# Fractal Schrödinger Equation: Implications for Fractal Sets

Alireza Khalili Golmankhaneh 1111Corresponding Author , Stergios Pellis 2,
Massimiliano Zingales 3
1 Department of Physics, Urmia Branch,
Islamic Azad University, Urmia, 63896, Iran
<EMAIL_ADDRESS>
2 Department of Physics, University of Ioannina, Ioannina, Greece
<EMAIL_ADDRESS>
3 Dipartimento di Ingegneria, Università degli Studi di Palermo,
Viale delle Scienze ed.8, Palermo, 90128, Italy
<EMAIL_ADDRESS>
###### Abstract

This paper delves into the world of fractal calculus, investigating its
implications for fractal sets. It introduces the Fractal Schrödinger Equation
and provides insights into its consequences. The study presents a General
Solution for the Time-Dependent Schrödinger Equation, unveiling its core
aspects. Exploring quantum mechanics in the context of fractals, the paper
analyzes the Probability Density of the Radial Hydrogen Atom, unveiling its
behavior within fractal dimensions. The investigation extends to deciphering
the intricate Energy Levels of the Hydrogen Atom, uncovering the interplay of
quantum mechanics and fractal geometry. Innovatively, the research applies the
Fractal Schrödinger Equation to Simple Harmonic Motion, leading to the
introduction of the Fractal Probability Density Function for the Harmonic
Oscillator. The paper employs a series of illustrative figures that enhance
the comprehension of the findings. By intertwining quantum mechanics and
fractal mathematics, this research paves the way for deeper insights into
their relationship.

Keywords: Fractal Calculus, Fractal Schrödinger Equation, Fractal Hydrogen
Atom, Fractal Simple Harmonic Motion
PACS Codes: 05.45.Df, 03.65.-w, 03.65.-w

##  1 Introduction

The concept of fractal geometry was pioneered by Benoit Mandelbrot [1]. This
revolutionary approach explores intricate geometric forms that defy
conventional boundaries, exhibiting fractal dimensions surpassing their
topological counterparts. These structures, known as fractals, possess
inherent self-similarity and frequently exhibit dimensions that transcend
integer values [2, 3]. Given that fractals possess distinct measures like the
Hausdorff measure, conventional metrics such as length, surface area, and
volume-typically applied to Euclidean geometric forms-prove inadequate for
fractal analysis [4, 5, 6, 7].
Numerous researchers have approached fractal analysis through various methods,
including harmonic analysis [8, 9], measure theory [10, 11, 12, 13, 14, 15,
16], probabilistic approaches [17], fractional space [18], fractional calculus
[19], and unconventional techniques [20].
Fractal calculus arises through an extension of conventional calculus,
encompassing differential equations whose solutions manifest as functions
characterized by fractal support, including fractal sets and curves. This type
of calculus is algorithmic in nature and offers a more streamlined approach
when compared to alternative methodologies [21, 22, 23]. Fractal calculus
found generalization and application within the realm of physics, extending to
stochastic equations and the establishment of Fourier and Laplace transforms
[24, 25, 26, 27, 28, 29, 30, 31]. The resolution of fractal delayed, neutral,
and renewal delay differential equations with constant coefficients was
accomplished using the method of steps and Laplace transforms [32]. Fractal
counterparts of Newtonian, Lagrangian, Hamiltonian, and Appellian mechanics
were proposed. Fractal $\alpha$-velocity and $\alpha$-acceleration were
defined, enabling the formulation of the Langevin equation on fractal curves
[33]. The fractal Frechet derivative and the fractal generalized Euler-
Lagrange equation and the fractal Du Bois-Reymond optimality condition were
introduced [34].
The Hermiticity of the fractional Hamilton operator was demonstrated, and the
conservation law for parity in fractional quantum mechanics was established.
Physical applications yield energy spectra for a hydrogen-like atom [35, 36].
The space-time fractional Schrödinger equation was investigated through the
incorporation of the Caputo fractional derivative and quantum Riesz fractional
operator [37]. The generalized fractional Schrödinger equation, incorporating
space and time fractional derivatives, was formulated and subsequently solved
for both a free particle and a square potential well. The solution techniques
involved integral transforms, Fourier transform, and Laplace transform,
resulting in expressions of solutions using the Mittag-Leffler function [38].
The spectral attributes of fractal structures were investigated, involving the
introduction of the concept of spectral dimensionality through a
straightforward scaling approach. Computations of the Schrödinger equation
spectrum were conducted for a regular fractal structure, specifically the
Sierpinski gasket [38]. The fractional Schrödinger equation for a particle
confined within an infinite potential well also for delta potentials were
studied [39, 40, 41]. A solution to the fractional Schrödinger equation for
the one-dimensional harmonic oscillator was provided [42]. The spectrum of the
Schrödinger equation on a regular fractal structure was explicitly calculated.
As illustrative examples, certain physical properties of percolating clusters
near the percolation threshold were discussed [43, 44, 45]. An asymptotic
eigenvalue counting function was formulated for the Schrödinger operator with
unbounded potentials on several types of unbounded fractal spaces [46]. The
Schrödinger equation was solved on a variety of fractal lattices using a
recursive technique and the resulting energy levels are discrete, very closely
spaced, and highly degenerate [47]. A versatile approach was proposed for
generating wave functions with predetermined fractal dimensions across a range
of quantum scenarios, such as the infinite potential well, harmonic
oscillator, linear potential, and free particle [48]. Fractal-like photonic
lattices were experimentally realized using the cw-laser-writing technique,
leading to the observation of distinct compact localized states-associated
with different flatbands within the same lattice configuration [49]. A
technique for enhancing the finite element method on the Sierpinski gasket was
demonstrated, enabling unrestricted space partitioning. This enhanced method
was applied to numerically analyze solutions of the Schrödinger equation with
well-type potentials, along with the wave equation [50]. Dirac operators and
magnetic Schrödinger Hamiltonians on fractals while demonstrating their
(essential) self-adjointness was suggested [51]. Regular and irregular
sampling for Sierpinski Gasket functions is examined, where ”bandlimited”
refers to finite expansions in the first Dirichlet eigenfunctions of the
Laplacian [52]. The analogue of the quantum-mechanical harmonic oscillator on
infinite blowups of the Sierpinski Gasket is studied using the standard Kigami
Laplacian [53]. Driven by these motivations, this paper undertakes the
generalization of the Schrödinger equation onto a fractal framework. Within
this context, we successfully addressed the Schrödinger equation for fractal
Hydrogen atoms and fractal simple harmonic oscillators.
The manuscript’s structure is outlined as follows:
In Section 2, we provide an overview of fractal calculus applied to fractal
sets. Section 3 introduces the Schrödinger equation in the context of fractal
space and time. We proceed to solve this equation for both the Hydrogen atom
and the simple harmonic oscillator, determining probability density functions
and energy levels, which are then compared to their standard counterparts. Our
concluding remarks are presented in Section 5.

##  2 Fractal calculus for fractal sets

In this context, we provide a concise overview of the fractal calculus applied
to sets $\mathbf{F}$ contained within the interval $[c_{1},c_{2}]$ of the real
number line, as described by Parvate et al. (2009) in their work on calculus
[21].

###### Definition 2.1

The flag function of $\mathbf{F}$ is introduced, defined by:

$\rho(\mathbf{F},I)=\left\\{\begin{array}[]{ll}1,&\textmd{if}~{}~{}\mathbf{F}\cap
I\neq\emptyset;\\\ 0,&\textmd{otherwise},\end{array}\right.$ (1)

where $I=[c_{1},c_{2}]\subset\mathbb{R}$.

###### Definition 2.2

The coarse-grained mass of $\mathbf{F}\cap[c_{1},c_{2}]$ is defined as:

$\xi_{\delta}^{\alpha}(\mathbf{F},c_{1},c_{2})=\inf_{|\mathfrak{P}|\leq\delta}\sum_{i=0}^{n-1}\Gamma(\alpha+1)(z_{i+1}-z_{i})^{\alpha}\rho(\mathbf{F},[z_{i},z_{i+1}]),$
(2)

where

$|\mathfrak{P}|=\max_{0\leq i\leq n-1}(z_{i+1}-z_{i}),$ (3)

and $0<\alpha\leq 1$.

###### Definition 2.3

The mass function of $\mathbf{F}$ is established as:

$\xi^{\alpha}(\mathbf{F},c_{1},c_{2})=\lim_{\delta\rightarrow
0}\xi_{\delta}^{\alpha}(\mathbf{F},c_{1},c_{2}).$ (4)

###### Definition 2.4

The fractal dimension of $\mathbf{F}\cap[c_{1},c_{2}]$ is defined as follows:

$\displaystyle\dim_{\xi}(\mathbf{F}\cap[c_{1},c_{2}])$
$\displaystyle=\inf\\{\alpha:\xi^{\alpha}(\mathbf{F},c_{1},c_{2})=0\\}$
$\displaystyle=\sup\\{\alpha:\xi^{\alpha}(\mathbf{F},c_{1},c_{2})=\infty\\}.$
(5)

In this manner, we succinctly encapsulate the fundamental concepts within the
realm of fractal calculus concerning sets $\mathbf{F}$ contained within the
interval $[c_{1},c_{2}]$. The definitions encompass the flag function, coarse-
grained mass, mass function, and fractal dimension, all of which play pivotal
roles in the analysis of these fractal structures.

###### Definition 2.5

The integral staircase function associated with $\mathbf{F}$ is introduced,
defined as follows:

$S_{\mathbf{F}}^{\alpha}(z)=\left\\{\begin{array}[]{ll}\xi^{\alpha}(\mathbf{F},c_{0},z),&if~{}z\geq
c_{0};\\\ -\xi^{\alpha}(\mathbf{F},z,c_{0}),&otherwise,\end{array}\right.$ (6)

where $c_{0}\in\mathbb{R}$ is a fixed number.

###### Definition 2.6

Definition 6: For a function $g$ defined on an $\alpha$-perfect fractal set,
the $F^{\alpha}$-derivative of $g$ at $x$ is defined as:

$D_{\mathbf{F}}^{\alpha}g(z)=\left\\{\begin{array}[]{ll}\underset{y\rightarrow
z}{\mathbf{F}_{-}lim}~{}\frac{g(y)-g(z)}{S_{\mathbf{F}}^{\alpha}(y)-S_{\mathbf{F}}^{\alpha}(z)},&if~{}z\in\mathbf{F};\\\
0,&otherwise,\end{array}\right.$ (7)

assuming the existence of the fractal limit $\mathbf{F}_{-}\text{lim}$ as
described in the work by Parvate et al. [21].

###### Definition 2.7

The $\mathbf{F}^{\alpha}$-integral of a bounded function $g(z)$, where $g$
belongs to the set $B(F)$ (indicating that $g$ is a bounded function of $F$),
is defined as follows:

$\displaystyle\int_{a}^{b}g(z)d_{\mathbf{F}}^{\alpha}z$
$\displaystyle=\sup_{\mathfrak{P}{[c{1},c_{2}]}}\sum_{i=0}^{n-1}\inf_{z\in\mathbf{F}\cap
I}g(z)(S_{\mathbf{F}}^{\alpha}(z_{i+1})-S_{F}^{\alpha}(z_{i}))$
$\displaystyle=\inf_{\mathfrak{P}{[c{1},c_{2}]}}\sum_{i=0}^{n-1}\sup_{z\in\mathbf{F}\cap
I}g(z)(S_{\mathbf{F}}^{\alpha}(z_{i+1})-S_{\mathbf{F}}^{\alpha}(z_{i})),$ (8)

where $z\in\mathbf{F}$, and the infimum or supremum is taken over all
subdivisions $\mathfrak{P}{[c{1},c_{2}]}$.

In this manner, we elucidate the essential concepts of the integral staircase
function, the $\mathbf{F}^{\alpha}$-derivative, and the
$\mathbf{F}^{\alpha}$-integral, which play integral roles in fractal calculus
applied to sets $\mathbf{F}$. These definitions underscore the mathematical
foundation required for analyzing and understanding fractal structures within
the specified context.

##  3 Fractal Schrödinger Equation

In this segment, we apply fractal calculus to modify the Schrödinger equations
of hydrogen atoms, revealing the manifestation of fractal properties within
the established models. This approach allows us to explore the incorporation
of fractal concepts in both spatial and temporal dimensions.

###  3.1 Fractal Schrödinger Equation of Hydrogen Atom

The time-dependent Schrödinger equation for the hydrogen atom within a fractal
space characterized by a dimension of $2+\alpha$, along with fractal time
characterized by a dimension of $\beta$, can be expressed in its entirety as
follows:

$i\hbar
D_{t}^{\beta}\Psi(r,\theta,\phi,t)=\left(-\frac{\hbar^{2}}{2m_{e}}\nabla^{2}-\frac{e^{2}}{4\pi\epsilon_{0}S_{\mathbf{F}}^{\alpha}(r)}\right)\Psi(r,\theta,\phi,t),~{}~{}r,t\in\mathbf{F},$
(9)

where $0\leq\theta\leq\pi,~{}~{}0\leq\phi\leq 2\pi$. In polar coordinates with
a dimensionality of $2+\alpha$, the Laplace operator (Laplacian) is
represented using fractal derivatives in the following manner:

$\nabla^{2}=\frac{1}{S_{\mathbf{F}}^{\alpha}(r)^{2}}D_{r}^{\alpha}\left(S_{\mathbf{F}}^{\alpha}(r)^{2}D_{r}^{\alpha}\right)+\frac{1}{S_{\mathbf{F}}^{\alpha}(r)^{2}\sin\theta}\frac{\partial}{\partial\theta}\left(\sin\theta\frac{\partial}{\partial\theta}\right)+\frac{1}{S_{\mathbf{F}}^{\alpha}(r)^{2}\sin^{2}\theta}\frac{\partial^{2}}{\partial\phi^{2}}$
(10)

where $D_{r}^{\alpha}$ is the fractal radial derivative operator. In
$2+\alpha$ polar coordinates employing fractal derivatives, the Laplacian
comprises three distinct segments: the fractal radial segment, the polar angle
segment, and the azimuthal angle segment. These segments correspond to
variations occurring in the radial, polar angle, and azimuthal angle
directions, respectively.

## General Solution for Time-Dependent Schrödinger Equation

A proposed solution for the time-dependent Schrödinger equation, which
incorporates fractal dimensions for both time and space in the context of the
hydrogen atom, is indicated by:

$\displaystyle\Psi(\mathbf{r},t)$
$\displaystyle=\sum_{n}c_{n}\psi_{n}(\mathbf{r})e^{-\frac{i}{\hbar}E_{n}^{\alpha}S_{\mathbf{F}}^{\beta}(t)}$
$\displaystyle\propto\sum_{n}c_{n}\psi_{n}(\mathbf{r})e^{-\frac{i}{\hbar}E_{n}^{\alpha}t^{\beta}}$
(11)

where $E_{n}^{\alpha}$ are the corresponding fractal energy eigenvalues. This
solution shows the fractal time evolution of the wave function and how it
changes over fractal time for a quantum system described by the Schrödinger
equation. The radial part fractal discretional equation of the Schrödinger
equation for the hydrogen atom with the replaced potential energy term is
given by:

$\displaystyle-\frac{\hbar^{2}}{2m}$
$\displaystyle\left(\frac{1}{S_{\mathbf{F}}^{\alpha}(r)^{2}}D_{r}^{\alpha}\left(S_{\mathbf{F}}^{\alpha}(r)^{2}D_{r}^{\alpha}R_{nl}(r)\right)-\frac{l(l+1)}{S_{\mathbf{F}}^{\alpha}(r)^{2}}R_{nl}(r)\right)$
$\displaystyle-\frac{e^{2}}{4\pi\epsilon_{0}}\frac{1}{S_{\mathbf{F}}^{\alpha}(r)}R_{nl}(r)=E_{n}^{\alpha}R_{nl}(r).$
(12)

Where $S_{F}^{\alpha}(r)$ is the radial staircase function, $m$ is the
electron mass, $l$ is the angular momentum quantum number,
$R_{nl}^{\alpha}(r)$ is the radial wave function, $e$ is the elementary
charge, $\epsilon_{0}$ is the vacuum permittivity, and $E_{n}^{\alpha}$ is the
fractal energy eigenvalue corresponding to the $n$th energy level. The
solution of Eq.(General Solution for Time-Dependent Schrödinger Equation) the
fractal radial wave function can be expressed as:

$\displaystyle R_{n\ell}^{\alpha}(r)$
$\displaystyle=A_{n\ell}\left(\frac{2}{nS_{\mathbf{F}}^{\alpha}(a_{0})}\right)^{\ell+1}\exp\left(\frac{-S_{\mathbf{F}}^{\alpha}(r)}{nS_{F}^{\alpha}(a_{0})}\right)\left(L_{n-\ell-1}^{2\ell+1}\left(\frac{2S_{\mathbf{F}}^{\alpha}(r)}{nS_{\mathbf{F}}^{\alpha}(a_{0})}\right)\right)$
$\displaystyle\propto
A_{n\ell}\left(\frac{2}{na_{0}^{\alpha}}\right)^{\ell+1}\exp\left(\frac{-r^{\alpha}}{na_{0}^{\alpha}}\right)\left(L_{n-\ell-1}^{2\ell+1}\left(\frac{2r^{\alpha}}{na_{0}^{\alpha}}\right)\right),$
(13)

where $L_{n-\ell-1}^{2\ell+1}$ represents the associated Laguerre polynomial.
The complete fractal wave function is given by the product of the radial and
angular parts:

$\psi_{n,l,m}^{\alpha}(r,\theta,\phi)=R_{n\ell}^{\alpha}(r)(r)Y_{l,m}(\theta,\phi),$
(14)

where $Y_{l,m}(\theta,\phi)$ spherical harmonics that determine the angular
behavior. For example, the fractal orbital function for the $1s$ state of the
hydrogen atom in spherical coordinates is given by

$\displaystyle\psi_{10}(r,\theta,\phi)$ $\displaystyle=R_{10}(r)\cdot
Y_{0,0}(\theta,\phi)$ (15)
$\displaystyle=\frac{2}{\sqrt{4\pi}}\left(\frac{1}{S_{F}^{\alpha}(a_{0})^{3}}\right)^{1/2}\exp\left(\frac{-S_{\mathbf{F}}^{\alpha}(r)}{S_{\mathbf{F}}^{\alpha}(a_{0})^{3}}\right)$
(16)
$\displaystyle\propto\frac{2}{\sqrt{4\pi}}\left(\frac{1}{a_{0}^{3\alpha}}\right)^{1/2}\exp\left(\frac{-r^{\alpha}}{a_{0}^{3\alpha}}\right).$
(17)

## Probability Density for Radial Hydrogen Atom

The probability density $P^{\alpha}(r)$ for the radial part of the hydrogen
atom wave function is given by the square of the radial wave function
$R_{n\ell}^{\alpha}(r)$:

$P^{\alpha}(r)=|\Psi^{\alpha}(\mathbf{r})|^{2}=R_{n\ell}^{\alpha}(r)^{2}.$
(18)

In terms of the provided expression for $R_{n\ell}^{\alpha}(r)$:

$R_{n\ell}^{\alpha}(r)\propto
A_{n\ell}\left(\frac{2}{na_{0}^{\alpha}}\right)^{\ell+1}\exp\left(-\frac{r^{\alpha}}{na_{0}^{\alpha}}\right)\left(L_{n-\ell-1}^{2\ell+1}\left(\frac{2r^{\alpha}}{na_{0}^{\alpha}}\right)\right).$
(19)

The fractal probability density $P^{\alpha}(r)$ can be calculated as:

$P^{\alpha}(r)=|R_{n\ell}^{\alpha}(r)|^{2}=|A_{n\ell}|^{2}\left(\frac{2}{na_{0}^{\alpha}}\right)^{2(\ell+1)}\exp\left(-\frac{r^{\alpha}}{na_{0}^{\alpha}}\right)\left|L_{n-\ell-1}^{2\ell+1}\left(\frac{2r^{\alpha}}{na_{0}^{\alpha}}\right)\right|^{2}.$
(20)

This fractal probability density provides information about the distribution
of the electron’s presence in the hydrogen atom as a function of the radial
distance $r$. In Figures 1(a), 1(b),1(c), and 1(d) we have shown how the
fractal dimension changes the fractal probability density function.

(a) Graph of Eq.(20) for the $n=1,~{}l=0,~{}a_{0}=1,~{}A_{n\ell}=1$.

(b) Graph of Eq.(20) for the $n=2,~{}l=1,~{}a_{0}=1,~{}A_{n\ell}=1$.

(c) Graph of Eq.(20) for the $n=3,~{}l=0,~{}a_{0}=1,~{}A_{n\ell}=1$.

(d) Graph of Eq.(20) for the $n=2,~{}l=0,~{}a_{0}=1,~{}A_{n\ell}=1$.

Figure 1: Probability Density Function of Hydrogen Atom

## Energy Levels of Hydrogen Atom

Within the Bohr model of the hydrogen atom, energy levels become quantized and
are determined by the equation:

$E_{n}=-\frac{13.6\,\text{eV}}{n^{2}}.$ (21)

In this equation, $n$ represents the principal quantum number. Each energy
level $n$ corresponds to a specific electron orbit or shell, with higher
energy levels associated with larger orbits. The average radius $r_{n}$ of the
$n$th energy level’s orbit can be connected to the energy level using the
formula:

$r_{n}=\frac{n^{2}\hbar^{2}}{me^{2}}.$ (22)

Here, $n$ denotes the principal quantum number, $\hbar$ denotes the reduced
Planck constant, $m$ stands for the mass of the electron, and $e$ represents
the elementary charge. By combining these equations, the energy levels of the
hydrogen atom can be expressed in terms of the average radius as:

$E_{n}=-\frac{me^{4}}{2\hbar^{2}}\frac{1}{r_{n}}.$ (23)

This relationship underscores the inverse relationship between energy levels
and the average radius of the electron’s orbit. When considering the context
of fractal space, we can adapt (23) as follows:

$\displaystyle E_{n}^{\alpha}$
$\displaystyle=-\frac{me^{4}}{2\hbar^{2}}\frac{1}{S_{\mathbf{F}}^{\alpha}(r_{n})}$
$\displaystyle\propto-\frac{me^{4}}{2\hbar^{2}}\frac{1}{r_{n}^{\alpha}}.$ (24)

In Figure 2, we have visualized the behavior described by Eq.(Energy Levels of
Hydrogen Atom) to illustrate the impact of the fractal dimension on energy
levels within the Hydrogen atom. The figure depicts that as the fractal
dimension decreases, the energy levels in the fractal space undergo an upward
shift.

Figure 2: Graph illustrating Eq.(Energy Levels of Hydrogen Atom) for different
values of the space dimension.

##  4 Fractal Schrödinger Equation of Simple Harmonic Motion

The fractal time-dependent harmonic Schrödinger equation, which includes the
time evolution of the wave function, is suggested by:

$i\hbar
D_{t}^{\beta}\Psi(x,t)=\left(-\frac{\hbar^{2}}{2m}D_{x}^{2\alpha}+\frac{1}{2}m(\omega^{\alpha})^{2}S_{\mathbf{F}}^{\alpha}(x)^{2}\right)\Psi(x,t),~{}~{}~{}x,t\in\mathbf{F},$
(25)

where $\omega^{\alpha}$ is the angular frequency of the fractal harmonic
oscillator. The fractal one-dimensional time-independent harmonic Schrödinger
equation is suggested as:

$-\frac{\hbar^{2}}{2m}D_{x}^{2\alpha}\psi_{n}^{\alpha}(x)+\frac{1}{2}m(\omega^{\alpha})^{2}S_{\mathbf{F}}^{\alpha}(x)^{2}\psi_{n}^{\alpha}(x)=E_{n}^{\alpha}\psi_{n}^{\alpha}(x),$
(26)

its solution is

$\displaystyle\psi_{n}^{\alpha}(x)$
$\displaystyle=\frac{1}{\sqrt{2^{n}n!}}\left(\frac{m\omega^{\alpha}}{\pi\hbar}\right)^{1/4}\exp\left(-\frac{m\omega^{\alpha}S_{\mathbf{F}}^{\alpha}(x)^{2}}{2\hbar}\right)H_{n}\left(\sqrt{\frac{m\omega^{\alpha}}{\hbar}}S_{\mathbf{F}}^{\alpha}(x)\right)$
$\displaystyle\propto\frac{1}{\sqrt{2^{n}n!}}\left(\frac{m\omega^{\alpha}}{\pi\hbar}\right)^{1/4}\exp\left(-\frac{m\omega^{\alpha}x^{2\alpha}}{2\hbar}\right)H_{n}\left(\sqrt{\frac{m\omega^{\alpha}}{\hbar}}x^{\alpha}\right),$
(27)

where $H_{n}$ stands for the $n$-th Hermite polynomial. Then the time-
dependent solution of Eq.(25) is

$\Psi(x,t)=\sum_{n}c_{n}\psi_{n}^{\alpha}(x)\exp\left(\frac{-iE_{n}^{\alpha}S_{\mathbf{F}}^{\beta}(t)}{\hbar}\right).$
(28)

## Fractal Probability Density Function of Harmonic Oscillator

The probability density function for a quantum harmonic oscillator is given by
the square of the absolute value of its wavefunction:

$\displaystyle P^{\alpha}(x)=|\psi_{n}^{\alpha}(x)|^{2}$
$\displaystyle=\left|\frac{1}{\sqrt{2^{n}n!}}\left(\frac{m\omega^{\alpha}}{\pi\hbar}\right)^{1/4}\exp\left(-\frac{m\omega^{\alpha}S_{\mathbf{F}}^{\alpha}(x)^{2}}{2\hbar}\right)H_{n}\left(\sqrt{\frac{m\omega^{\alpha}}{\hbar}}S_{\mathbf{F}}^{\alpha}(x)\right)\right|^{2}$
$\displaystyle\propto\left|\frac{1}{\sqrt{2^{n}n!}}\left(\frac{m\omega^{\alpha}}{\pi\hbar}\right)^{1/4}\exp\left(-\frac{m\omega^{\alpha}x^{2\alpha}}{2\hbar}\right)H_{n}\left(\sqrt{\frac{m\omega^{\alpha}}{\hbar}}x^{\alpha}\right)\right|^{2}.$
(29)

In Figure 3, we’ve depicted the probability density function of simple
harmonic motion. This visualization effectively illustrates how the
wavefunction is influenced by the dimensionality of fractal space.

(a) Caption for Image n=0

(b) Caption for Image n=1

(c) Caption for Image n=2

(d) Caption for Image n=3

Figure 3: Probability Density Function

## Fractal Energy Levels of Simple Harmonic Oscillator

In the realm of quantum mechanics, the wave-like nature of particles gives
rise to a fascinating phenomenon: energy levels become quantized. This
signifies that the energy associated with a quantum harmonic oscillator can
exclusively adopt specific, distinct values. These discrete energy levels
unveil a remarkable feature when viewed through the lens of fractal space, as
outlined below:
The energy levels in this context are denoted as $E_{n}^{\alpha}$ and are
coined ”fractal energy levels.” They take the form:

$\displaystyle E_{n}^{\alpha}$
$\displaystyle=\hbar\omega^{\alpha}\left(n+\frac{1}{2}\right),~{}~{}~{}n=0,1,2,...$
(30)

Another representation arises from the exploration of fractal space, yielding:

$\displaystyle E_{n}^{\alpha}$
$\displaystyle=\hbar\sqrt{\frac{S_{F}^{\alpha}(x)}{m}}\left(n+\frac{1}{2}\right)$
$\displaystyle\propto\hbar\sqrt{\frac{x^{\alpha}}{m}}\left(n+\frac{1}{2}\right).$
(31)

This intriguing relationship is linked to the fractal harmonic oscillator’s
angular frequency $\omega^{\alpha}$.

Figure 4: Graph of Eq.(30)

Figure 4 reveals that as the fractal dimension of space decreases, the energy
level within the harmonic oscillator increases.

##  5 Conclusion

In conclusion, this study demonstrates the profound interconnection between
fractal calculus and quantum mechanics. By embarking on an exploration of
fractal sets, we have illuminated the applications and implications of the
Fractal Schrödinger Equation. The unveiled General Solution for the Time-
Dependent Schrödinger Equation provides a foundational understanding of its
dynamics in the fractal realm. Analyzing the Probability Density of the Radial
Hydrogen Atom within the fractal framework reveals the intricate influence of
dimensionality on quantum behavior. Furthermore, the investigation into the
Energy Levels of the Hydrogen Atom establishes a bridge between fractal
geometry and quantum states. The extension of the Fractal Schrödinger Equation
to Simple Harmonic Motion introduces the Fractal Probability Density Function
for the Harmonic Oscillator, expanding our insights into the quantum realm
within a fractal context. The visual aids incorporated throughout the research
enhance the clarity of the findings. In synthesizing quantum mechanics and
fractal mathematics, this study not only unravels their synergy but also lays
the groundwork for deeper explorations at the nexus of these captivating
fields.
Declaration of Competing Interest:
The authors declare that they have no known competing financial interests or
personal relationships that could have appeared to influence the work reported
in this paper.
Declaration of generative AI and AI-assisted technologies in the writing
process. During the preparation of this work, the authors used GPT in order to
correct grammar and writing. After using this GPT, the authors reviewed and
edited the content as needed and takes full responsibility for the content of
the publication.

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# On the complex valued metric-like spaces

A. Hosseini and M. Mohammadzadeh Karizaki  Amin Hosseini, Kashmar Higher
Education Institute, Kashmar, Iran<EMAIL_ADDRESS>M. Mohammadzadeh
Karizaki, Department of Computer Engineering, University of Torbat Heydarieh,
Torbat Heydarieh, Iran<EMAIL_ADDRESS>
(Date: Received: xxxxxx; Revised: yyyyyy; Accepted: zzzzzz.
∗ Corresponding author)

###### Abstract.

The main purpose of this paper is to study complex valued metric-like spaces
as an extension of metric-like spaces, complex valued partial metric spaces,
partial metric spaces, complex valued metric spaces and metric spaces. In this
article, the concepts such as quasi-equal points, completely separate points,
convergence of a sequence, Cauchy sequence, cluster points and complex
diameter of a set are defined in a complex valued metric-like space.

###### Key words and phrases:

Complex valued metric-like space, metric-like space, complex valued partial
metric space, partial metric space, metric space

###### 2010 Mathematics Subject Classification:

54A05, 46A19

##  1\. Introduction and preliminaries

Distance is an important and fundamental notion in mathematics and there exist
many generalizations of this concept in the literature (see [6]). One of such
generalizations is the partial metric which was introduced by Matthews (see
[11]). It differs from a metric in that points are allowed to have non-zero
”self-distances” (i.e., $d(x,x)\geq 0$), and the triangle inequality is
modified to account for positive self-distances. O’Neill [12] extended
Matthews definition to partial metrics with ”negative distances”. Before
describing the material of this paper, let us recall some definitions and set
the notations which we use in what follows.

######  Definition 1.1.

A mapping $\mathfrak{p}:X\times X\rightarrow\mathbb{R}^{+}$, where $X$ is a
non-empty set, is said to be a partial metric on $X$ if for any $x,y,z\in X$,
the following four conditions hold true:
(i) $x=y$ if and only if
$\mathfrak{p}(x,x)=\mathfrak{p}(y,y)=\mathfrak{p}(x,y)$;
(ii) $\mathfrak{p}(x,x)\leq\mathfrak{p}(x,y)$;
(iii) $\mathfrak{p}(x,y)=\mathfrak{p}(y,x)$;
(iv)
$\mathfrak{p}(x,z)\leq\mathfrak{p}(x,y)+\mathfrak{p}(y,z)-\mathfrak{p}(y,y)$.

The pair $(X,\mathfrak{p})$ is then called a _partial metric space._ A
sequence $\\{x_{n}\\}$ in a partial metric space $(X,\mathfrak{p})$ converges
to a point $x_{0}\in X$ if
$\lim_{n\rightarrow\infty}\mathfrak{p}(x_{n},x_{0})=\mathfrak{p}(x_{0},x_{0})$.
A sequence $\\{x_{n}\\}$ of elements of $X$ is called _Cauchy_ if the limit
$\lim_{m,n\rightarrow\infty}\mathfrak{p}(x_{n},x_{m})$ exists and is finite.
The partial metric space $(X,\mathfrak{p})$ is called complete if for each
Cauchy sequence $\\{x_{n}\\}$, there is some $x\in X$ such that

$\lim_{n\rightarrow\infty}\mathfrak{p}(x_{n},x)=\mathfrak{p}(x,x)=\lim_{m,n\rightarrow\infty}\mathfrak{p}(x_{n},x_{m}).$

An example of a partial metric space is the pair
$(\mathbb{R}^{+},\mathfrak{p})$, where $\mathfrak{p}(x,y)=\max\\{x,y\\}$ for
all $x,y\in\mathbb{R}^{+}$. For more material about the partial metric spaces,
see, e.g. [3, 5, 9] and references therein.

In 2012, A. Amini-Harandi [1] introduced a new extension of the concept of
partial metric space, called a _metric-like space_. After that, the concept of
$b$-metric-like space which generalizes the notions of partial metric space,
metric-like space, and $b$-metric space was introduced by Alghamdi et al. in
[4]. Recently, Zidan and Mostefaoui [14] introduced the double controlled
quasi metric-like spaces and studied some topological properties of this
space. Here, we state the concept of a metric-like space.

######  Definition 1.2.

A mapping $\mathfrak{D}:X\times X\rightarrow\mathbb{R}^{+}$, where $X$ is a
non-empty set, is said to be a metric-like on $X$ if for any $x,y,z\in X$, the
following three conditions hold true:
(i) $\mathfrak{D}(x,y)=0\Rightarrow x=y$;
(ii) $\mathfrak{D}(x,y)=\mathfrak{D}(y,x)$;
(iii) $\mathfrak{D}(x,y)\leq\mathfrak{D}(x,z)+\mathfrak{D}(z,y)$.

The pair $(X,\mathfrak{D})$ is called a metric-like space. A metric-like on
$X$ satisfies all of the conditions of a metric except that
$\mathfrak{D}(x,x)$ may be positive for some $x\in X$. The study of partial
metric spaces has wide area of application, especially in computer sciences,
see, e.g. [5, 10, 13] and references therein. That is why working on this
topic can be very useful in practice. Since metric-likes are generalizations
of partial metrics, knowing them can therefore provide us more applicable
fields. In fact, this is our motivation to study the metric-like spaces. Each
metric-like $\mathfrak{D}$ on $X$ generates a topology $\tau_{\mathfrak{D}}$
on $X$ whose base is the family of open balls. An open ball in a metric-like
space $(X,\mathfrak{D})$, with center $x$ and radius $r>0$, is the set

$B(x,r)=\\{y\in X:|\mathfrak{D}(x,y)-\mathfrak{D}(x,x)|<r\\}.$

It is clear that a sequence $\\{x_{n}\\}$ in the metric-like space
$(X,\mathfrak{D})$ converges to a point $x\in X$ if and only if
$\lim_{n\rightarrow\infty}\mathfrak{D}(x_{n},x)=\mathfrak{D}(x,x)$. A sequence
$\\{x_{n}\\}$ of elements of a metric-like space $(X,\mathfrak{D})$ is called
Cauchy if the limit $\lim_{n,m\rightarrow\infty}\mathfrak{D}(x_{n},x_{m})$
exists and is finite. The metric-like space $(X,\mathfrak{D})$ is called
complete if for each Cauchy sequence $\\{x_{n}\\}$, there is some $x_{0}\in X$
such that

$\lim_{n\rightarrow\infty}\mathfrak{D}(x_{n},x_{0})=\mathfrak{D}(x_{0},x_{0})=\lim_{n,m\rightarrow\infty}\mathfrak{D}(x_{n},x_{m}).$

For more details on this topic, see, e.g. [1, 7]. Note that every partial
metric space is a metric-like space. But, the converse is not true in general.
For example, let $X=\mathbb{R}$, and let
$\mathfrak{D}(x,y)=\max\\{|x-5|,|y-5|\\}$ for all $x,y\in\mathbb{R}$. Then
$(X,\mathfrak{D})$ is a metric-like space, but since
$\mathfrak{D}(0,0)\nleq\mathfrak{D}(1,2)$, then $(X,\mathfrak{D})$ is not a
partial metric space. We now state another extension of the notion of distance
that allows distance to be a complex value. Azam et al., [2] introduced the
concept of a complex valued metric space and obtained sufficient conditions
for the existence of common fixed points of a pair of mappings satisfying
contractive type conditions. In that article, they consider a partial order
$\precsim$ on the set of complex numbers $\mathbb{C}$ and then introduce a
complex valued metric. The partial order $\precsim$ is as follows:

$\displaystyle z_{1}\precsim z_{2}\Leftrightarrow\ Re(z_{1})\leq Re(z_{2}),\
Im(z_{1})\leq Im(z_{2}).$

Let $X$ be a non-empty set. Suppose that the mapping $d:X\times
X\rightarrow\mathbb{C}$ satisfies the following conditions:
(i) $0\precsim d(x,y)$ for all $x,y\in X$;
(ii) $d(x,y)=0\Leftrightarrow x=y$;
(iii) $d(x,y)=d(y,x)$ for all $x,y\in X$;
(iv) $d(x,y)\precsim d(x,z)+d(z,y)$ for all $x,y,z\in X$.
Then $d$ is called a complex valued metric on $X$ , and $(X,d)$ is called a
complex valued metric space. Combining the two concepts complex valued metric
spaces and metric-like spaces, we get complex valued metric-like spaces. Also,
we introduce the notion of a complex valued partial metric space. As will be
seen, the notion of complex valued metric-like space is a generalization of
the notions of metric-like space, complex valued metric space, partial metric
space, complex valued partial metric space and metric space. Therefore, it is
interesting to investigate this general notion.

In this article, we focus on the structure of complex valued metric-like
spaces and study some topological properties of this space. For instance, we
introduce some concepts such as quasi-equal points, completely separate
points, convergence of a sequence, Cauchy sequence, cluster point, limit
point, complex absolute value and complex diameter of a subset of a complex
valued metric-like space. Additionally, we present several results about
complex valued metric-like spaces.

##  2\. Results and proofs

Let $\mathbb{C}$ be the set of complex numbers and $z_{1},z_{2}\in\mathbb{C}$.
Following [2], we define a partial order $\precsim$ on $\mathbb{C}$ as
follows:

$\displaystyle z_{1}\precsim z_{2}\Leftrightarrow Re(z_{1})\leq Re(z_{2}),\
Im(z_{1})\leq Im(z_{2}).$

Hence, $z_{1}\precsim z_{2}$ if one of the following conditions is satisfied:
(i) $Re(z_{1})=Re(z_{2})$ and $Im(z_{1})<Im(z_{2})$;
(ii) $Re(z_{1})<Re(z_{2})$ and $\ Im(z_{1})=Im(z_{2})$;
(iii) $Re(z_{1})<Re(z_{2})$ and $\ Im(z_{1})<Im(z_{2})$;
(iv) $Re(z_{1})=Re(z_{2})$ and $\ Im(z_{1})=Im(z_{2})$.

We write $z_{1}\precnsim z_{2}$ if $z_{1}\neq z_{2}$ and one of (i), (ii), and
(iii) is satisfied. Also, we write $z_{1}\prec z_{2}$ if only (iii) is
satisfied. Note that

$\displaystyle 0\precsim z_{1}\precnsim z_{2}\Rightarrow|z_{1}|<|z_{2}|,$
$\displaystyle z_{1}\precsim z_{2}\prec z_{3}\Rightarrow z_{1}\prec z_{3}.$

Also, $z_{2}\succsim z_{1}$ (resp. $z_{2}\succ z_{1}$) means that
$z_{1}\precsim z_{2}$ (resp. $z_{1}\prec z_{2}$). Throughout the paper, the
set $\\{z\in\mathbb{C}\ |\ z\succsim 0\\}$ is denoted by $\mathbb{C}\succsim
0$, i.e. $\mathbb{C}\succsim 0=\\{z\in\mathbb{C}\ |\ z\succsim 0\\}$. A
complex number $z$ is called positive if $0\prec z$.

######  Definition 2.1.

Let $X$ be a non-empty set. A mapping $d:X\times
X\rightarrow\mathbb{C}\succsim 0$ is called a complex valued metric-like on
$X$ if for any $x,y,z\in X$, the following conditions hold:
(D1) $d(x,y)=0\Rightarrow x=y$;
(D2) $d(x,y)=d(y,x)$;
(D3)$d(x,y)\precsim d(x,z)+d(z,y).$

The pair $(X,d)$ is then called a complex valued metric-like space. Indeed, a
complex valued metric-like on $X$ satisfies all of the conditions of complex
valued metric except that may be $0\precnsim d(x,x)$ for some $x\in X$. For
convenience, we write (CVML) for ”complex valued metric-like”.

In the following, we provide an example of a CVML space.

######  Example 2.2.

Let $X=\mathbb{C}$ . A mapping
$d:\mathbb{C}\times\mathbb{C}\rightarrow\mathbb{C}$ defined by
$d\left(z_{1},z_{2}\right)=e^{i\theta}\left(|z_{1}|+\left|z_{2}\right|\right)$,
where $0\leqslant\theta\leqslant\frac{\pi}{2}$ is a CVML on $\mathbb{C}$.

In the following, we introduce the complex absolute value of $z\in\mathbb{C}$
which is denoted by $|\cdot|_{c}$.

$|z|_{c}=\left|\operatorname{Re(z)}\right|+i\left|Im(z)\right|$

for any $z\in\mathbb{C}$. Clearly, $0\precsim|z|_{c}$ for all
$z\in\mathbb{C}$.

######  Definition 2.3.

Let $(X,d)$ be a CVML space and $A\subseteq X$. An open ball with center
$x_{0}\in X$ and radius $0\prec r\in\mathbb{C}$ is the set

$\displaystyle N(x_{0},r)=\left\\{y\in X:|d(x_{0},y)-d(x_{0},x_{0})|_{c}\prec
r\right\\}.$

######  Definition 2.4.

Let $X$ be a non-empty set. A mapping $p:X\times
X\rightarrow\mathbb{C}\succsim 0$ is said to be a complex valued partial
metric on $X$ if for all $x,y,z\in X,$ the following conditions hold:

  * •

$x=y\Leftrightarrow p(x,x)=p(y,y)=p(x,y)$;

  * •

$\max\\{p(x,x),p(y,y)\\}\precsim p(x,y)$;

  * •

$p(x,y)=p(y,x)$;

  * •

$p(x,z)\precsim p(x,y)+p(y,z)-p(y,y)$

The pair $(X,p)$ is called a complex valued partial metric space.

######  Definition 2.5.

Let $(X,d)$ be a CVML space, let $\\{x_{n}\\}_{n\geq 1}$ be a sequence of $X$
and let $x_{0}\in X$. We say that

  * •

The sequence $\\{x_{n}\\}_{n\geqslant 1}$ converges to $x_{0}$ if for every
$0\prec r\in\mathbb{C}$ there exists $n_{0}\in\mathbb{N}$ such that $x_{n}\in
N(x_{0},r)$ for all $n>n_{0}$, i.e. $|d(x_{n},x_{0})-d(x_{0},x_{0})|_{c}\prec
r$ for all $n>n_{0}$. We denote this by
$\lim_{n\rightarrow+\infty}x_{n}=x_{0},$ or $x_{n}\rightarrow x_{0}$ as
$n\rightarrow+\infty$.

  * •

The sequence $\\{x_{n}\\}_{n\geqslant 1}$ is called Cauchy if
$\lim_{m,n\rightarrow+\infty}d(x_{n},x_{m})$ exists and is finite. It means
that sequence $\\{x_{n}\\}_{n\geqslant 1}$ is Cauchy if and only if
$\lim_{m,n\rightarrow+\infty}d(x_{n},x_{m})=z_{0}$ for some
$z_{0}\in\mathbb{C}$.

  * •

The complex valued metric-like space $(X,d)$ is complete if every Cauchy
sequence of $X$ is convergent. On the other hand, the complex valued metric-
like space $(X,d)$ is complete if for each Cauchy sequence $\\{x_{n}\\}$,
there is some $x\in X$ such that

$\displaystyle\lim_{n\rightarrow+\infty}d(x_{n},x)=d(x,x)=\lim_{m,n\rightarrow+\infty}d(x_{n},x_{m})$

######  Remark 2.6.

Note that in CVML spaces the limit of a convergent sequence is not necessarily
unique and this means that topology of these spaces is not necessarily a
Hausdorff topology. For instance, suppose that $X=\mathbb{C}$ and
$d(x,y)=i\max\\{|x|,|y|\\}$ for each $x,y\in X$. Putting $x_{n}=\frac{i}{n}$,
we have
$\lim_{n\rightarrow\infty}d(\frac{i}{n},i)=\lim_{n\rightarrow\infty}i\max\\{|\frac{i}{n}|,|i|\\}=i=d(i,i)$.
It means that the sequence $\\{\frac{i}{n}\\}$ converses to $i$, i.e.
$\frac{i}{n}\rightarrow i$. Moreover, we have
$\lim_{n\rightarrow\infty}d(\frac{i}{n},2i)=\lim_{n\rightarrow\infty}i\max\\{|\frac{i}{n}|,|2i|\\}=2i=d(2i,2i)$,
and consequently, $\frac{i}{n}\rightarrow 2i$ as well. This demonstrates that
the sequence $\\{\frac{i}{n}\\}$ converges to two different points.

The above example leads us to the next definition.

######  Definition 2.7.

(Quasi-equal points) Let $(X,d)$ be a CVML space. The points $x,y\in X$ are
called _quasi-equal points_ if there exists a sequence $\\{x_{n}\\}$ of $X$
converging to both $x$ and $y$, i.e. $x_{n}\rightarrow x$ and
$x_{n}\rightarrow y$.

From the previous remark, one can easily deduce that if $X=\mathbb{C}$ and
$d(x,y)=i\max\\{|x|,|y|\\}$ for each $x,y\in\mathbb{C}$, then $i$ and $2i$ are
two quasi-equal points.

######  Definition 2.8.

(Completely separate points) Let $(X,d)$ be a CVML space. The points $x,y$ of
$X$ are called _completely separate points_ if the following condition holds
true:

$d(x,x)+d(y,y)\prec d(x,y)$

######  Theorem 2.9.

Let $(X,d)$ be a CVML space. Then there are no convergent sequences to two
completely separate points.

###### Proof.

Suppose that $x,y$ are two completely separate points. To obtain a
contradiction, let $\\{x_{n}\\}$ be a sequence of $X$ converging to both
$x,y$. Put $r_{1}=\frac{1}{3}r$, where $r=d(x,y)-d(x,x)-d(y,y)\succ 0$. Since
$x_{n}\rightarrow x$ and $x_{n}\rightarrow y$, for $\varepsilon=r_{1}$, there
exist two positive integers $N_{1}$ and $N_{2}$ such that

$\max\\{|d(x_{N},x)-d(x,x)|,|d(x_{N},y)-d(y,y)|\\}\prec r_{1}$

for all $n\geq N=\max\\{N_{1},N_{2}\\}$. Therefore,

$\displaystyle d(x,y)=|d(x,y)|_{c}$
$\displaystyle\precsim|d(x,x_{N})+d(x_{N},y)|_{c}$
$\displaystyle=|d(x,x_{N})+d(x_{N},y)-d(x,x)+d(x,x)-d(y,y)+d(y,y)|_{c}$
$\displaystyle\precsim|d(x,x_{N})-d(x,x)|_{c}+|d(x_{N},y)-d(y,y)|_{c}+d(x,x)+d(y,y)$
$\displaystyle\prec r_{1}+r_{1}+d(x,x)+d(y,y)$
$\displaystyle=2r_{1}+d(x,x)+d(y,y)$
$\displaystyle=\frac{2}{3}\Big{[}d(x,y)-d(x,x)-d(y,y)\Big{]}+d(x,x)+d(y,y)$
$\displaystyle=\frac{2}{3}d(x,y)+\frac{1}{3}d(x,x)+\frac{1}{3}d(y,y).$

So we obtain that $d(x,y)\prec d(x,x)+d(y,y)$, a contradiction. This
contradiction proves our claim. ∎

Immediate conclusion from the above theorem demonstrates that completely
separate points are not quasi-equal.

######  Definition 2.10.

(Cluster points) Let $(X,d)$ be a CVML space and let $A$ be a subset of $X$. A
point $x_{0}\in X$ is said to be a cluster point of $A$ if for every positive
complex number $r$ there exists an element $a\in A$ such that
$|d(a,x_{0})-d(x_{0},x_{0})|_{c}\prec r$.

As usual, the set of all _cluster points_ of $A$ is called the closure of $A$
and is denoted by $\overline{A}$. Note that $x_{0}\in\overline{A}$ if and only
if $N(x_{0},r)\cap A\neq\phi$ for all $0\prec r\in\mathbb{C}$. Indeed, we have

$\displaystyle\overline{A}=\Big{\\{}x_{0}\in X:N(x_{0},r)\cap A\neq\phi\ \
for\ all\ 0\prec r\in\mathbb{C}\Big{\\}}$

It is clear that if $(X,d)$ is a CVML space and $A$ is a subset of $X$, then
$A\subseteq\overline{A}$. In the following, we establish a theorem to present
a necessary and sufficient condition for cluster points in the CVML spaces.
First, we prove a theorem about the convergence of sequences in CVML spaces.

######  Theorem 2.11.

Let $(X,d)$ be a CVML space and let $\left\\{x_{n}\right\\}$ be a sequence of
$X$. Then $\left\\{x_{n}\right\\}$ converges to $x$ if and only if
$\lim_{n\rightarrow+\infty}\Big{|}d\left(x_{n},x\right)-d(x,x)\Big{|}=0$.

###### Proof.

Suppose that $\left\\{x_{n}\right\\}$ converges to $x$. Let
$\varepsilon=\frac{\alpha}{\sqrt{2}}+i\frac{\alpha}{\sqrt{2}}=\frac{\alpha}{\sqrt{2}}(1+i)$,
where $\alpha$ is a given positive real number. Since $\left\\{x_{n}\right\\}$
converges to $x$, there exists a positive integer $N$ such that
$\left|d\left(x_{n},x\right)-d(x,x)\right|_{c}\prec\varepsilon$ for all $n>N$.
Therefore,

$\Big{|}|d\left(x_{n},x\right)-d(x,x)|_{c}\Big{|}<|\varepsilon|=\alpha$

for all $n>N$, which means that
$\left|d\left(x_{n},x\right)-d(x,x)\right|<\alpha$ for all $n>N$. So

$\lim_{n\rightarrow+\infty}\left|d\left(x_{n},x\right)-d(x,x)\right|=0.$

Conversely, suppose that
$\lim_{n\rightarrow+\infty}|d\left(x_{n},x\right)-d(x,x)|=0$. Therefore, for
each positive real number $\alpha$, there exists a positive integer $N$ such
that $\left|d(x_{n},x)-d(x,x)\right|<\alpha$ for all $n>N$. It is clear that
for any $0\prec\varepsilon\in\mathbb{C}$, there exists a positive real number
$\alpha$ such that $\alpha+i\alpha=(1+i)\alpha\prec\varepsilon.$ Since
$\lim_{n\rightarrow+\infty}\left|d\left(x_{n},x\right)-d(x,x)\right|=0,$ there
exists a positive integer $n_{0}\in\mathbb{N}$ such that
$\left|d\left(x_{n},x\right)-d(x,x)\right|<\alpha$ for all $n\geq n_{0}$.
Hence, for all $n\geq n_{0}$, we have

$\displaystyle\left|d\left(x_{n},x\right)-d(x,x)\right|_{c}\precsim\left(1+i\right)|d\left(x_{n},x\right)-d\left(x,x\right)|\prec(1+i)\alpha\prec\varepsilon$

Indeed, we get that for all $0\prec\varepsilon\in\mathbb{C}$ there exists a
positive integer $n_{0}$ such that
$|d\left(x_{n},x\right)-\left.d(x,n)\right|_{c}\prec\varepsilon$ for all
$n\geq n_{0}$, and this means that $\lim_{n\rightarrow+\infty}x_{n}=x$, as
desired. ∎

######  Theorem 2.12.

Let $(X,d)$ be a CVML space and let $A$ be a subset of $X$. Then
$x_{0}\in\overline{\mathcal{A}}$ if and only if there is a sequence
$\\{a_{n}\\}\subseteq\mathcal{A}$ converging to $x_{0}$.

###### Proof.

Suppose that $x_{0}\in\overline{A}$. So for each $r_{n}=\frac{1+i}{n}$
($n\in\mathbb{N}$), there is an element $a_{n}\in\mathcal{A}$ such that
$|d(a_{n},x_{0})-d(x_{0},x_{0})|_{c}\prec\frac{1+i}{n}$. It implies that
$\lim_{n\rightarrow+\infty}|d(a_{n},x_{0})-d(x_{0},x_{0})|=0$, and it follows
from Theorem 2.11 that the sequence $\\{a_{n}\\}\subseteq A$ converges to
$x_{0}$. Conversely, assume that $\\{a_{n}\\}$ is a sequence of $A$ converging
to $x_{0}$. We must to show that $x_{0}\in\overline{A}$. Let $r$ be an
arbitrary positive complex number. Therefore, there exists a positive number
$N$ such that $a_{n}\in N(x_{0},r)$ for all $n\geq N$. It means that $A\cap
N(x_{0},r)\neq\phi$. Since we are assuming that $r$ is an arbitrary positive
complex number, it is deduced that $x_{0}\in\overline{A}$, as desired. ∎

######  Definition 2.13.

(Limit points) Let $(X,d)$ be a CVML space and let $A$ be a subset of $X$. A
point $x_{0}\in X$ is said to be a limit point of $A$ if
$N(x_{0},r)\cap(A-\\{x_{0}\\})\neq\phi$ for all positive complex numbers $r$.

As usual, the set of all _limit points_ of $A$ is denoted by $A^{\prime}$. A
subset $A$ of $X$ is called a closed set, whenever each limit point of $A$
belongs to itself, i.e. $A^{\prime}\subseteq A.$ One can easily prove the
following theorem.

######  Theorem 2.14.

Let $(X,d)$ be a CVML space and let $A$ be a subset of $X$. Then
$\overline{A}=A\cup A^{\prime}$.

######  Definition 2.15.

Let $(X,d)$ be a CVML space and let $A$ be a subset of $X$. The complex
diameter of $A$ is defined as follows:

$diam_{c}(A):=\sup\Big{\\{}|d(x,y)-d(x,x)|_{c},|d(x,y)-d(y,y)|_{c}:x,y\in
A\Big{\\}}.$

If $|diam_{c}(A)|<\infty$, then the subset $A\subseteq X$ is said to be
bounded.

######  Example 2.16.

Let $X=\mathbb{C}$, $d(x,y)=(1+i)(|z_{1}|+|z_{2}|)$, and let
$A=\\{z\in\mathbb{C}:3<|z|<5\\}$. In this case, we have

$\displaystyle diam_{c}(A)$
$\displaystyle=\sup\Big{\\{}|d(z_{1},z_{2})-d(z_{1},z_{1})|_{c},|d(z_{1},z_{2})-d(z_{2},z_{2})|_{c}:z_{1},z_{2}\in
A\Big{\\}}$
$\displaystyle=\sup\Big{\\{}\big{|}|z_{1}|-|z_{2}|\big{|}+i\big{|}|z_{1}|-|z_{2}|\big{|}:z_{1},z_{2}\in
A\Big{\\}}$
$\displaystyle=\sup\Big{\\{}\big{|}|z_{1}|-|z_{2}|\big{|}(1+i):z_{1},z_{2}\in
A\Big{\\}}$ $\displaystyle=2(1+i).$

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