Bound states in the continuum are universal under the effect of minimal length

We consider a particle moving in a time-independent potential $V(X)$. The quantities $X$ and $P$ with minimal length can be represented as $X=x,P=p(1+\beta^{\prime}{p}^{2})$ 11 , where $x$ and $p$ are the standard quantities without the minimal length and $\beta^{\prime}=\beta/3$. Substituting these into the Hamiltonian

and using the correspondence relation of the momentum $p\rightarrow-i\hbar(\mathrm{d}/\mathrm{d}x)$, the dynamical evolutions with the minimal length of the wave functions (the Schrödinger equation with the minimal length) are given by

There are two main kinds of methods to reduce the above fourth-order Schrödinger equation to a second-order one at present:

1. (i)

   omitting the terms with $\beta^{2}$ to reduce the order of the fourth-order Schrödinger equation after using some substitution of wave functions 50 ; 54 ;

2. (ii)

   studying the fourth-order Schrödinger equation of some very particular potentials in the momentum representation (by employing Fourier transform) 12 ; 57 .

However, using these methods, to some extent, only some particular solutions of the fourth-order Schrödinger equation can possibly be obtained (see App. A for more details) in some particular situations. Thus, we should be very careful with this method since it would influence the solution space of the equation and hence alter the properties of the solution states. We have shown its influence on the solutions in two different situations in App. A.

Since there exists the parameter $\beta^{\prime}$ in the highest order term $[\beta^{\prime}\hbar^{4}\mathrm{d}^{4}\varphi(x)]/(m\mathrm{d}x^{4})$ of the above Schrödinger equation Eq. (2), the equation is a singular perturbation system 55 ; 56 . If we truncate the term with $\beta^{\prime}$ or set $\beta^{\prime}=0$ in Eq. 2, the fourth-order equation is reduced to a second-order one and this may change the space solution and alter the properties of the solution states. This fourth-order term is essential to the equations and cannot be simply omitted in some particular situations, because it not only accounts for the effect of the quantum-gravitational fluctuations of the background metric 53 , but also determines the solution space of the equation 55 ; 56 .

This kind of system is sensitive to its the highest derivative terms; transforming it may cause exotic phase transitions in its solution space 55 ; 56 . This causes exotic results when we study the system using the above two methods in some situations: for example, only some particular solutions are obtained; the states caused by the singular perturbation are ignored; complete wave functions can not be obtained; or certain phenomena are ignored 58 ; 59 (see App. A for more details). In particular, Ref. 4 showed that we can only obtain part of the solutions of the Schrödinger equation of the minimal length in the linear potential in the momentum representation (when the parameter $\alpha=1$ of the Schrödinger equation in Ref. 4 is Eq. (2)). Therefore, we should be very careful with this method since it would influence the solution space of the equation and hence alter the properties of the solution states. (For the linear potential and quantum harmonic oscillator, if we transform the time-independent Schrödinger equation in the minimal length from the position representation to the momentum representation, the equations become regular perturbation systems from the singular perturbation systems, because not all the highest order derivative terms of the two equations include the parameters $\beta$ 55 ; 56 ). In this article, we provide a method to directly analyze the fourth-order Schrödinger equation in the position representation, and the results indicate some counterintuitive conclusions.

Next, we will address the method to analyze the spatial structure of a particle’s bound states and find BICs in most ordinary potentials with a single particle under the effect of the minimal length, more precisely obtaining the energy and wave functions of the systems. For simplicity, we rewrite Eq. (2) in matrix form:

where

$A(x)$ describes the Hamiltonian of the system, and $\Phi(x)$ represents the particle’s motion.

First, we show that the space of the solutions of Eq. (2) has $4$ DFs in the following mild mathematical derivations. This means that the general solutions of Eq. (2) have $4$ free undetermined coefficients (that is, $C_{1},\cdots,C_{4}$ in Eq. (4)). The $4$ DFs are illustrated in Fig. 2(a) and (b) as $\omega_{1},\cdots,\omega_{4}$, namely, any linear combinations of $\omega_{1},\cdots,\omega_{4}$ are solutions of Eq. (2). However, the standard 1-D Schrödinger equation has $2$ DFs at most, which are illustrated as $\omega_{1}$ and $\omega_{2}$ in Fig. 2(c).

For any four linearly independent vectors $\varepsilon_{i}\in\mathds{R}^{4}$, where $i=1,\cdots,4$, and a point $x_{0}\in\mathds{R}$, there exist solutions $\Phi_{1}(x),\cdots,\Phi_{4}(x)$ of Eq. (3) satisfying $\Phi_{i}(x_{0})=\varepsilon_{i}$ 19 . Let $\varphi_{i}(x)$ represents the first component of $\Phi_{i}(x)$; then, there exists a non-zero $x=x_{0}$ of the Wronskian determinant $W(\varphi_{1}(x),\varphi_{2}(x),\varphi_{3}(x),\varphi_{4}(x))$:

Thus, $\varphi_{1}(x),\cdots,\varphi_{4}(x)$ are linearly independent. Any solutions of Eq. (2) can be represented by them:

where $C_{1},\cdots,C_{4}$ are $4$ arbitrary coefficients. This shows that the space of the solutions of Eq. (2) is $4$ dimensional, and $\varphi(x)$ has $4$ DFs.

Second, we propose a method to determine whether BICs exist in a system under the effect of the minimal length. Applying the method, we show that BICs can be found in many ordinary potentials. To build this method, we divide all the boundary conditions that limit the motion of particles into two types. One type can ensure states being bounded, such as $\lim\limits_{x\rightarrow\pm\infty}\varphi(x)=0$; we call them key boundary conditions (KBCs). The other type is an unnecessary condition for states being bounded, such as $\lim\limits_{x\rightarrow 0}\varphi(x)=0$ in the Coulomb potential; we call them non-key boundary conditions (non-KBCs). Then, we divide the three cases up to discuss the energy and bound states, therein being one of the main results in this work.

Case I–The KBCs satisfy the following conditions: there exist two different $x_{a},x_{b}(x_{a}<x_{b})$ such that the wave function $\varphi(x)=0$ in the regions $(-\infty,x_{a}]$ and $[x_{b},+\infty)$, as shown in Fig. 1(a). If there are fewer than $2$ non-KBCs, combining the KBCs, there are $3$ conditions at most, which cannot determine all $4$ undetermined coefficients of Eq. (4)

Thus, the energy $E$ is arbitrary. The KBCs ensure the states of the system to be bounded so that the BICs exist in these potential under the effect of the minimal length. Many potentials satisfy the conditions of Case I, e.g., the infinite potential well, asymmetric infinite well 16 , Dirac delta function in the infinite square well, Poschl-Teller potential 14 and quantum bouncer in a closed court 16 .

Case II–The KBCs satisfy the conditions $\lim\limits_{x\rightarrow+\infty}\varphi(x)=0$ (or $\lim\limits_{x\rightarrow-\infty}\varphi(x)=0$), and there exists one point $x=x_{a}$ (or $x_{b}$) such that wave function $\varphi(x)=0$ in the region $(-\infty,x_{a}]$ (or $[x_{b},+\infty)$), as shown in Fig. 1(b). This type of KBC would determine $3$ undetermined coefficients.

If there are no non-KBCs, $3$ of $4$ undetermined coefficients in Eq. (4) are determined. Thus, the energy $E$ is also arbitrary. Again, we can find the BICs in these potentials under the effect of the minimal length. Many potentials satisfy the above-mentioned conditions, e.g., linear potential, quantum bouncer 16 , half oscillator 16 and some other half infinite potential wells.

Case III–The KBCs satisfy the conditions $\lim\limits_{x\rightarrow\pm\infty}\varphi(x)=0$, as shown in Fig. 1(c). They can only determine $2$ undetermined coefficients.

If there are fewer than $2$ non-KBCs, combining the KBCs, they can determine $3$ of $4$ coefficients in Eq. (4) at most. Thus, the energy $E$ is arbitrary. Again, we can find the BICs in these potentials under the effect of the minimal length. Many potentials, such as quantum harmonic oscillator, harmonic oscillator plus Dirac delta function 16 and Coulomb potential, satisfy the mentioned boundary conditions.

As a result, we obtain BICs in these three cases. Most of potentials that have bound states belong to these three cases; thus, a counterintuitive conclusion is obtained: BICs are universal phenomena under the effect of a minimal length.

The time-independent Schrödinger equation (Eq. (2)) in the presence of the minimal length is

Let $\eta^{4}=\beta^{\prime-1}$, $a=\eta^{2}/(4\hbar^{2})$, and $b(x)=(V(x)-E)m/\hbar^{4}$; the time-independent Schrödinger equation with the minimal length becomes

If the energy $E>0$, solving the above Eq. (5), we have the solutions

where $C_{1},\cdots,C_{4}$ are arbitrary coefficients, and

where $x_{0}$ is an arbitrary point of the particle’s position and

$\omega_{1}(x),\omega_{2}(x),\cdots,\omega_{4}(x)$ determine the solution space of the Schrödiner equation with the minimal length; thus, there are $4$ DFs, as shown in Fig. 2 (a) and (b). The standard Schrödiner equation, namely, when it is not perturbed by the effect of the minimal length, has $2$ DFs, as shown in Fig. 2 (c). From the results in Sec. 2.1, the extra DFs caused by the effect of the minimal length lead to the energy of the bound states being continuous. Thus, the effect of the minimal length leads to the existence of BICs in universal potentials. When the minimal length has a prominent influence on the system, BICs are easily found; otherwise, when the influence of the minimal length is negligible, BICs can hardly be observed. Next, we provide the condition for determining whether BICs are easily found.

In the Schrödinger equation, Eq. (2), the fourth-order item $(\beta^{\prime}\hbar^{4}/m)[\mathrm{d}^{4}\varphi(x)/\mathrm{d}x^{4}]$ is produced by the perturbation of the minimal length, which causes the extra DFs. By the GUP:

the fourth-order item is determined by the extra item $0.5*\hbar\beta[(\Delta P)^{2}+\langle P\rangle^{2}]$ in the GUP. Compared with the HUP $\Delta X\Delta P\geq 0.5*\hbar$, if the magnitude of the extra item perturbed by the minimal length is close to the magnitude of the item determined by the Planck length, then

The influence of the minimal length is prominent and provides extra DFs. Thus, the BICs are readily found, as shown in Fig. 2 (a).

If the magnitude of the item perturbed the minimal length is much less than the magnitude of the item determined by the Planck length, namely, satisfies that

The influence of the minimal length is negligible, and the probabilities of two states $\omega_{1},\omega_{2}$ that provide $2$ DFs are much smaller than the others probabilities of the superposition states. The system is close to the state with $2$ DFs (without the minimal length); thus, the BICs are very inconspicuous, that is, they are difficult to observe, as shown in Fig. 2 (b). The grey lines represent the states that are difficult to find at certain energy levels. However, since there are still $4$ DFs of the system, BICs still exist.

If the effect of the minimal length completely vanishes, that is, $\beta=0$, there is once again $2$ DFs of the system. Since there is no extra DF, the energy of the bound states is not free to take any value; thus, it is discrete, as shown in Fig. 2 (c).

As a result, BICs are universal under the effect of a minimal length; however, they may be difficult to observe due to the intensity of the perturbation caused by the minimal length.

Moreover, a mechanism of the BICs is provided. Figure 2 illustrates the mechanism of the BICs: the BICs are universal under the effect of a minimal length. The intensity of the effect of the minimal length determines whether the BICs can easily be found. When $\hbar\beta[(\Delta P)^{2}+\langle P\rangle^{2}]$ is close to Planck constant $\hbar$, the system has $4$ DFs at most; thus, the particle cannot be restricted to move only at certain special energy levels, causing a continuous energy. When $\beta[(\Delta P)^{2}+\langle P\rangle^{2}]\ll 1$, the system is close to the state with $2$ DFs (without the minimal length), causing some states (gray lines in Fig. 2(b)) at certain energies to be difficult to observe. Although the energy is still continuous, it is difficult to find since these energy levels’ states (gray lines in Fig. 2(b)) are difficult to observe. Thus, it is often considered that particles cannot move at these energies, i.e., they have discrete energies; it is also often considered that the BICs do not exist without a minimal length. When the minimal length completely vanishes ($\beta=0$), there are $2$ DFs; there are insufficient DFs to produce extra states such that the energy is discrete, namely, no BICs exist. This is consistent with the current results: the bound discrete states are typical; however, the BICs are always found in certain particular environments when a minimal length vanishes.

We consider a particle moving in the potential $V(x)$ satisfying $V(x)=0$ in the region $(-a,a)$ and $V(x)=+\infty$ in the other regions, where $a>0$. The time-independent Schrödinger equation with minimal length in the region $(-a,a)$ is

where the boundary conditions are $\varphi(\pm a)=0$. If the energy $E>0$, solving the above Eq. (12), we have

where $C_{1},\cdots,C_{4}$ are arbitrary coefficients, and

By the method in Case I, the conditions $\varphi(\pm a)=0$ cannot determine all the coefficients of Eq. (13), and BICs exist.

More precisely, to show the motion of particles with continuous energies, we provide the exact wave functions of each $E$ in the following.

1. (i)

   At the energy $E=k^{4}\pi^{4}\hbar^{4}\beta^{\prime}/(16ma^{4})+k^{2}\pi^{2}\hbar^{2}/(8ma^{2})$ (blue parts in Fig. 3(a)), where $k=1,2,\cdots$, the system is $2$-degree degenerate. When $\beta\rightarrow 0$, these energies are equal to the standard discrete energy levels of the infinite potential well without a minimal length. For each $E$, the two wave functions are

   	$\displaystyle\varphi_{1}(x)$	$\displaystyle=\frac{1}{\sqrt{a}}\sin[\frac{k\pi}{2a}(x+a)];$
   	$\displaystyle\varphi_{2}(x)$	$\displaystyle=D_{1}\exp(\lambda_{1}x)+D_{2}\exp(\lambda_{2}x)+D_{3}\cos[\frac{k\pi}{2a}(x+a)],$

   where we let $w_{1}(x)=\exp(\lambda_{1}x)$, $w_{2}(x)=\exp(\lambda_{2}x)$, and $w_{3}(x)=\cos[\frac{k\pi}{2a}(x+a)]$, then

   $\displaystyle\left\{\begin{array}[]{ll}D_{1}=\widetilde{D_{1}}s\\ D_{2}=\widetilde{D_{2}}s\\ D_{3}=\widetilde{D_{3}}s\end{array}\right.;$

   $\displaystyle\left\{\begin{array}[]{ll}\widetilde{D_{1}}=w_{3}(a)w_{2}(-a)-w_{3}(-a)w_{2}(a)\\ \widetilde{D_{2}}=w_{3}(-a)w_{1}(a)-w_{3}(a)w_{1}(-a)\\ \widetilde{D_{3}}=w_{2}(a)w_{1}(-a)-w_{2}(-a)w_{1}(a)\end{array}\right.,$

   (20)

   and the parameter $s$ is determined by the equation below:

   $$\begin{split}s&=\pm(\widetilde{D_{1}}^{2}F_{1,1}+\widetilde{D_{2}}^{2}F_{2,2}+\widetilde{D_{3}}^{2}F_{3,3}+\widetilde{D_{1}}\widetilde{D_{2}}F_{1,2}+\widetilde{D_{2}}\widetilde{D_{1}}F_{2,1}\\ &+\widetilde{D_{2}}\widetilde{D_{3}}F_{2,3}+\widetilde{D_{3}}\widetilde{D_{2}}F_{3,2}+\widetilde{D_{1}}\widetilde{D_{3}}F_{1,3}+\widetilde{D_{3}}\widetilde{D_{1}}F_{3,1})^{-0.5},\end{split}$$

   (21)

   where $F_{i,j}=\int_{-a}^{a}w_{i}(x)w_{j}^{*}(x)\mathrm{d}x$. Figures 3(c1) and (c2) illustrate the $2$-degree degenerate states of wave functions for $k=1,2,3$.

2. (ii)

   At the other energy $E$ (yellow parts in Fig. 3(a)), the states are also $2$-degree degenerate. These energies are extra energy under the effect of the minimal length when $\beta\rightarrow 0$, which is equal to the energy gaps between discrete energy levels of the infinite potential well without the minimal length. For each $E$, the two wave functions are

   	$\displaystyle\varphi_{1}(x)$	$\displaystyle=D_{1}\exp(\lambda_{1}x)+D_{2}\exp(\lambda_{2}x)+D_{3}\cos(\lambda_{3}x);$		(22)
   	$\displaystyle\varphi_{2}(x)$	$\displaystyle=D_{4}\exp(\lambda_{2}x)+D_{5}\cos(\lambda_{3}x)+D_{6}\sin(\lambda_{3}x),$		(23)

   where the two coefficient sets $D_{1},D_{2},D_{3}$ and $D_{4},D_{5},D_{6}$ are decided by the same method as the process Eqs. (20)-(21). Figures 3(b1) and (b2) illustrate the $2$-degree degenerate states of the wave functions for $E=10^{-18}$ J, $5*10^{-18}$ J and $10^{-17}$ J.

We can see that the wave functions in Figs. 3(b1), (b2) and (c2) are extra states caused by the perturbation of the minimal length, and the wave functions in Fig. 3(c1) are the same as the standard case ($\beta^{\prime}=0$). Compared to the particle that can only move with the energy in the blue parts without the minimal length, the effects of the minimal length make the particle move with the energy in the yellow parts to make the energy continuous. In particular, the quantity $k^{4}\pi^{4}\hbar^{4}\beta^{\prime}/(16ma^{4})$ of $E$ in (i) is the energy shift, which is discussed in many papers about the minimal length 11 ; 20 ; 21 ; 48 ; 49 .

For the environmental variables of the infinite potential well set in tpyical theoretical and experimental studies, namely, the width of the potential well $a=10^{-10}$ m and the mass of the particle $m=9.1095610^{-31}$ kg, we can calculate that when the magnitude of $\beta$ is close to $47$, which is still within the allowable upper bound of $\beta$ 22 ; 23 , $\hbar\beta[(\Delta P)^{2}+\langle P\rangle^{2}]$ is close to the Planck constant $\hbar$. By the method in Sec. 2.2, the BICs are obvious.

Although $\beta$ has a large upper bound 11 ; 22 ; 23 , $\beta$ is often considered a small number in some researches 20 ; 21 ; 48 ; 49 ; 51 ; 52 ; 50 , which means that $\beta[(\Delta P)^{2}+\langle P\rangle^{2}]\ll 1$. Based on the method in Sec. 2.2, the influence of the minimal length is negligible in these researches, and the system is close to the state with $2$ DFs (without the minimal length). Thus, under the environmental variables of the infinite potential well set as usual, the BICs exist but are very inconspicuous in the infinite potential well. This explains why BICs are not found in the infinite potential well with a single particle when the minimal length is not considered in the previous researches. On the other hand, this method in Secs. 2.1 and 2.2 suggests that we can adjust the environmental variables of the infinite potential well to obtain obvious BICs in the infinite potential well.

In this example, we assume a particle moving in a potential $V(x)$ satisfying $V(x)=Lx$ in the region $(0,+\infty)$ and $V(x)=+\infty$ in the other region, where $L>0$. Let $\eta^{4}=\beta^{\prime-1}$, $a=\eta^{2}/(4\hbar^{2})$, and $b(x)=(Lx-E)m/\hbar^{4}$. In the region $(0,+\infty)$, the time-independent Schrödinger equation with the minimal length becomes

where the boundary conditions are $\varphi(0)=0$ and $\lim\limits_{x\rightarrow+\infty}\varphi(x)=0$. If the energy $E>0$, solving the above Eq. (24), we have the approximate solutions

where $C_{1},\cdots,C_{4}$ are arbitrary coefficients, and

where

We study the asymptotic behavior of $\omega_{j}(x)$. When $x\rightarrow+\infty$, we have

where $j=1,2,3,4$, and

Thus, when $x\rightarrow+\infty$, $\omega_{1},\omega_{3}\rightarrow+\infty$ (omitting them due to physical significance) and $\omega_{2},\omega_{4}\rightarrow 0$. The wave functions Eq. (25) become

Substituting another boundary condition $\varphi(0)=0$ into Eq. (28), we obtain the wave functions for any $E>0$:

where $C_{2}$ can be determined after $\varphi(x)$ be normalized. These are non-degenerate since the condition $\varphi(0)=0$ reduces the DFs by $1$. By the results of case II in Sec. 2.1, the energy $E$ is still continuous because the DFs remain more than zero. Thus, for any $E>0$ with continuous energy, the bound wave function is Eq. (29). This indicates that the BICs exist in the potential under the effect of the minimal length.

For the environmental variables of the linear potential set in typical theoretical and experimental studies, namely, the linear parameter of the potential $l=mg$ and the mass of the particle $m=9.1095610^{-31}$ kg, we can calculate that when the magnitude of $\beta$ is close to $37$, which is still within the allowable upper bound of $\beta$ 22 ; 23 , $\hbar\beta[(\Delta P)^{2}+\langle P\rangle^{2}]$ is close to the Planck constant $\hbar$. By the method in Sec. 2.2, the BICs are obvious.

Although $\beta$ has a large upper bound 11 ; 22 ; 23 , $\beta$ is often considered a small number in some researches 20 ; 21 ; 48 ; 49 ; 51 ; 52 ; 50 , which makes $\beta[(\Delta P)^{2}+\langle P\rangle^{2}]\ll 1$. By the method in Sec. 2.2, the influence of the minimal length is negligible, and the system is close to the state with $2$ DFs (without the minimal length). Thus, under the environmental variables of the linear potential typically used, the BICs exist but are very inconspicuous in the linear potential. This explains why BICs are not found in the linear potential with a single particle when the minimal length is not considered in the previous researches. On the other hand, the method in Secs. 2.1 and 2.2 suggests that we can adjust the environmental variables of the linear potential to obtain obvious BICs in the linear potential.

In Ref. 20 ; 48 ; 49 , the authors studied this potential with the minimal length, ignoring the DFs of the energy; thus, they did not find that the energy was continuous.

In this example, we assume a particle moving in the potential $V(x)$ satisfying $V(x)=0.5m\omega^{2}x^{2}$, where $\omega$ is the vibrational frequency. Let $\eta^{4}=\beta^{\prime-1}$, $a=\eta^{2}/(4\hbar^{2})$, and $b(x)=(0.5m\omega^{2}x^{2}-E)m/\hbar^{4}$; the time-independent Schrödinger equation with the minimal length becomes

where the boundary conditions are $\lim\limits_{x\rightarrow\pm\infty}\varphi(x)=0$. If the energy $E>0$, solving the above Eq. (30), we have the approximate solutions

where $C_{1},\cdots,C_{4}$ are arbitrary coefficients, and

where

We study the asymptotic behavior of $\omega_{j}(x)$ when $x\rightarrow+\infty$. Similar to the above section, we have $\omega_{1},\omega_{3}\rightarrow+\infty$ (omitting them due to physical significance) and $\omega_{2},\omega_{4}\rightarrow 0$ when $x\rightarrow\pm\infty$. Thus, the wave functions for any $E>0$ with continuous energy are

where the coefficients $C_{2}$ and $C_{4}$ cannot be determined after $\varphi(x)$ is normalized. This causes the energy to be $2$-degree degenerate. The two degenerate wave functions are $\varphi_{1}(x)=\omega_{2}(x)$ and $\varphi_{2}(x)=\omega_{4}(x)$. For any $E>0$ with continuous energy, we have obtained the bound wave functions. This indicates that the BICs exist in the potential under the effect of the minimal length.

For the environmental variables of the quantum harmonic oscillator set as in typical theoretical and experimental studies, namely, the vibrational frequency $\omega=10^{30}$ Hz and the mass of the particle $m=9.1095610^{-31}$ kg, we can calculate that when the magnitude of $\beta$ is close to $33$, which is still within the allowable upper bound of $\beta$ 22 ; 23 , $\hbar\beta[(\Delta P)^{2}+\langle P\rangle^{2}]$ is close to Planck constant $\hbar$. By the method in Sec. 2.2, the BICs are obvious.

Although $\beta$ has a large upper bound 11 ; 22 ; 23 , $\beta$ is often considered a small number in some researches 20 ; 21 ; 48 ; 49 ; 51 ; 52 ; 50 , which makes $\beta[(\Delta P)^{2}+\langle P\rangle^{2}]\ll 1$. By the method in Sec. 2.2, the influence of the minimal length is negligible in these researches, and the system is close to the state with $2$ DFs (without the minimal length). Thus, under the environmental variables of the quantum harmonic oscillator typical used, the BICs exist but are very inconspicuous. This explains why BICs are not found in the quantum harmonic oscillator with a single particle when the minimal length is not considered in the previous researches. On the other hand, the method in Secs. 2.1 and 2.2 suggests that we can adjust the environmental variables of the quantum harmonic oscillator to obtain obvious BICs.

In Ref. 21 , the authors studied the same potential as the minimal length; they followed the method of quantum mechanics without the minimal length, and they ignored the DFs and degeneracy of the energy; thus, they did not find that the energy was continuous.