# Exact solution of the position-dependent mass Schrödinger equation with the
completely positive oscillator-shaped quantum well potential

E.I. Jafarov Corresponding author<EMAIL_ADDRESS>Institute of
Physics, State Agency for Science and Higher Education
Javid av. 131, AZ1143, Baku, Azerbaijan  S.M. Nagiyev
<EMAIL_ADDRESS>Institute of Physics, State Agency for Science
and Higher Education
Javid av. 131, AZ1143, Baku, Azerbaijan

###### Abstract

Two exactly-solvable confined models of the completely positive oscillator-
shaped quantum well are proposed. Exact solutions of the position-dependent
mass Schrödinger equation corresponding to the proposed quantum well
potentials are presented. It is shown that the discrete energy spectrum
expressions of both models depend on certain positive confinement parameters.
The spectrum exhibits positive equidistant behavior for the model confined
only with one infinitely high wall and non-equidistant behavior for the model
confined with the infinitely high wall from both sides. Wavefunctions of the
stationary states of the models under construction are expressed through the
Laguerre and Jacobi polynomials. In general, the Jacobi polynomials appearing
in wavefunctions depend on parameters $a$ and $b$, but the Laguerre
polynomials depend only on the parameter $a$. Some limits and special cases of
the constructed models are discussed.

Keywords: Completely positive quantum well, Position-dependent mass, Exact
solution, Laguerre and Jacobi polynomials

##  1 Introduction

The study of the harmonic oscillator problem is of great importance in both
classical and quantum mechanics due to the problem plays a flagship role in
the explanation of a huge number of physical phenomena. The harmonic
oscillator problem in one dimension of the macroscopic sizes can be exactly
solved within Newtonian mechanics. A block with mass vibrating on a spring or
a simple pendulum is the best example of it. As a result of the well-known
solution, one observes that the classical system moves about the equilibrium
position and has two turning points, at which its kinetic energy becomes zero
and the object starts to recover its initial equilibrium position. In other
words, a confined motion that the classical harmonic oscillator exhibits
within Newtonian mechanics can be considered one of its main features [1].

The harmonic oscillator problem of the sub-micron sizes within the non-
relativistic quantum mechanics drastically differs from similar classical
mechanics problem [2, 3]. First of all, one needs to take into account that
within the quantum theory the momentum and position operators don’t commute
[4, 5]. Therefore, we are dealing with probabilities instead of precise joint
measurements of position and momentum. The appearance of the probabilistic
nature of the observations requires that the obtained expressions of the
distributions will be correct only if they are valid within the whole range of
the position or momentum from $-\infty$ to $+\infty$. This solution of the
quantum harmonic oscillator is well known. It has a discrete energy spectrum
with equidistant energy levels, its wavefunctions of the stationary states are
analytically expressed through the Hermite polynomials and vanish at
$x=\pm\infty$. This does not mean that it is impossible to consider the
quantum oscillator problem that is confined similar to its classical analog.
Such a problem also is studied intensively due to its necessity in the
explanation of many phenomena mainly coming from nanotechnologies. The quantum
approach for such a problem consists of coupling together two of its exactly-
solvable problems, namely the quantum harmonic oscillator and the infinite
potential well problems. In general, these two coupled problems don’t lead to
the exact solution of the corresponding Schrödinger equation. As we are aware,
[6] can be considered as a first discussion of the problem, where, energy
levels of the confined quantum harmonic oscillator are computed approximately
with further application to the proton-deuteron transformation as a source of
energy in dense stars. Some exactly soluble quantum harmonic oscillator models
are known for the case, if one replaces continuous position and momentum
representations with their finite-discrete analogues [7, 8, 9, 10, 11, 12].
However, the problem still can be exactly soluble in the continuous
configuration representation, if one assumes that the mass of the quantum
system is not constant, but varies with position [13, 14, 15, 16, 17, 18, 19,
20, 21, 22, 23, 24, 25, 26].

We are going to discuss the exact solution of the confinement model of the
non-rectangular quantum well, a profile of which consists of the harmonic
oscillator potential valid within the finite or semi-infinite positive range.
Recently, an exact quantum-mechanical solution for the one-dimensional
harmonic oscillator model asymmetrically confined into the infinite well was
introduced [27]. Its wavefunctions of the stationary states are expressed
through the Jacobi polynomials. Unlike the model with the approximate
solutions from [28], the exact solution in [27] is achieved thanks to the
replacement of the constant mass $m_{0}$ with the mass $M\left(x\right)$
specifically varying with the position. It also generalizes the
semiconfinement oscillator model with the wavefunctions expressed through the
Laguerre polynomials [29, 30, 31]. Then, we decided that one also can
successfully generate the confinement model of a completely positive
oscillator-shaped quantum well by using the less-studied orthogonality
properties of the Jacobi polynomials. The main feature of the model under
construction will be the exhibition of the confinement effect at infinitely
positive values of the potential. In other words, the quantum system under
consideration will be in the cavity between positive $a$ and $b$ values of
position with initial condition $a<b$. Also, the special case can be
considered here, when the wall located at position $x=b$ will be driven to the
$+\infty$.

We structured the paper as follows: Section 2 presents some basic information
about the Jacobi polynomials and generalization of their orthogonality
relation from $\left(-1;1\right)$ to $\left(a;b\right)$ with the condition
$a<x<b$ being satisfied. Here, also some less-known properties of the Laguerre
polynomials and the connection of these properties with the Jacobi polynomials
are discussed. Further, Section 3 consists of the main result of the current
paper – exact solutions to the confinement model of the non-relativistic one-
dimensional completely positive oscillator-shaped quantum well problems in
terms of the Laguerre and Jacobi polynomials are presented in this section.
Detailed discussions and conclusions are given in the final section. We
discuss the special cases as well as present some plots of the energy spectrum
and probability distributions corresponding to certain discrete energy levels
of the models under construction.

##  2 Jacobi polynomials, generalized orthogonality relation on the interval
$\left(a,b\right)$ and their slightly modified direct limit to the Laguerre
polynomials

Jacobi polynomials are one of the well-known and thoroughly studied continuous
classical polynomials of ${}_{2}F_{1}$ hypergeometric type belonging to the
Askey scheme of the orthogonal polynomials. They appear as a result of the
positive-definite exact solution of the following second-order differential
equation:

${\scriptstyle\left({z^{2}-1}\right)\frac{{d^{2}}}{{dz^{2}}}P_{n}^{\left({\alpha,\beta}\right)}\left(\pm
z\right)+\left[\left({\alpha+\beta+2}\right)z\pm\left({\alpha-\beta}\right)\right]\frac{d}{{dz}}P_{n}^{\left({\alpha,\beta}\right)}\left(\pm
z\right)=n\left({n+\alpha+\beta+1}\right)P_{n}^{\left({\alpha,\beta}\right)}\left(\pm
z\right),\quad n=0,1,2,\ldots,}$ (1)

where, the Jacobi polynomials
$P_{n}^{\left({\alpha,\beta}\right)}\left(z\right)$ are defined through the
${}_{2}F_{1}$ hypergeometric function by the following manner:

$P_{n}^{\left({\alpha,\beta}\right)}\left(z\right)=\frac{{\left({\alpha+1}\right)_{n}}}{{n!}}{\kern
1.0pt}_{2}F_{1}\left({\begin{array}[]{*{20}c}{-n,n+\alpha+\beta+1}\\\
{\alpha+1}\\\ \end{array};\frac{{1-z}}{2}}\right),$ (2)

or it can be written down as a finite sum

$P_{n}^{\left({\alpha,\beta}\right)}\left(z\right)=\frac{{\left({\alpha+1}\right)_{n}}}{{n!}}\sum\limits_{k=0}^{n}{\frac{{\left({-n}\right)_{k}\left({n+\alpha+\beta+1}\right)_{k}}}{{\left({\alpha+1}\right)_{k}2^{k}k!}}\left({1-z}\right)^{k}}.$
(3)

Under condition $\alpha>-1$ and $\beta>-1$ as well as $-1<z<1$ the following
continuous orthogonality relation in the finite interval holds for them:

$\scriptstyle\int\limits_{-1}^{1}{\left({1\mp z}\right)^{\alpha}\left({1\pm
z}\right)^{\beta}P_{m}^{\left({\alpha,\beta}\right)}\left(\pm
z\right)P_{n}^{\left({\alpha,\beta}\right)}\left(\pm
z\right)dz}=\frac{{2^{\alpha+\beta+1}}}{{2n+\alpha+\beta+1}}\frac{{\Gamma\left({n+\alpha+1}\right)\Gamma\left({n+\beta+1}\right)}}{{\Gamma\left({n+\alpha+\beta+1}\right)n!}}\delta_{mn}.$
(4)

Let’s introduce two real parameters $a$ and $b$ in such a manner that the
condition $a<b$ would be satisfied. Further, via the introduction of a new
variable $x$ of the following behavior:

$x=\frac{{a+b}}{2}\pm\frac{{a-b}}{2}z,$ (5)

one obtains that eq.(1) will be slightly changed as follows:

$\displaystyle\scriptstyle\left({x-a}\right)\left({x-b}\right)\frac{{d^{2}}}{{dx^{2}}}P_{n}^{\left({\alpha,\beta}\right)}\left({\frac{{a+b-2x}}{{b-a}}}\right)+\left[{\left({\alpha+1}\right)\left({x-b}\right)+\left({\beta+1}\right)\left({x-a}\right)}\right]\frac{d}{{dx}}P_{n}^{\left({\alpha,\beta}\right)}\left({\frac{{a+b-2x}}{{b-a}}}\right)=$
(6)
$\displaystyle\scriptstyle=n\left({n+\alpha+\beta+1}\right)P_{n}^{\left({\alpha,\beta}\right)}\left({\frac{{a+b-2x}}{{b-a}}}\right).$

One observes that due to the quasi-arbitrary nature of the parameters $\alpha$
and $\beta$, the condition $a<b$ will not be violated through the introduction
of new variable $x$ via both manners shown in (5). Therefore, taking into
account that the following symmetry relation holds for the Jacobi polynomials:

$P_{n}^{\left({\alpha,\beta}\right)}\left({\frac{{a+b-2x}}{{b-a}}}\right)=\left({-1}\right)^{n}P_{n}^{\left({\beta,\alpha}\right)}\left({\frac{{2x-a-b}}{{b-a}}}\right)$
(7)

then, eq.(6) can also be written as follows:

$\displaystyle\scriptstyle\left({x-a}\right)\left({x-b}\right)\frac{{d^{2}}}{{dx^{2}}}P_{n}^{\left({\beta,\alpha}\right)}\left({\frac{{2x-a-b}}{{b-a}}}\right)+\left[{\left({\alpha+1}\right)\left({x-b}\right)+\left({\beta+1}\right)\left({x-a}\right)}\right]\frac{d}{{dx}}P_{n}^{\left({\beta,\alpha}\right)}\left({\frac{{2x-a-b}}{{b-a}}}\right)=$
(8)
$\displaystyle\scriptstyle=n\left({n+\alpha+\beta+1}\right)P_{n}^{\left({\beta,\alpha}\right)}\left({\frac{{2x-a-b}}{{b-a}}}\right).$

Orthogonality relation (4) also changes and now holds for Jacobi polynomials
$P_{n}^{\left({\beta,\alpha}\right)}\left({\frac{{2x-a-b}}{{a-b}}}\right)$
within the interval $\left(a,b\right)$, i.e.

${\scriptstyle\int\limits_{a}^{b}{\left({x-a}\right)^{\alpha}\left({b-x}\right)^{\beta}P_{m}^{\left({\beta,\alpha}\right)}\left({\frac{{2x-a-b}}{{b-a}}}\right)P_{n}^{\left({\beta,\alpha}\right)}\left({\frac{{2x-a-b}}{{b-a}}}\right)dx}=\frac{{\left({b-a}\right)^{\alpha+\beta+1}}}{{2n+\alpha+\beta+1}}\frac{{\Gamma\left({n+\alpha+1}\right)\Gamma\left({n+\beta+1}\right)}}{{\Gamma\left({n+\alpha+\beta+1}\right)n!}}\delta_{mn},\quad
a<x<b.}$ (9)

The above-described generalized case has been discussed thoroughly in [32]. We
are going to apply such a generalization to our computations in the next
section. [32] also presents the direct limit relation between the Jacobi
polynomials defined through (2) or (3) and the Laguerre polynomials defined as
follows:

$L_{n}^{\left(\alpha\right)}\left(z\right)=\frac{{\left({\alpha+1}\right)_{n}}}{{n!}}\,_{1}F_{1}\left({\begin{array}[]{*{20}c}{-n}\\\
{\alpha+1}\\\
\end{array};z}\right)=\frac{1}{{n!}}\sum\limits_{k=0}^{n}{\frac{{\left({-n}\right)_{k}}}{{k!}}\left({\alpha+k+1}\right)_{n-k}z^{k}}.$
(10)

Then, if we take $z=1-2\beta^{-1}x$ in the definition of the Jacobi
polynomials (2) and let $\beta\to\infty$, one obtains that

$\mathop{\lim}\limits_{\beta\to\infty}P_{n}^{\left({\alpha,\beta}\right)}\left({1-\frac{{2x}}{\beta}}\right)=L_{n}^{\left(\alpha\right)}\left(x\right).$
(11)

We are going to change slightly this known limit relation between the Jacobi
and Laguerre polynomials. First of all, we connect $\alpha$ and $\beta$
parameters of the Jacobi polynomials as follows:

$\beta=\frac{b}{a}\alpha.$ (12)

Next, letting $b\to\infty$ one obtains that the following limit relation
between the Jacobi and Laguerre polynomials holds:

$\mathop{\lim}\limits_{b\to\infty}P_{n}^{\left({\alpha,\frac{b}{a}\alpha}\right)}\left({1-2\frac{{x-a}}{{b-a}}}\right)=L_{n}^{\left(\alpha\right)}\left({\frac{\alpha}{a}x-\alpha}\right).$
(13)

One observes that limit relation (13) is a slightly modified version of (11)
and it is valid for the Jacobi polynomials being appeared in the differential
equation (6). Its application to this equation yields the following
differential equation for the Laguerre polynomials
$L_{n}^{\left(\alpha\right)}\left({\frac{\alpha}{a}x-\alpha}\right)$:

$\left({x-a}\right)\frac{{d^{2}}}{{dx^{2}}}L_{n}^{\left(\alpha\right)}\left({\frac{\alpha}{a}x-\alpha}\right)+\left({2\alpha+1-\frac{\alpha}{a}x}\right)\frac{d}{{dx}}L_{n}^{\left(\alpha\right)}\left({\frac{\alpha}{a}x-\alpha}\right)=-\frac{\alpha}{a}nL_{n}^{\left(\alpha\right)}\left({\frac{\alpha}{a}x-\alpha}\right).$
(14)

We are going to apply such a generalization to our computations in the next
section, too. This solution in terms of the Laguerre polynomials will be
directly related to our oscillator-shaped quantum well model being confined
only from the right side with the infinitely high wall.

##  3 Position-dependent mass Schrödinger equation with the positive
oscillator-shaped quantum well potential

Let’s consider a quantum system confined only from the right side at a
position positive value $x=a$ with an infinitely high wall. In fact, this wall
forms a model of the semi-infinitely deep quantum well of the arbitrary
behavior of one of its walls. Attractivity of the problem under consideration
increases enormously if one manages to succeed with an analytical solution to
its Schrödinger equation under a certain type of potential energies. We are
going to demonstrate how one can obtain an exact solution to the Schrödinger
equation with the completely positive oscillator-shaped quantum well
potential. The notion of ’the completely positive oscillator-shaped quantum
well potential’ means that the non-relativistic quantum system described by
the Schrödinger equation is valid only for certain positive values of the
position.

Our starting point is the Schrödinger equation in one dimension:

$\left[{\frac{{\hat{p}_{x}}}{{2m_{0}}}+V\left(x\right)}\right]\psi\left(x\right)=E\psi\left(x\right).$
(15)

Here, $\psi\left(x\right)$ is the wavefunction of the stationary states of the
quantum system under study, $m_{0}$ is the constant mass and $E$ is its
energy. One-dimensional momentum operator $\hat{p}_{x}$ can be defined within
the canonical or non-canonical approach. We are going to deal with the
canonical approach, within which the momentum operator is defined as

$\hat{p}_{x}=-i\hbar\frac{d}{{dx}}.$

Its substitution at (15) leads to the following second-order differential
equation:

$\left[{-\frac{{\hbar^{2}}}{{2m_{0}}}\frac{{d^{2}}}{{dx^{2}}}+V\left(x\right)}\right]\psi\left(x\right)=E\psi\left(x\right).$
(16)

Now, one needs to introduce the analytical expression of the positive
oscillator-shaped quantum well potential $V\left(x\right)$. One of the ways is
a direct substitution of the well-known harmonic oscillator potential with the
constant mass $m_{0}$ and angular frequency $\omega$ at eq.(16) and then, to
solve corresponding second-order equation within the finite positive values of
the position $a<x<+\infty$ with additional confinement condition of existence
of an infinitely high wall at position $x=a$. It is well known that in this
case eq.(16) will not be solved exactly. Therefore, we are going with a
different way through the replacement of the constant mass with a position-
dependent one: $m_{0}\to M\left(x\right)$. The mass varying with position is a
well-known conception in physics. Its origin can be traced back to seminal
electron tunneling experiments [33, 34] and their theoretical explanation
within the approach of the band structure varying with position [35].
Moreover, the approach was developed to the assumption that varying band
structure of the solid compounds can lead also to the effective mass varying
with position and the following replacement for the non-relativistic kinetic
energy operator was proposed [36]:

$-\frac{{\hbar^{2}}}{{2m_{0}}}\frac{{d^{2}}}{{dx^{2}}}\to-\frac{{\hbar^{2}}}{2}\frac{d}{{dx}}\frac{1}{{M\left(x\right)}}\frac{d}{{dx}}.$

Such a replacement also preserves the Hermiticity property of the kinetic
energy operator. Its substitution at (16) yields:

$\left[{-\frac{{\hbar^{2}}}{2}\frac{d}{{dx}}\frac{1}{{M\left(x\right)}}\frac{d}{{dx}}+V\left(x\right)}\right]\psi\left(x\right)=E\psi\left(x\right).$
(17)

Recently, it was shown that this equation can be considered as a powerful tool
for simultaneous study of the movement of an electron in a semiconductor
quantum well and the propagation of electromagnetic waves in a dielectric
guide [37].

Taking into account that

$\frac{d}{{dx}}\frac{1}{M}\frac{d}{{dx}}=\frac{1}{M}\frac{{d^{2}}}{{dx^{2}}}-\frac{{M^{\prime}}}{{M^{2}}}\frac{d}{{dx}},\quad
M\equiv M\left(x\right),$

one can write down eq.(17) as follows:

$\left[{-\frac{{\hbar^{2}}}{{2M}}\frac{{d^{2}}}{{dx^{2}}}+\frac{{\hbar^{2}}}{2}\frac{{M^{\prime}}}{{M^{2}}}\frac{d}{{dx}}+V\left(x\right)}\right]\psi\left(x\right)=E\psi\left(x\right).$
(18)

This second-order differential equation can be solved exactly if one defines
analytical expressions of the potential $V\left(x\right)$ and position-
dependent mass $M\left(x\right)$. As we already highlighted that our potential
is almost the same as one-dimensional harmonic oscillator potential, but with
the mass varying with the position. Analytical expression of the mass should
be of such form that it will go to positive infinity at values of position
$a$, creating an infinitely high impenetrable wall as well as preserving
completely positive harmonic behavior of the oscillator for position values
$x>a$. The main feature of this oscillator is that it never approaches its
equilibrium state being completely positive. Of course, another important
condition for its introduction is the exact solubility of eq.(18).

Briefly, the following analytical expression of the position-dependent mass
$M\left(x\right)$

$M\left(x\right)=\frac{{am_{0}}}{{{x-a}}},$

and potential energy $V\left(x\right)$

$V\left(x\right)=\frac{{M\omega^{2}x^{2}}}{2},\quad a<x,$

is proposed.

Their substitution at (18) and further simple mathematical tricks yields:

$\psi^{\prime\prime}+\frac{1}{{x-a}}\psi^{\prime}-\frac{{\lambda_{0}^{4}a^{2}x^{2}-c_{0}x+ac_{0}}}{{\left({x-a}\right)^{2}}}\psi=0,$
(19)

where the notations $\frac{{d\psi}}{{dx}}\equiv\psi^{\prime}$ and
$\frac{{d^{2}\psi}}{{dx^{2}}}\equiv\psi^{\prime\prime}$ are introduced for
simplicity.

Here, also

$c_{0}=\frac{{2m_{0}aE}}{{\hbar^{2}}},\qquad\lambda_{0}=\sqrt{\frac{{m_{0}\omega}}{{\hbar}}},$
(20)

are introduced.

We look for exact solution of eq.(19) in the form of

$\psi\left(x\right)=\varphi\left(x\right)y\left(x\right),$ (21)

where $y\left(x\right)$ is the hypergeometric function part of the
wavefunction that defines its positive oscillator behavior at position values
$x>a$ as well as $\varphi\left(x\right)$ is its weight function that is going
to suppress the wavefunction to vanish under the boundary condition $x=a$.
Substitution of (21) at eq.(19) yields:

$\varphi
y^{\prime\prime}+\left[{2\varphi^{\prime}+\frac{1}{{x-a}}\varphi}\right]y^{\prime}+\left[{\varphi^{\prime\prime}+\frac{1}{{x-a}}\varphi^{\prime}-\frac{{\lambda_{0}^{4}a^{2}x^{2}-c_{0}x+ac_{0}}}{{\left({x-a}\right)^{2}}}\varphi}\right]y=0.$
(22)

Taking into account vanishing behavior of the wavefunction at values $x=a$ and
$x\to+\infty$, one introduces $\varphi\left(x\right)$ of the following semi-
general form:

$\varphi\left(x\right)=\left({x-a}\right)^{A}e^{Bx}.$ (23)

Here, $A$ and $B$ are arbitrary parameters, which values will be defined below
being related to the general behavior of the wavefunction both between walls
and at borders.

Simple computations allow to obtain that,

$\varphi^{\prime}=\frac{{Bx+A-aB}}{{x-a}}\varphi$ (24)

and

$\varphi^{\prime\prime}=\frac{{B^{2}x^{2}+2B\left({A-aB}\right)x+\left({A-aB}\right)^{2}-A}}{{\left({x-a}\right)^{2}}}\varphi.$
(25)

One observes from (25) that the positive oscillator behavior of the
wavefunction between two walls or polynomial solution of eq.(22) is possible
only if the following system of equations hold:

$\left\\{\begin{array}[]{l}B^{2}-\lambda_{0}^{4}a^{2}=0,\\\
c_{0}+2B\left({A-aB}\right)+B=\mu,\\\
c_{0}+B-\frac{{\left({A-aB}\right)^{2}}}{a}=\mu.\\\ \end{array}\right.$

Here, $\mu$ is an arbitrary real parameter. The above system of equations can
be easily solved leading to the following values for $A$, $B$, and $\mu$:

$A=\varepsilon_{1}\lambda_{0}^{2}a^{2},\;B=\varepsilon_{2}\lambda_{0}^{2}a,\;\mu=c_{0}+2\varepsilon_{2}\lambda_{0}^{2}a\left({\varepsilon_{1}-\varepsilon_{2}}\right)\lambda_{0}^{2}a^{2}+\varepsilon_{2}\lambda_{0}^{2}a,\quad\varepsilon_{1,2}=\pm
1.$

Then, one observes for (23) that

$\varphi\left(x\right)=\left({x-a}\right)^{\varepsilon_{1}\lambda_{0}^{2}a^{2}}e^{\varepsilon_{2}\lambda_{0}^{2}ax},$
(26)

and as a consequence of it eq.(22) reduces to the equation for
$y\left(x\right)$ of the following form:

$\scriptstyle\left({x-a}\right)y^{\prime\prime}+\left[{2\left({\varepsilon_{1}-\varepsilon_{2}}\right)\lambda_{0}^{2}a^{2}+1+2\varepsilon_{2}\lambda_{0}^{2}ax}\right]y^{\prime}+\left[{c_{0}+2\varepsilon_{2}\lambda_{0}^{2}a\left({\varepsilon_{1}-\varepsilon_{2}}\right)\lambda_{0}^{2}a^{2}+\varepsilon_{2}\lambda_{0}^{2}a}\right]y=0.$
(27)

Taking into account that parameters $\varepsilon_{1,2}$ can be both positive
and negative, then the four possible scenarios can be constructed. However,
only the vanishing behavior of the wavefunction with one of these values will
be satisfied. This values are $\varepsilon_{1}=+1$ and $\varepsilon_{2}=-1$.
One simplifies for (26) as follows:

$\varphi\left(x\right)=\left({x-a}\right)^{\lambda_{0}^{2}a^{2}}e^{-\lambda_{0}^{2}ax},$
(28)

as well as eq.(27) also slightly simplifies to the following form:

$\left({x-a}\right)y^{\prime\prime}+\left({4\lambda_{0}^{2}a^{2}+1-2\lambda_{0}^{2}ax}\right)y^{\prime}+\left({c_{0}-4\lambda_{0}^{4}a^{3}-\lambda_{0}^{2}a}\right)y=0.$
(29)

This second-order differential equation for $y\left(x\right)$ is already known
to us from the previous section. One observes its similarity with eq.(14).
Then, the comparison of both (29) and (14) yields the following analytical
expression of the energy spectrum

$E_{n}=\hbar\omega\left({n+\frac{1}{2}}\right)+2m_{0}\omega^{2}a^{2},$ (30)

and orthonormalized wavefunctions in terms of the Laguerre polynomials as
follows:

$\psi_{n}\left(x\right)=C_{n}\left({x-a}\right)^{\lambda_{0}^{2}a^{2}}e^{-\lambda_{0}^{2}ax}L_{n}^{\left({2\lambda_{0}^{2}a^{2}}\right)}\left({2\lambda_{0}^{2}a\left({x-a}\right)}\right),$
(31)

where the orthonormalized coefficient of the following analytical expression

$C_{n}=\left(-1\right)^{n}e^{\lambda_{0}^{2}a^{2}}\left({2\lambda_{0}^{2}a}\right)^{\lambda_{0}^{2}a^{2}+\frac{1}{2}}\sqrt{\frac{{n!}}{{\Gamma\left({n+2\lambda_{0}^{2}a^{2}+1}\right)}}}$
(32)

can be easily deduced from the following orthogonality relation for the
Laguerre polynomials:

$\scriptstyle\int\limits_{a}^{\infty}{e^{-2\lambda_{0}^{2}ax}\left({x-a}\right)^{2\lambda_{0}^{2}a^{2}}L_{m}^{\left({2\lambda_{0}^{2}a^{2}}\right)}\left({2\lambda_{0}^{2}a\left({x-a}\right)}\right)L_{n}^{\left({2\lambda_{0}^{2}a^{2}}\right)}\left({2\lambda_{0}^{2}a\left({x-a}\right)}\right)dx}=\frac{{\Gamma\left({n+2\lambda_{0}^{2}a^{2}+1}\right)}}{{e^{2\lambda_{0}^{2}a^{2}}\left({2\lambda_{0}^{2}a}\right)^{2\lambda_{0}^{2}a^{2}+1}n!}}\delta_{mn}.$

Next, let’s consider a more general quantum system, i.e. the system confined
to the cavity within the two positively infinite high walls. These walls are
located in positive position values $x=a$ and $x=b$ with initial condition
$a<b$. One deduces that these two walls are separated by a distance $b-a$ and
in fact, they form a model of the infinitely deep quantum well of the
arbitrary behavior. In other words, such behavior can have an arbitrary
profile depending on the analytical expression of the potential energy being
applied between two walls. We are going to demonstrate how one can obtain an
exact solution to the Schrödinger equation with the completely positive
oscillator-shaped quantum well potential for the case, if the quantum system
under construction is confined within the two infinitely high walls of
positive values and behaves itself as an oscillator between these walls.

In this general case, the following analytical expression of the position-
dependent mass $M\left(x\right)$

$M\left(x\right)=\frac{{abm_{0}}}{{\left({x-a}\right)\left({b-x}\right)}},$

and potential energy $V\left(x\right)$

$V\left(x\right)=\frac{{M\omega^{2}x^{2}}}{2},\quad a<x<b,$

is proposed.

Again, their substitution at (18) and further simple mathematical tricks
yields:

$\psi^{\prime\prime}-\frac{{2x-\left({a+b}\right)}}{{\left({x-a}\right)\left({b-x}\right)}}\psi^{\prime}-\frac{{c_{2}x^{2}-\left({a+b}\right)c_{1}x+abc_{1}}}{{\left({x-a}\right)^{2}\left({b-x}\right)^{2}}}\psi=0,$
(33)

where,

$\displaystyle c_{1}=$ $\displaystyle bc_{0}=\frac{{2m_{0}abE}}{{\hbar^{2}}},$
(34) $\displaystyle c_{2}=$ $\displaystyle c_{1}+\lambda_{0}^{4}a^{2}b^{2},$

are introduced for simplicity.

We look for exact solution of eq.(33) in the form of

$\psi\left(x\right)=\varphi\left(x\right)y\left(x\right),$ (35)

where $y\left(x\right)$ is the hypergeometric function part of the
wavefunction that defines its positive oscillator behavior between two walls
and $\varphi\left(x\right)$ is its weight function that is going to suppress
the wavefunction to vanish under the boundary conditions $x=a$ and $x=b$.
Substitution of (35) at eq.(33) yields:

$\scriptstyle\varphi
y^{\prime\prime}+\left[{2\varphi^{\prime}-\frac{{2x-\left({a+b}\right)}}{{\left({x-a}\right)\left({b-x}\right)}}\varphi}\right]y^{\prime}+\left[{\varphi^{\prime\prime}-\frac{{2x-\left({a+b}\right)}}{{\left({x-a}\right)\left({b-x}\right)}}\varphi^{\prime}-\frac{{c_{2}x^{2}-\left({a+b}\right)c_{1}x+abc_{1}}}{{\left({x-a}\right)^{2}\left({b-x}\right)^{2}}}\varphi}\right]y=0.$
(36)

Taking into account vanishing behavior of the wavefunction at values $x=a$ and
$x=b$, one introduces $\varphi\left(x\right)$ of the following semi-general
form:

$\varphi\left(x\right)=\left({x-a}\right)^{A}\left({b-x}\right)^{B}.$ (37)

Here, $A$ and $B$ again are arbitrary parameters, which values will be defined
below being related to the general behavior of the wavefunction both between
walls and at borders.

Simple computations allow to obtain that,

$\varphi^{\prime}=\frac{{\left({bA+aB}\right)-\left({A+B}\right)x}}{{\left({x-a}\right)\left({b-x}\right)}}\varphi$
(38)

and

$\scriptstyle\varphi^{\prime\prime}=\frac{{\left[{\left({bA+aB}\right)-\left({A+B}\right)x}\right]^{2}+\left[{\left({bA+aB}\right)-\left({A+B}\right)x}\right]\left[{2x-\left({a+b}\right)}\right]-\left({A+B}\right)\left({x-a}\right)\left({b-x}\right)}}{{\left({x-a}\right)^{2}\left({b-x}\right)^{2}}}\varphi.$
(39)

One observes from (39) that the positive oscillator behavior of the
wavefunction between two walls or polynomial solution of eq.(36) is possible
only if the following system of equations hold:

$\left\\{\begin{array}[]{l}c_{2}-\left({A+B}\right)^{2}-\left({A+B}\right)=\mu,\\\
c_{1}-\left({A+B}\right)-\frac{{2\left({bA+aB}\right)\left({A+B}\right)}}{{a+b}}=\mu,\\\
c_{1}-\left({A+B}\right)-\frac{{\left({bA+aB}\right)^{2}}}{{ab}}=\mu.\\\
\end{array}\right.$

Here, $\mu$ again is an arbitrary real parameter. The above system of
equations again can be easily solved leading to the following values for $A$,
$B$ and $\mu$:

$\scriptstyle
A=\varepsilon_{1}\lambda_{0}^{2}\frac{{a^{2}b}}{{a-b}},\;B=\varepsilon_{2}\lambda_{0}^{2}\frac{{ab^{2}}}{{a-b}},\;\mu=c_{1}-\lambda_{0}^{2}\frac{{ab}}{{a-b}}\left({\varepsilon_{1}a+\varepsilon_{2}b}\right)-2\lambda_{0}^{4}\frac{{a^{3}b^{3}}}{{\left({a-b}\right)^{2}}}\left({1+\varepsilon_{1}\varepsilon_{2}}\right),\quad\varepsilon_{1,2}=\pm
1.$

Then, one observes for (37) that

$\varphi\left(x\right)=\left({x-a}\right)^{\varepsilon_{1}\lambda_{0}^{2}\frac{{a^{2}b}}{{a-b}}}\left({b-x}\right)^{\varepsilon_{2}\lambda_{0}^{2}\frac{{ab^{2}}}{{a-b}}},$
(40)

and as a consequence of it eq.(36) reduces to the equation for
$y\left(x\right)$ of the following form:

$\displaystyle\left({x-a}\right)\left({b-x}\right)y^{\prime\prime}+$
$\displaystyle\left[{2\lambda_{0}^{2}\frac{{a^{2}b^{2}}}{{a-b}}\left({\varepsilon_{1}+\varepsilon_{2}}\right)+\left({a+b}\right)-2\left({\varepsilon_{1}\lambda_{0}^{2}\frac{{a^{2}b}}{{a-b}}+\varepsilon_{2}\lambda_{0}^{2}\frac{{ab^{2}}}{{a-b}}+1}\right)x}\right]y^{\prime}$
(41) $\displaystyle+$
$\displaystyle\left[{c_{1}-\lambda_{0}^{2}\frac{{ab}}{{a-b}}\left({\varepsilon_{1}a+\varepsilon_{2}b}\right)-2\lambda_{0}^{4}\frac{{a^{3}b^{3}}}{{\left({a-b}\right)^{2}}}\left({1+\varepsilon_{1}\varepsilon_{2}}\right)}\right]y=0.$

Now, still the parameters $\varepsilon_{1,2}$ can be both positive and
negative, hence the four possible scenarios can be constructed. However, it is
clear that only the vanishing behavior of the wavefunction with one of these
values will be satisfied. This value is $\varepsilon_{1,2}=-1$. One simplifies
for (40) as follows:

$\varphi\left(x\right)=\left({x-a}\right)^{\lambda_{0}^{2}\frac{{a^{2}b}}{{b-a}}}\left({b-x}\right)^{\lambda_{0}^{2}\frac{{ab^{2}}}{{b-a}}},$
(42)

as well as eq.(41) also slightly will be simplified to the following form:

$\scriptstyle\left({x-a}\right)\left({b-x}\right)y^{\prime\prime}+\left[{2\left({\lambda_{0}^{2}ab\frac{{a+b}}{{a-b}}-1}\right)x-4\lambda_{0}^{2}\frac{{a^{2}b^{2}}}{{a-b}}+\left({a+b}\right)}\right]y^{\prime}+\left[{c_{1}+\lambda_{0}^{2}ab\frac{{a+b}}{{a-b}}-4\lambda_{0}^{4}\frac{{a^{3}b^{3}}}{{\left({a-b}\right)^{2}}}}\right]y=0.$
(43)

This second-order differential equation for $y\left(x\right)$ is also already
known to us from the previous section. One observes its similarity with
eq.(8). Then, the comparison of both (43) and (8) yields the following
analytical expression of the energy spectrum

$E_{n}=\frac{{b+a}}{{b-a}}\hbar\omega\left({n+\frac{1}{2}}\right)+\frac{{\hbar^{2}}}{{2m_{0}ab}}n\left({n+1}\right)+2m_{0}\omega^{2}\frac{{a^{2}b^{2}}}{{\left({b-a}\right)^{2}}},$
(44)

and orthonormalized wavefunctions in terms of the Jacobi polynomials as
follows:

$\psi_{n}\left(x\right)=C_{n}\left({x-a}\right)^{\lambda_{0}^{2}\frac{{a^{2}b}}{{b-a}}}\left({b-x}\right)^{\lambda_{0}^{2}\frac{{ab^{2}}}{{b-a}}}P_{n}^{\left({2\lambda_{0}^{2}\frac{{ab^{2}}}{{b-a}},2\lambda_{0}^{2}\frac{{a^{2}b}}{{b-a}}}\right)}\left({\frac{{2x-a-b}}{{b-a}}}\right),$
(45)

where the orthonormalization coefficient of the following analytical
expression can be easily deduced from (9):

$C_{n}=\left(-1\right)^{n}\left({b-a}\right)^{-\lambda_{0}^{2}ab\frac{{b+a}}{{b-a}}-\frac{1}{2}}\sqrt{\frac{{\left({2n+2\lambda_{0}^{2}ab\frac{{b+a}}{{b-a}}+1}\right)\Gamma\left({n+2\lambda_{0}^{2}ab\frac{{b+a}}{{b-a}}+1}\right)n!}}{{\Gamma\left({n+2\lambda_{0}^{2}\frac{{a^{2}b}}{{b-a}}+1}\right)\Gamma\left({n+2\lambda_{0}^{2}\frac{{ab^{2}}}{{b-a}}+1}\right)}}}.$
(46)

Exact analytical expressions of the wavefunctions (31) and (45), as well as
analytical expressions of the energy spectrum (30) and (44), are the main goal
that we needed to achieve. Now, we are going to discuss briefly the possible
connection between these analytical expressions as well as their behavior
under special conditions.

##  4 Discussions and Conclusions

We managed to solve exactly both equations (19) and (33). One observes from
obtained analytical expressions of the eigenvalues of these second-order
differential equations, the behavior of both of them is drastically different.
First of all, the energy spectrum (30) corresponding to the model with only
one infinitely high wall is equidistant. However, the energy spectrum (44)
corresponding to the model located within two infinitely high walls exhibits
non-equidistant behavior. Also, despite that the first model is confined to
the positive region of position $a<x$, its energy spectrum (30) is almost
similar to the well-known non-relativistic quantum harmonic oscillator with
the constant mass [1, 2, 3]. Its first term completely overlaps with the known
oscillator energy spectrum. The only difference in energy spectrum (30) is the
existence of its additional second term that depends on the so-called
confinement parameter $a$. It seems that if letting to $a\to 0$, then the non-
relativistic quantum harmonic oscillator model will be directly recovered
through simple mathematical computations. However, it is a misassumption.
Therefore, one needs to analyze some details of the energy spectrum (44), too.
It consists of three terms. The first term does not overlap but is similar to
the known oscillator energy spectrum. The second term adds non-equidistant
behavior and the third term is simply some constant that does not contribute
anything to the general non-equidistant behavior of the energy spectrum (44).
In general, all three terms depend on parameters $a$ and $b$. Then, the case
$a\to 0$ violates its such a general behavior through the second term.
Parameter $a$ is in its denominator. Therefore, this term goes to infinity.
From a physics viewpoint, masses of both models changing by position simply
become zero if $a\to 0$. Therefore, both models restricted within the positive
region have a number of singularities, which never allow them to recover the
well-known non-relativistic quantum harmonic oscillator model with the
wavefunctions expressed through the Hermite polynomials.

Figure 1: Dependence of the probability densities
$\left|\psi_{n}\left(x\right)\right|^{2}$ generated via (45) and non-
equidistant energy levels (44) from the confinement parameters $a$ and $b$ for
the ground and a) 10 excited states if $a=1$ and $b=2$; b) 7 excited states if
$a=1.5$ and $b=2.5$ ($m_{0}=\omega=\hbar=1$).

In fig.1, we depicted dependence of the probability densities
$\left|\psi_{n}\left(x\right)\right|^{2}$ generated via (45) and non-
equidistant energy levels (44) corresponding to them from the confinement
parameters $a$ and $b$. In order to keep the overlapping scales of the plot,
we visualize only a restricted number of the stationary state. There are
ground plus $10$ excited states if $a=1$ and $b=2$ as well as ground plus $7$
excited states if $a=1.5$ and $b=2.5$. For simplicity, the measurement system
$m_{0}=\omega=\hbar=1$ is chosen, too. One observes Gaussian-like behavior for
the ground state probability densities
$\left|\psi_{0}\left(x\right)\right|^{2}$ for both values of $a$ and $b$.
Another feature that can be observed from these plots is the exhibition by the
system more probability close to $x\to a$ than $x\to b$. Of course, the main
feature of the model under construction is that its equilibrium state is
always different than zero. This is a consequence of a completely positive
definition of the model in the initial state.

Figure 2: Dependence of the probability densities
$\left|\psi_{n}\left(x\right)\right|^{2}$ generated via (31) and equidistant
energy levels (30) from the confinement parameter $a$ for the ground and a) 5
excited states if $a=1$; b) 3 excited states if $a=1.5$
($m_{0}=\omega=\hbar=1$).

The case, when $b\to+\infty$ is depicted in fig.2. We visualize dependence of
the probability densities $\left|\psi_{n}\left(x\right)\right|^{2}$ generated
via (31) and equidistant energy levels (30) corresponding to these probability
densities from the confinement parameter $a$. Mathematically, the limit
$b\to+\infty$ results in the disappearance of the second term in (44) as well
as simplifications of its first and third terms. One observes equidistant
energy levels as well as higher probability densities close to the value of
the position $x\to a$. Also, Gaussian-like behavior for the ground state
probability densities $\left|\psi_{0}\left(x\right)\right|^{2}$ still can be
observed here.

Recovery of the polynomial part of the wavefunction (31) from the wavefunction
(45) under the limit $b\to+\infty$ is based on the slightly modified limit
relation between Jacobi and Laguerre polynomials, introduced via (13). The
recovery of the weight function of the wavefunction (31) can be performed
through the following basic limit relations:

$\mathop{\lim}\limits_{b\to\infty}\left({x-a}\right)^{\lambda_{0}^{2}\frac{{a^{2}b}}{{b-a}}}=\left({x-a}\right)^{\lambda_{0}^{2}a^{2}},$
$\mathop{\lim}\limits_{b\to\infty}\Gamma\left({n+2\lambda_{0}^{2}\frac{{a^{2}b}}{{b-a}}+1}\right)=\Gamma\left({n+2\lambda_{0}^{2}a^{2}+1}\right),$

and

$\mathop{\lim}\limits_{b\to\infty}\sqrt{\frac{{2n+2\lambda_{0}^{2}ab+1}}{{b-a}}}=\sqrt{2\lambda_{0}^{2}a},$

as well as by applying Stirling’s approximation for the gamma function
$\Gamma\left({z+1}\right)\approx\sqrt{2\pi}x^{x+\frac{1}{2}}e^{-x}$, which
leads to

$\mathop{\lim}\limits_{b\to\infty}\left({b-a}\right)^{-\lambda_{0}^{2}ab\frac{{b+a}}{{b-a}}}\sqrt{\frac{{\Gamma\left({n+2\lambda_{0}^{2}ab\frac{{b+a}}{{b-a}}+1}\right)}}{{\Gamma\left({n+2\lambda_{0}^{2}\frac{{ab^{2}}}{{b-a}}+1}\right)}}}\left({b-x}\right)^{\lambda_{0}^{2}\frac{{ab^{2}}}{{b-a}}}=\left({2\lambda_{0}^{2}a}\right)^{\lambda_{0}^{2}a^{2}}e^{-\lambda_{0}^{2}a\left({x-a}\right)}.$

Concluding, we want to highlight that a special approach to the quantum well
problem is used for the construction of its two unique models. These models
are completely defined within the positive position representation. They
behave as oscillator-shaped, however, their mass is not constant, but varies
with position. This feature of the mass allows for considering these models as
confined quantum wells. Also, if one attempts to extend the definition of
these models to the negative position values, then an interesting singularity
arises changing the mass from positive to the negative one. Such negativity of
the mass also changes potential itself from a positive to a negative
definition. At present, this point of discussions is out of the aim of the
paper, however, in the future, this property can be the attractive initial
point of discussion regarding exactly solvable models of the inverted harmonic
oscillator problem (cf. with [38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49,
50, 51], where similar oscillator problem is discussed via replacement
$\omega\to i\omega$ in the potential energy).

## Acknowledgement

This work was supported by the Azerbaijan Science Foundation through Grant No.
AEF-MCG-2022-1(42)-12/01/1-M-01.

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