mathit{ћ}t}}}{\int }_{-\infty }^{\infty }d{x}^{\prime }{\mathsf{\Phi }}_{{\mu }_x}(x^{\prime },\overline{t})e^{-{\displaystyle \frac{m{(x-x^{\prime })}^2}{2i\mathit{ћ}t}}},\hfill \\ \hfill {\mathsf{\Phi }}_{{\mu }_x}(x,\overline{t})& ={\displaystyle \frac{1}{\left|x\right|}}{H}_{2,2}^{1,1}\left[{\displaystyle \frac{\mathit{ћ}}{\overline{t}{\mathfrak{K}}_x}}\sqrt{{\displaystyle \frac{2\mathit{ћ}}{mi}}}{\left|x\right|}^{{\mu }_x}\left|\begin{array}{c}(1,1),(1,{\displaystyle \frac{{\mu }_x}{2}})\hfill \\ (1,{\mu }_x),(1,{\displaystyle \frac{{\mu }_x}{2}})\hfill \end{array}\right.\right],\hfill \end{array}\end{array}$

and

(21)$\begin{array}{c}\hfill \begin{array}{cc}\hfill {\mathcal{G}}_{f,z,{\mu }_z}(z,\overline{t},t)& =\sqrt{{\displaystyle \frac{m}{2\pi i\mathit{ћ}t}}}{\int }_{-\infty }^{\infty }d{x}^{\prime }{\mathsf{\Phi }}_{{\mu }_z}(z^{\prime },\overline{t})e^{-{\displaystyle \frac{m{(z-z^{\prime })}^2}{2i\mathit{ћ}t}}},\hfill \\ \hfill {\mathsf{\Phi }}_{{\mu }_z}(z,\overline{t})& ={\displaystyle \frac{1}{\left|z\right|}}{H}_{2,2}^{1,1}\left[{\displaystyle \frac{\mathit{ћ}}{\overline{t}{\mathfrak{K}}_z}}\sqrt{{\displaystyle \frac{2\mathit{ћ}}{mi}}}{\left|z\right|}^{{\mu }_z}\left|\begin{array}{c}(1,1),(1,{\displaystyle \frac{{\mu }_z}{2}})\hfill \\ (1,{\mu }_z),(1,{\displaystyle \frac{{\mu }_z}{2}})\hfill \end{array}\right.\right].\hfill \end{array}\end{array}$

Note that in previous results, we used the Fox H-function [43,44]:

(22)$\begin{array}{c}\hfill \begin{array}{cc}\hfill H_{p,q}^{m,n}\left[z{|}_{(b_q,B_q)}^{(a_p,A_p)}\right]& =H_{p,q}^{m,n}\left[z{|}_{(b_1,B_1),\cdots ,(b_q,B_q)}^{(a_1,A_1),\cdots ,(a_p,A_p)}\right]={\displaystyle \frac{1}{2\pi i}}{\int }_{L}h\left(s\right)z^{-s}ds,\hfill \\ \hfill h\left(s\right)& ={\displaystyle \frac{{\mathsf{\Pi }}_{j=1}^m\mathsf{\Gamma }\left(b_j-B_{j}s\right){\mathsf{\Pi }}_{j=1}^n\mathsf{\Gamma }\left(1-a_j+A_{j}s\right)}{{\mathsf{\Pi }}_{j=m+1}^q\mathsf{\Gamma }\left(1-b_j+B_{j}s\right){\mathsf{\Pi }}_{j=n+1}^p\mathsf{\Gamma }\left(a_j-A_{j}s\right)}},\hfill \end{array}\end{array}$

which, for the current cases, is reduced to the Lévy stable distribution, see Ref. [45],

(23)$\begin{array}{cc}\hfill L(\xi ;\alpha )={\mathcal{F}}^{-1}\left\{e^{-|k_{\xi }{|}^{\alpha }}\right\}& ={\displaystyle \frac{1}{\left|\xi \right|}}{H}_{2,2}^{1,1}\left[{\left|\xi \right|}^{\alpha }\left|\begin{array}{c}\left(1.1\right),(1,{\displaystyle \frac{\alpha }{2}})\\ (1,\alpha ),(1,{\displaystyle \frac{\alpha }{2}})\end{array}\right.\right]\hfill \\ & ={\displaystyle \frac{1}{\pi \alpha }}\sum _{l=0}^{\infty }{\displaystyle \frac{{(-1)}^l\mathsf{\Gamma }\left(\frac{2l+1}{\alpha }\right)}{\left(2l\right)!}}{\xi }^{2l}.\hfill \end{array}$

From the solution, which is represented through Fox H-functions, i.e., through Lévy stable distributions, the emergence of anomalous dynamics in the system due to the geometrically constrained long-range nonlocal term is evident. Note that for ${\mu }_x={\mu }_z=2$, the Fox H-functions in ${\mathsf{\Phi }}_{{\mu }_x}(x,\overline{t})$ and ${\mathsf{\Phi }}_{{\mu }_z}(z,\overline{t})$ in Equations (20) and (21), respectively, reduce to the Gaussian distribution, as (see, for example, Ref. [43])

(24)$\begin{array}{cc}\hfill L(\xi ;2)& ={\displaystyle \frac{1}{\left|\xi \right|}}{H}_{2,2}^{1,1}\left[{\displaystyle \frac{{\left|\xi \right|}^2}{Dt}}\left|\begin{array}{c}(1,1),(1,1)\hfill \\ (1,2),(1,1)\hfill \end{array}\right.\right]\hfill \\ & ={\displaystyle \frac{1}{2\left|\xi \right|}}{H}_{1,1}^{1,0}\left[{\displaystyle \frac{\left|\xi \right|}{\sqrt{Dt}}}\left|\begin{array}{c}(1,{\displaystyle \frac{1}{2}})\hfill \\ (1,1)\hfill \end{array}\right.\right]={\displaystyle \frac{1}{\sqrt{4\pi Dt}}}{e}^{-{\displaystyle \frac{{\xi }^2}{4Dt}}}.\hfill \end{array}$

Graphical representation of the solution (18) for ${\mu }_x={\mu }_z=2$ is given in Figure 1.

For the second case with the nonlocal term of form (13), after substituting the kernel $\mathcal{K}(k_x,k_z,s)$ in Equation (11), we have

(25)$\begin{array}{c}\hfill \begin{array}{cc}\hfill \mathcal{G}(k_x,y,y^{\prime },& k_z,s)={\mathcal{G}}_f(k_x,y-y^{\prime },k_z,s)\hfill \\ & -{\displaystyle \frac{(i/\mathit{ћ})\mathfrak{K}{\left[s+i\mathit{ћ}{k}_{xz}^2/\left(2m\right)\right]}^{\gamma }{k}_{xz}^2{\mathcal{G}}_f(k_x,y=0,k_z,s)}{1+(i/\mathit{ћ})\mathfrak{K}{k}_{xz}^2{\left[s+i\mathit{ћ}{k}_{xz}^2/\left(2m\right)\right]}^{\gamma }{\mathcal{G}}_f(k_x,y=0,k_z,s)}}{\mathcal{G}}_f(k_x,\left|y\right|+|y^{\prime }|,s).\hfill \end{array}\end{array}$

Using Equation (12), the previous equation can be written as follows

(26)$\begin{array}{c}\hfill \begin{array}{cc}\hfill \mathcal{G}(k_x,y,y^{\prime },k_z,s)=& {\mathcal{G}}_f(k_x,y-y^{\prime },z,s)-{\mathcal{G}}_f(k_x,\left|y\right|+|y^{\prime }|,k_z,s)\hfill \\ & +\sqrt{{\displaystyle \frac{m}{2i\mathit{ћ}}}}{\displaystyle \frac{e^{-\sqrt{s+\frac{i\mathit{ћ}{k}_{xz}^2}{2m}}\sqrt{\frac{2m}{i\mathit{ћ}}}\left(\right|y|+|y^{\prime }\left|\right)}}{\sqrt{s+\frac{i\mathit{ћ}{k}_{xz}^2}{2m}}+\frac{i}{\mathit{ћ}}\mathfrak{K}\sqrt{\frac{m}{2i\mathit{ћ}}}{k}_{xz}^2{\left[s+i\mathit{ћ}{k}_{xz}^2/\left(2m\right)\right]}^{\gamma }}}.\hfill \end{array}\end{array}$

By performing the inverse Laplace transform, we can obtain that

(27)$\begin{array}{c}\hfill \begin{array}{cc}\hfill \mathcal{G}(k_x,y,y^{\prime },& k_z,t)={\mathcal{G}}_f(k_x,y-y^{\prime },k_z,t)-{\mathcal{G}}_f(k_x,\left|y\right|+|y^{\prime }|,k_z,t)\hfill \\ & +e^{-{\displaystyle \frac{i\mathit{ћ}}{2m}}t{k}_{xz}^2}{\int }_0^t{\displaystyle \frac{d{t}^{\prime }}{\sqrt{t^{\prime }}}}{E}_{{\displaystyle \frac{1}{2}}-\gamma ,{\displaystyle \frac{1}{2}}}\left(-{\displaystyle \frac{i}{\mathit{ћ}}}\mathfrak{K}\sqrt{{\displaystyle \frac{m}{2i\mathit{ћ}}}}{\left|k_{xz}\right|}^2{t}^{\prime {\displaystyle \frac{1}{2}}-\gamma }\right){\mathcal{G}}_{f,y}(y,y^{\prime },t-t^{\prime }),\hfill \end{array}\end{array}$

for $0<\gamma <1/2$, where

(28)$\begin{array}{c}\hfill {\mathcal{G}}_{f,y}(y,y^{\prime },t)=\sqrt{{\displaystyle \frac{m}{8\pi i\mathit{ћ}{t}^3}}}\left(\left|y\right|+|y^{\prime }|\right)e^{-{\displaystyle \frac{m}{2i\mathit{ћ}t}}{\left(\left|y\right|+|y^{\prime }|\right)}^2},\end{array}$

and

(29)$E_{\alpha ,\beta }\left(z\right)=\sum _{k=0}^{\infty }{\displaystyle \frac{z^k}{\mathsf{\Gamma }(\alpha k+\beta )}},$

with $z,\beta \in \mathbb{C}$, $\Re \left(\alpha \right)>0$ is the two-parameter Mittag–Leffler function [46].

The inverse Fourier transform of the Equation (27) yields

(30)$\begin{array}{c}\hfill \begin{array}{cc}\hfill \mathcal{G}(x,y,y^{\prime },z,t)={\mathcal{G}}_f(x,y& -y^{\prime },z,t)-{\mathcal{G}}_f(x,|y|+|y^{\prime }|,z,t)\hfill \\ & +{\int }_0^{t}d{t}^{\prime }{\mathcal{G}}_{f,xz}^{(\mu ,\gamma )}(x,z,t,t^{\prime }){\mathcal{G}}_{f,y}(y,y^{\prime },t-t^{\prime }),\hfill \end{array}\end{array}$

where

$\begin{array}{c}\hfill \begin{array}{c}\hfill {\mathcal{G}}_{f,xz}^{(\mu ,\gamma )}(x,z,t,t^{\prime })={\displaystyle \frac{1}{\sqrt{t^{\prime }}}}{\int }_{-\infty }^{\infty }{\displaystyle \frac{d{k}_x}{2\pi }}{e}^{i{k}_{x}x}{\int }_{-\infty }^{\infty }{\displaystyle \frac{d{k}_z}{2\pi }}{e}^{i{k}_{z}z}{E}_{{\displaystyle \frac{1}{2}}-\gamma ,{\displaystyle \frac{1}{2}}}\left(-{\displaystyle \frac{i}{\mathit{ћ}}}\mathfrak{K}\sqrt{{\displaystyle \frac{m}{2i\mathit{ћ}}}}{k}_{xz}^2{t}^{\prime {\displaystyle \frac{1}{2}}-\gamma }\right)e^{-{\displaystyle \frac{i\mathit{ћ}}{2m}}t{k}_{xz}^2}.\end{array}\end{array}$

Equation (31) combines two behaviors, one characterized by exponential relaxation and the other described in terms of a Mittag–Leffler function, which asymptotically has a power-law behavior, i.e., anomalous dynamics in the system. In this manner, the wave package has different spreading regimes when subjected to these dynamics. We note that for $\mathfrak{K}=0$, the Green function (31) reduces to the one for a free particle.

3. Reduced Green Functions

Now, we consider the reduced Green functions $G_R(x,y,z,t)$ to analyze how the nonlocal term with the dependence on space influences the behavior in each direction. For this, let us consider the case of Equation (2) with initial condition $\psi (x,y,z,t=0)=\delta \left(x\right)\delta \left(y\right)\delta \left(z\right)$, where we have a nonlocal term only in space, i.e., $\mathcal{K}(x,z,t)=\overline{\mathcal{K}}(x,z)\delta \left(t\right)$. By performing the Fourier–Laplace transform of Equation (2), we have that

(31)$\begin{array}{cc}\hfill i\mathit{ћ}\left(s{G}_R(k_x,k_y,k_z,s)-1\right)=& {\displaystyle \frac{{\mathit{ћ}}^2}{2m}}\left(k_x^2+k_y^2+k_z^2\right)G_R(k_x,k_y,k_z,s)\hfill \\ & -\overline{\mathcal{K}}\left(k_x,k_z\right)G_R(k_x,y=0,k_z,s),\hfill \end{array}$

from where we find

(32)$\begin{array}{c}\hfill G_R(k_x,k_y,k_z,s)={\displaystyle \frac{1}{s+\frac{i\mathit{ћ}}{2m}\left(k_x^2+k_y^2+k_z^2\right)}}+{\displaystyle \frac{i}{\mathit{ћ}}}{\displaystyle \frac{\overline{\mathcal{K}}\left(k_x,k_z\right)}{s+\frac{i\mathit{ћ}}{2m}\left(k_x^2+k_y^2+k_z^2\right)}}{G}_R(k_x,y=0,k_z,s),\end{array}$

i.e.,

(33)$\begin{array}{c}\hfill G_R(k_x,k_y,k_z,s)={\displaystyle \frac{\frac{2m}{i\mathit{ћ}}}{\left[\frac{2ms}{i\mathit{ћ}}+\left(k_x^2+k_z^2\right)\right]+k_y^2}}+{\displaystyle \frac{\frac{2m}{{\mathit{ћ}}^2}\overline{\mathcal{K}}\left(k_x,k_z\right)}{\left[\frac{2ms}{i\mathit{ћ}}+\left(k_x^2+k_z^2\right)\right]+k_y^2}}{G}_R(k_x,y=0,k_z,s).\end{array}$

The inverse Fourier transform with respect to $k_y$ yields

(34)$\begin{array}{cc}\hfill G_R(k_x,y,k_z,s)=& {\displaystyle \frac{\frac{2m}{i\mathit{ћ}}}{2\sqrt{\frac{2ms}{i\mathit{ћ}}+\left(k_x^2+k_z^2\right)}}}{e}^{-\sqrt{{\displaystyle \frac{2ms}{i\mathit{ћ}}}+\left(k_x^2+k_z^2\right)}\left|y\right|}\hfill \\ & +{\displaystyle \frac{\frac{2m}{{\mathit{ћ}}^2}\overline{\mathcal{K}}\left(k_x,k_z\right)}{2\sqrt{\frac{2ms}{i\mathit{ћ}}+\left(k_x^2+k_z^2\right)}}}{e}^{-\sqrt{{\displaystyle \frac{2ms}{i\mathit{ћ}}}+\left(k_x^2+k_z^2\right)}\left|y\right|}{G}_R(k_x,y=0,k_z,s),\hfill \end{array}$

from where we have

(35)$\begin{array}{c}\hfill G_R(k_x,y=0,k_z,s)={\displaystyle \frac{\frac{2m}{i\mathit{ћ}}}{2\sqrt{\frac{2ms}{i\mathit{ћ}}+\left(k_x^2+k_z^2\right)}}}+{\displaystyle \frac{\frac{2m}{{\mathit{ћ}}^2}\overline{\mathcal{K}}\left(k_x,k_z\right)}{2\sqrt{\frac{2ms}{i\mathit{ћ}}+\left(k_x^2+k_z^2\right)}}}{G}_R(k_x,y=0,k_z,s),\end{array}$

i.e.,

(36)$\begin{array}{cc}\hfill G_R(k_x,y=0,k_z,s)=& {\displaystyle \frac{\frac{2m}{i\mathit{ћ}}}{2\sqrt{\frac{2ms}{i\mathit{ћ}}+\left(k_x^2+k_z^2\right)}}}{\displaystyle \frac{1}{1-\frac{\frac{2m}{{\mathit{ћ}}^2}\overline{\mathcal{K}}\left(k_x,k_z\right)}{2\sqrt{\frac{2ms}{i\mathit{ћ}}+\left(k_x^2+k_z^2\right)}}}}.\hfill \end{array}$

If we exchange this relation in (32), we find

(37)$\begin{array}{c}\hfill G_R(k_x,k_y,k_z,s)={\displaystyle \frac{1}{s+\frac{i\mathit{ћ}}{2m}\left(k_x^2+k_y^2+k_z^2\right)}}{\displaystyle \frac{1}{1+\frac{2m/{\left(i\mathit{ћ}\right)}^2\overline{\mathcal{K}}\left(k_x,k_z\right)}{2\sqrt{\frac{2ms}{i\mathit{ћ}}+\left(k_x^2+k_z^2\right)}}}}.\end{array}$

From here (37), we see that the Green function is normalized if $\mathcal{K}\left(k_x,k_z\right)=0$ for $\{k_x,k_z\}\to 0$, for which $G_R(k_x=0,k_y=0,k_z=0,s)=1/s$, i.e., ${\int }_{-\infty }^{\infty }{\int }_{-\infty }^{\infty }{\int }_{-\infty }^{\infty }dxdydz{G}_R(x,y,z,t)=1$.

From this point, we can analyze the reduced Green functions, which are given by $G_{R,x}(k_x,s)=G_R(k_x,k_y=0,k_z=0,s)$, $G_{R,y}(k_y,s)=G_R(k_x=0,k_y,k_z=0,s)$ and $G_{R,z}(k_z,s)=G_R(k_x=0,k_y=0,k_z,s)$. If we consider the nonlocal term of form (13), which was analyzed before, $\overline{\mathcal{K}}\left(k_x,k_z\right)=-{\mathfrak{K}}_x|k_x{|}^{{\mu }_x}-{\mathfrak{K}}_z{\left|k_z\right|}^{{\mu }_z}$, and thus

(38)$\begin{array}{c}\hfill G_R(k_x,k_y,k_z,s)={\displaystyle \frac{1}{s+\frac{i\mathit{ћ}}{2m}\left(k_x^2+k_y^2+k_z^2\right)}}{\displaystyle \frac{1}{1-\frac{[2m/{\left(i\mathit{ћ}\right)}^2]\left({\mathfrak{K}}_x|k_x{|}^{{\mu }_x}+{\mathfrak{K}}_z{\left|k_z\right|}^{{\mu }_z}\right)}{2\sqrt{\frac{2ms}{i\mathit{ћ}}+\left(k_x^2+k_z^2\right)}}}}.\end{array}$

Thus,

(39)$\begin{array}{c}\hfill G_{R,x}(k_x,s)={\displaystyle \frac{1}{s+\frac{i\mathit{ћ}}{2m}{k}_x^2}}{\displaystyle \frac{1}{1-\frac{[2m/{\left(i\mathit{ћ}\right)}^2]{\mathfrak{K}}_x{\left|k_x\right|}^{{\mu }_x}}{2\sqrt{\frac{2ms}{i\mathit{ћ}}+k_x^2}}}}.\end{array}$

(40)$\begin{array}{c}\hfill G_{R,y}(k_y,s)={\displaystyle \frac{1}{s+\frac{i\mathit{ћ}}{2m}{k}_y^2}},\end{array}$

(41)$\begin{array}{c}\hfill G_{R,z}(k_z,s)={\displaystyle \frac{1}{s+\frac{i\mathit{ћ}}{2m}{k}_z^2}}{\displaystyle \frac{1}{1-\frac{[2m/{\left(i\mathit{ћ}\right)}^2]{\mathfrak{K}}_z{\left|k_z\right|}^{{\mu }_z}}{2\sqrt{\frac{2ms}{i\mathit{ћ}}+k_z^2}}}}.\end{array}$

By performing the inverse of the Fourier and Laplace transform, the previous equations are written as follows:

(42)$\begin{array}{c}\hfill \begin{array}{cc}\hfill G_{R,x}(x,t)& ={\int }_{-\infty }^{\infty }{\displaystyle \frac{d{k}_x}{2\pi }}{e}^{i{k}_{x}x}{E}_{{\displaystyle \frac{1}{2}}}\left(-{\displaystyle \frac{i}{\mathit{ћ}}}{\mathfrak{K}}_x\sqrt{{\displaystyle \frac{mt}{2i\mathit{ћ}}}}{\left|k_x\right|}^{{\mu }_x}\right)e^{-{\displaystyle \frac{i\mathit{ћ}}{2m}}t{k}_x^2}\hfill \\ & =\sqrt{{\displaystyle \frac{m}{2\pi i\mathit{ћ}t}}}{\int }_{-\infty }^{\infty }d{x}^{\prime }{\mathsf{\Omega }}_{{\mu }_x}(x^{\prime },t)e^{-{\displaystyle \frac{m{(x-x^{\prime })}^2}{2i\mathit{ћ}t}}},\hfill \end{array}\end{array}$

$\begin{array}{c}\hfill {\mathsf{\Omega }}_{{\mu }_x}(x,t)={\displaystyle \frac{1}{|x^{\prime }|}}{H}_{3,3}^{2,1}\left[{\displaystyle \frac{\mathit{ћ}}{\sqrt{t}{\mathfrak{K}}_x}}\sqrt{{\displaystyle \frac{2\mathit{ћ}}{mi}}}{\left|x^{\prime }\right|}^{{\mu }_x}\left|\begin{array}{c}(1,1),(1,{\displaystyle \frac{1}{2}}),(1,{\displaystyle \frac{{\mu }_x}{2}})\hfill \\ (1,{\mu }_x),(1,1),(1,{\displaystyle \frac{{\mu }_x}{2}})\hfill \end{array}\right.\right],\end{array}$

(43)$\begin{array}{c}\hfill G_{R,y}(y,t)=\sqrt{{\displaystyle \frac{m}{2\pi i\mathit{ћ}t}}}{e}^{-{\displaystyle \frac{m}{2i\mathit{ћ}t}}{y}^2},\end{array}$

and

(44)$\begin{array}{cc}\hfill G_{R,z}(z,t)& ={\int }_{-\infty }^{\infty }{\displaystyle \frac{d{k}_z}{2\pi }}{e}^{i{k}_{z}z}{E}_{{\displaystyle \frac{1}{2}}}\left(-{\displaystyle \frac{i}{\mathit{ћ}}}{\mathfrak{K}}_z\sqrt{{\displaystyle \frac{mt}{2i\mathit{ћ}}}}{\left|k_z\right|}^{{\mu }_z}\right)e^{-{\displaystyle \frac{i\mathit{ћ}}{2m}}t{k}_x^2}\hfill \end{array}$

(45)$\begin{array}{cc}& =\sqrt{{\displaystyle \frac{m}{2\pi i\mathit{ћ}t}}}{\int }_{-\infty }^{\infty }d{z}^{\prime }{\mathsf{\Omega }}_{{\mu }_z}(z^{\prime },t)e^{-{\displaystyle \frac{m{(z-z^{\prime })}^2}{2i\mathit{ћ}t}}},\hfill \end{array}$

$\begin{array}{c}\hfill {\mathsf{\Omega }}_{{\mu }_z}(z,t)={\displaystyle \frac{1}{|z^{\prime }|}}{H}_{3,3}^{2,1}\left[{\displaystyle \frac{\mathit{ћ}}{\sqrt{t}{\mathfrak{K}}_z}}\sqrt{{\displaystyle \frac{2\mathit{ћ}}{mi}}}{\left|z^{\prime }\right|}^{{\mu }_z}\left|\begin{array}{c}(1,1),(1,{\displaystyle \frac{1}{2}}),(1,{\displaystyle \frac{{\mu }_z}{2}})\hfill \\ (1,{\mu }_z),(1,1),(1,{\displaystyle \frac{{\mu }_z}{2}})\hfill \end{array}\right.\right].\end{array}$

Note that the reduced Green functions for the x and z variables have mixing between two terms, one from the kinetic energy and another from the nonlocal term, which was introduced in the Schrödinger equation. This feature indicates that spreading the wave package governed by these Green functions may present different behaviors depending on the choice of the parameters in the nonlocal term. These Fox H-functions appear due to long range nonlocal effects geometrically constrained in the $xOz$-plane, represented by the $\delta \left(y\right)$ function in front of the nonlocal term, and are signature of anomalous dynamics in the system. Figure 2 illustrates the behavior of Equation (42) for different values of ${\mu }_x$ for the real and imaginary parts to show the effect of the combination of different effects.

For the other kernel of form (14), performing a similar calculation by considering the reduced Green function is also possible. In particular, we have that

(46)$\begin{array}{c}\hfill \begin{array}{c}\hfill G_R(k_x,y,k_z,s)=\sqrt{{\displaystyle \frac{m}{2i\mathit{ћ}}}}{\displaystyle \frac{e^{-\sqrt{s+\frac{i\mathit{ћ}{k}_{xz}^2}{2m}}\sqrt{\frac{2m}{i\mathit{ћ}}}\left|y\right|}}{\sqrt{s+\frac{i\mathit{ћ}{k}_{xz}^2}{2m}}+\frac{i}{\mathit{ћ}}\mathfrak{K}\sqrt{\frac{m}{2i\mathit{ћ}}}{k}_{xz}^2{\left[s+i\mathit{ћ}{k}_{xz}^2/\left(2m\right)\right]}^{\gamma }}},\end{array}\end{array}$

which yields,

(47)$\begin{array}{c}\hfill \begin{array}{cc}\hfill G_{R,x}(x,t)& ={\int }_{-\infty }^{\infty }{\displaystyle \frac{d{k}_x}{2\pi }}{e}^{i{k}_{x}x}{E}_{{\displaystyle \frac{1}{2}}-\gamma }\left(-{\displaystyle \frac{i}{\mathit{ћ}}}\mathfrak{K}\sqrt{{\displaystyle \frac{m}{2i\mathit{ћ}}}}{k}_x^2{t}^{{\displaystyle \frac{1}{2}}-\gamma }\right)e^{-{\displaystyle \frac{i\mathit{ћ}}{2m}}t{k}_x^2}\hfill \\ & =\sqrt{{\displaystyle \frac{m}{2\pi i\mathit{ћ}t}}}{\int }_{-\infty }^{\infty }d{x}^{\prime }{\mathsf{\Omega }}_{2,\gamma }(x^{\prime },t)e^{-{\displaystyle \frac{m{(x-x^{\prime })}^2}{2i\mathit{ћ}t}}},\hfill \end{array}\end{array}$

$\begin{array}{c}\hfill {\mathsf{\Omega }}_{2,\gamma }(x,t)={\displaystyle \frac{1}{|x^{\prime }|}}{H}_{1,1}^{1,0}\left[{\displaystyle \frac{\mathit{ћ}}{\sqrt{t^{1-2\gamma }}\mathfrak{K}}}\sqrt{{\displaystyle \frac{2\mathit{ћ}}{mi}}}{\left|x^{\prime }\right|}^2\left|\begin{array}{c}(1,{\displaystyle \frac{1}{2}}-\gamma )\hfill \\ (1,2)\hfill \end{array}\right.\right],\end{array}$

(48)$\begin{array}{c}\hfill \begin{array}{c}\hfill G_{R,y}(y,t)=\sqrt{{\displaystyle \frac{m}{2\pi i\mathit{ћ}t}}}{e}^{-{\displaystyle \frac{m}{2i\mathit{ћ}t}}{y}^2},\end{array}\end{array}$

and

(49)$\begin{array}{c}\hfill \begin{array}{cc}\hfill G_{R,z}(z,t)& ={\int }_{-\infty }^{\infty }{\displaystyle \frac{d{k}_z}{2\pi }}{e}^{i{k}_{z}z}{E}_{{\displaystyle \frac{1}{2}}-\gamma }\left(-{\displaystyle \frac{i}{\mathit{ћ}}}\mathfrak{K}\sqrt{{\displaystyle \frac{m}{2i\mathit{ћ}}}}{k}_z^2{t}^{{\displaystyle \frac{1}{2}}-\gamma }\right)e^{-{\displaystyle \frac{i\mathit{ћ}}{2m}}t{k}_z^2}\hfill \\ & =\sqrt{{\displaystyle \frac{m}{2\pi i\mathit{ћ}t}}}{\int }_{-\infty }^{\infty }d{z}^{\prime }{\mathsf{\Omega }}_{2,\gamma }(z^{\prime },t)e^{-{\displaystyle \frac{m{(z-z^{\prime })}^2}{2i\mathit{ћ}t}}},\hfill \end{array}\end{array}$

$\begin{array}{c}\hfill {\mathsf{\Omega }}_{2,\gamma }(z,t)={\displaystyle \frac{1}{|z^{\prime }|}}{H}_{1,1}^{1,0}\left[{\displaystyle \frac{\mathit{ћ}}{\sqrt{t^{1-2\gamma }}\mathfrak{K}}}\sqrt{{\displaystyle \frac{2\mathit{ћ}}{mi}}}{\left|z^{\prime }\right|}^2\left|\begin{array}{c}(1,{\displaystyle \frac{1}{2}}-\gamma )\hfill \\ (1,2)\hfill \end{array}\right.\right].\end{array}$

Similarly to the analysis performed for the kernel with spatial dependence, the kernel affected only the directions x and z. In addition, the time dependence on the kernel in combination with the geometric constraint of the nonlocal effects in the $xOz$-plane introduce anomalous dynamics in the system, represented by the Fox H-functions in the solution. This point can be verified by the Mittag-Leffler function in the solutions, which presents an asymptotic behavior characterized by power laws for the relaxation process. It is clear from these results that for $\mathfrak{K}=0$ the Green functions $G_{R,x}(x,t)$ and $G_{R,z}(z,t)$ reduce to those for a free particle.

4. Discussion and Conclusions

We investigated an analytically solvable model based on the Schrödinger equation in the presence of a nonlocal term, dependent on spatial and time coordinates. Along with nonlocality, the model also reflects anisotropy in a way that is suitable for describing correlations that take effect only in one plane, keeping the free-particle conditions outside of this plane. We employed the Green’s function method along with integral Fourier–Laplace transforms to derive closed-form solutions for the Green’s functions. Further, we considered different forms of the nonlocal kernel and analyzed their effects. It was shown that the nonlocal term influenced the system’s spreading behavior, varying based on the selected kernel. This point is evident in the first case when the spatial dependence of the kernel is considered. In this case, we consider the convolution between the solutions of Gaussian and Lévy types, as shown for the reduced distributions. In this manner, the kernel with a spatial dependence on Lévy solutions is asymptotically characterized by long-tailed behavior. This shows that two different behaviors govern the behavior of the wave package. Similar situations have been analyzed in a diffusion scenario [47,48,49], where different diffusive terms have been considered. We then considered the time dependence of the kernel, which can be linked to fractional derivatives in time within the framework of the Riemann–Liouville definition. This fractional operator introduces an anomalous relaxation in time, asymptotically governed by a power law with long-tailed behavior related to the asymptotic behavior of the Mittag–Leffler function. In this case, we have two behaviors, one connected to the exponential and another to the Mittag–Leffler function during the relaxation of the wave package.

In conclusion, we can say that the model proposed in this work provides a general simplified approach for a better understanding of the anomalous dynamics in layered two-dimensional nanomaterials, where the long-range interactions are dominant in one plane [50,51,52] and cannot be treated using the classical diffusion models. Memory effects in such complex low-dimensional materials also play a crucial role in predicting the system’s behavior. The results show that temporal nonlocality can likewise be addressed through an appropriate selection of the kernel within the same model. In our research, we extend previous generalizations of the Schrödinger equation by considering the nonlocality and anisotropy in the interparticle correlations. In the future, the model established in this work could be further extended to include interactions in additional planes perpendicular to the y-axis. This will mimic layered two-dimensional materials, which can be achieved by introducing another term with the Dirac delta function positioned elsewhere along the y-axis.

Author Contributions

Conceptualization, I.P., P.T., T.S., J.A.M.A.R., A.S.M.d.C. and E.K.L.; methodology, I.P., P.T., T.S., J.A.M.A.R., A.S.M.d.C. and E.K.L.; formal analysis, I.P., P.T., T.S., J.A.M.A.R., A.S.M.d.C. and E.K.L.; investigation, I.P., P.T., T.S., J.A.M.A.R., A.S.M.d.C. and E.K.L.; writing—original draft preparation, I.P., P.T., T.S., J.A.M.A.R., A.S.M.d.C. and E.K.L.; writing—review and editing, I.P., P.T., T.S., J.A.M.A.R., A.S.M.d.C. and E.K.L. All authors have read and agreed to the published version of the manuscript.

Data Availability Statement

No new data were created or analyzed in this study. Data sharing is not applicable to this article.

Conflicts of Interest

The authors declare no conflicts of interest.

Footnotes

Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content.

Figures

Figure 1. This figure shows [Forumla omitted. See PDF.] for the the wave function obtained from initial condition [Forumla omitted. See PDF.] with the Green function given by Equation (18). We consider, for simplicity, [Forumla omitted. See PDF.], [Forumla omitted. See PDF.], [Forumla omitted. See PDF.], [Forumla omitted. See PDF.], and [Forumla omitted. See PDF.], in arbitrary units.

Figure 2. This figure shows the real and imaginary part of the reduced green function [Forumla omitted. See PDF.] for different values of [Forumla omitted. See PDF.]. We consider, for simplicity, [Forumla omitted. See PDF.], [Forumla omitted. See PDF.], and [Forumla omitted. See PDF.], in arbitrary unities.

Figure 2. This figure shows the real and imaginary part of the reduced green function [Forumla omitted. See PDF.] for different values of [Forumla omitted. See PDF.]. We consider, for simplicity, [Forumla omitted. See PDF.], [Forumla omitted. See PDF.], and [Forumla omitted. See PDF.], in arbitrary unities.

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