Aharonov-Bohm-Like Scattering in the Generalized Uncertainty Principle-corrected Quantum Mechanics

The most theories of quantum gravity predict the existence of a minimal length mead64 ; townsend76 ; amati89 ; garay94 at the Planck scale. It appears as various different expressions in loop quantum gravityrovelli98 ; carlip01 , string theorykonishi90 ; kato90 , path-integral quantum gravitypadmanabhan85 ; padmanabhan87 ; greensite91 , and black hole physicsmaggiore93 . From the aspect of quantum mechanics the existence of a minimal length results in the modification of the Heisenberg uncertainty principle (HUP)uncertainty ; robertson1929 $\Delta P\Delta Q\geq\frac{\hbar}{2}$, because $\Delta Q$ should be larger than the minimal length. Various modification of HUP, called the generalized uncertainty principle (GUP), were suggested in Ref. kempf93 ; kempf94 . The GUP has been used to explore the various branches of physics such as micro-black holescar99-1 , gravityadler99-1 , cosmological constantokamura02-1 , and classical central potential problemokamura02-2 . It is also used in the low-energy regimescar10-1 and the emergence of (doubly) special relativityscar12-1 . The experimental detection of GUP was emphasized in Ref. scar15-1 , where GUP is directly linked to the deformation of the spacetime metric. In this way the existence of GUP can be experimentally verified by measuring the light deflection and perihelion procession. As we will show in this paper, the existence of GUP also can be verified by measuring the cross section of the Aharonov-Bohm-Like scattering.

In this paper²²2Although the effect of the presence of a minimal length should be discussed in a relativistic fashion, we will examine it in the non-relativistic quantum mechanics when the Aharonov-Bohm potential is involved. Therefore, the results presented in this paper should be modified when the relativistic effect is included. we will choose the $d$-dimensional GUP in a form

$$\Delta P_{i}\Delta X_{i}\geq\frac{\hbar}{2}\left[1+\beta\left(\Delta{\bf P}^{2}+\langle{\bf P}\rangle^{2}\right)+2\beta\left(\Delta P_{i}^{2}+\langle P_{i}\rangle^{2}\right)\right]\hskip 28.45274pt(i=1,2,\cdots,d)$$

(1)

where $\beta$ is a GUP parameter, which has a dimension $(\mbox{momentum})^{-2}$. Using $\Delta A\Delta B\geq\frac{1}{2}|\langle[A,B]\rangle|$, Eq. (1) induces the modification of the commutation relations as³³3One may wonder whether the commutation relations (2) are inconsistent with each other. If, in fact, we impose $[P_{i},P_{j}]=0$, the Jacobi identity determines $[X_{i},X_{j}]$ in a form $$[X_{i},X_{j}]=\frac{4\beta^{2}{\bf P}^{2}}{1+\beta{\bf P}^{2}}(P_{i}X_{j}-P_{j}X_{i})={\cal O}(\beta^{2}).$$ Since we will explore the AB-like phenomenon up to first of $\beta$, Eq. (2) is valid for this reason.

	$\displaystyle\left[X_{i},P_{j}\right]=i\hbar\left(\delta_{ij}+\beta\delta_{ij}{\bf P}^{2}+2\beta P_{i}P_{j}\right)$		(2)
	$\displaystyle\hskip 28.45274pt\left[X_{i},X_{j}\right]=\left[P_{i},P_{j}\right]=0.$

The existence of the minimal length is easily shown at $d=1$. In this case Eq. (1) is expressed as

$$\Delta P\Delta X\geq\frac{\hbar}{2}\left(1+3\beta\Delta P^{2}\right)$$

(3)

if $\langle P\rangle=0$. Then, the equality of Eq. (3) yields

$$\Delta X^{2}\geq\Delta X_{min}^{2}=3\beta\hbar^{2}.$$

(4)

In Fig. 1 the allowed region and minimal length of Eq. (3) is plotted when $\hbar=\beta=1$.

If $\beta$ is small, Eq. (2) can be solved as

$$P_{i}=p_{i}\left(1+\beta{\bf p}^{2}\right)+{\cal O}(\beta^{2})\hskip 28.45274ptX_{i}=x_{i}$$

(5)

where $p_{i}$ and $x_{i}$ obey the usual HUP. Using Eq. (5) and Feynman’s path-integral techniquefeynman ; kleinert the Feynman propagator (or kernel) was exactly derived up to ${\cal O}(\beta)$ for $d=1$ free particle casedas2012 ; gangop2019 . Also the propagator for $d$-dimensional simple harmonic oscillator system was also derived recently in Ref. comment-1 ; park20-1 .

The main purpose of this paper is to examine how the Aharonov-Bohm (AB) effectAB-1 ; hagen-91 is modified when the GUP (1) is introduced. The AB effect is a pure quantum mechanical phenomenon, which predicts that the electromagnetic vector potential plays a role of observable at the quantum level when the charged particle moves around an infinitely thin magnetic flux tube. The experimental realization of this effect was discussed in Ref. peshkin . The effect of the particle spin in the AB-scattering was examined a few years ago in Ref. hagen-91 ; hagen-90-2 ; park-95 . In particular, when the spin is $1/2$, the corresponding Schrödinger-like equation derived from Dirac equation involves the $\delta$-function potentialhagen-91 ; jackiw as a Zeeman interaction. In order to make the theory finite a mathematically-oriented self-adjoint extensioncapri or the physically-oriented renormalizationhuang can be adopted. The equivalence of both methods was discussed in Ref. jackiw ; park97-1 .

The paper is organized as follows. In the next section we derive the classical equation of motion up to ${\cal O}(\beta)$ by making use of the Poisson bracket formalism when the minimal length (4) exists. Also, the classical Lagrangian is explicitly derived in this section. In section III we discuss the AB-scattering in the presence of GUP (5). Unlike the usual AB-effect with HUP it is shown that the irregularity at the origin cannot be avoided because of the effect of GUP. The scattering cross section is shown to be discontinuous at every integer $\alpha^{\prime}=\alpha/\hbar$, where $\alpha$ is a magnetic flux parameter. The various symmetries of the cross section in the usual AB-scattering are explicitly broken. In section IV a brief conclusion is given.

In this section we discuss how the classical electrodynamics is modified if the minimal length (4) exists. We start with a classical Hamiltonian

$$H_{cl}=\frac{1}{2M}\left(\bm{P}-q\bm{A}\right)^{2}+qV=H_{0,cl}+\frac{\beta}{M}\bm{p}^{2}\left(\bm{p}-q\bm{A}\right)\cdot\bm{p}+{\cal O}(\beta^{2})$$

(6)

where $\bm{A}$ and $V$ are the vector and scalar potentials, and $H_{0,cl}$ is the classical Hamiltonian when there is no minimal length, which is explicitly expressed by

$$H_{0,cl}=\frac{1}{2M}\left(\bm{p}-q\bm{A}\right)^{2}+qV.$$

(7)

In Eq. (6) we used Eq. (5). Of course, we have not considered the ordering problem of $\bm{p}$ and $\bm{x}$ because we deal with the classical Hamiltonian.

In order to derive a classical equation of motion we use the Poisson bracket

$$\dot{x}_{j}\equiv\{H_{cl},x_{j}\}=\frac{\partial H_{cl}}{\partial p_{j}}\frac{\partial x_{j}}{\partial x_{j}}-\frac{\partial H_{cl}}{\partial x_{j}}\frac{\partial x_{j}}{\partial p_{j}},$$

(8)

which yields

$$M\dot{\bm{x}}=\bm{p}-q\bm{A}+\beta\left[4\bm{p}^{2}\bm{p}-2q(\bm{A}\cdot\bm{p})\bm{p}-q\bm{p}^{2}\bm{A}\right]+{\cal O}(\beta^{2}).$$

(9)

Also, one can compute $\dot{p}_{j}=\left\{H_{cl},p_{j}\right\}$, which gives

$$\dot{p}_{j}=\left[\frac{q}{M}(\bm{p}-q\bm{A})\cdot\frac{\partial\bm{A}}{\partial x_{j}}-q\frac{\partial V}{\partial x_{j}}\right]+\frac{\beta q}{M}\bm{p}^{2}\left(\frac{\partial\bm{A}}{\partial x_{j}}\cdot\bm{p}\right)+{\cal O}(\beta^{2}).$$

(10)

Combining Eqs. (9) and (10) with long and tedious calculation, it is possible to derive

$$M\ddot{x}_{j}=q(\bm{E}+\bm{v}\times\bm{B})_{j}+\beta q\Gamma_{j}+{\cal O}(\beta^{2})$$

(11)

where $\bm{E}=-\bm{\nabla}V-\frac{\partial}{\partial t}\bm{A}$ and $\bm{B}=\bm{\nabla}\times\bm{A}$, which are the usual electric and magnetic fields. The correction term $\Gamma_{j}$ at the first order of $\beta$ is expressed as

	$\displaystyle\Gamma_{j}=\bm{H}_{11}^{2}(\bm{E}+\bm{v}\times\bm{B})_{j}+2H_{11,j}(\bm{H}_{11}\cdot\bm{E})-2mv_{j}(\bm{H}_{32}\cdot\bm{\nabla}V)-2qA_{j}(\bm{H}_{21}\cdot\bm{\nabla}V)$
	$\displaystyle\hskip 56.9055pt-\frac{\partial V}{\partial x_{j}}(\bm{H}_{11}\cdot\bm{H}_{31})+2qH_{32,j}\left(\bm{A}\cdot(\bm{v}\times\bm{B})\right)+2mH_{32,j}(\bm{G}\cdot\bm{v})$		(12)
	$\displaystyle\hskip 85.35826pt+2qH_{21,j}(\bm{G}\cdot\bm{A})-q(\bm{v}\cdot\bm{H}_{11})\frac{\partial}{\partial x_{j}}\bm{A}^{2}$

where

$$\bm{H}_{ab}=am\bm{v}+bq\bm{A}\hskip 28.45274pt\bm{G}=\sum_{i}v_{i}\frac{\partial\bm{A}}{\partial x_{i}}.$$

(13)

The equation of motion (11) can be derived as an Euler-Lagrange equation from the Lagrangian

$$L(\bm{x},\dot{\bm{x}})=\frac{1}{2}m\dot{\bm{x}}^{2}+q\dot{\bm{x}}\cdot\bm{A}-qV-\beta\bm{H}_{11}^{2}(\dot{\bm{x}}\cdot\bm{H}_{11})+{\cal O}(\beta^{2}),$$

(14)

where $\bm{v}$ in $\bm{H}_{ab}$ should be replaced by $\dot{\bm{x}}$ in Eq. (14). Unlike the classical equation of motion in the absence of the minimal length, the scalar and vector potentials explicitly appear in Eq. (11) at the first order of $\beta$. This means that if the minimal length exists, the potentials $V$ and $\bm{A}$ are not merely mathematical tools for the derivation of $\bm{E}$ and $\bm{B}$ even at the classical level. Of course, the classical equation of motion (11) indicates that the usual gauge symmetry ${\bf A}\rightarrow{\bf A}+\frac{1}{q}\nabla\Lambda$ and $V\rightarrow V-\frac{1}{q}\frac{\partial\Lambda}{\partial t}$ does not hold at the classical level. However, one can show that this theory has a modified symmetry up to ${\cal O}(\beta)$ in a form:

$${\bf A}\rightarrow{\bf A}+\frac{1}{q}\left(\nabla\Lambda+\beta{\bf F}\right)\hskip 28.45274ptV\rightarrow V-\frac{1}{q}\left(\frac{\partial\Lambda}{\partial t}+\beta F_{0}\right)$$

(15)

where ${\bf F}$ and $F_{0}$ satisfy

$${\bf F}-2({\bf H}_{11}\cdot\nabla\Lambda){\bf H}_{11}-{\bf H}_{11}^{2}\nabla\Lambda\equiv\nabla\Lambda_{1}\hskip 28.45274ptF_{0}=\frac{\partial\Lambda_{1}}{\partial t}.$$

(16)

Under the transformation the Lagrangian (14) transforms $L\rightarrow L+\frac{d}{dt}\left(\Lambda+\beta\Lambda_{1}\right)$. Of course, the symmetry (15) reduces to the usual gauge symmetry at $\beta=0$. However, this symmetry is completely different from the usual one because the vector ${\bf H}_{11}$ contains not only particle’s velocity but also the vector potential itself. The Lagrangian (14) can be used to explore the quantum electrodynamics in the presence of the minimal length by applying the path-integral techniquefeynman ; kleinert .

In this section we examine how the AB effectAB-1 is modified when the GUP (5) is introduced. The Hamiltonian with AB system can be written as

$$\hat{H}=\frac{1}{2M}\left(\bm{P}-e\bm{A}\right)^{2}=\frac{1}{2M}\left[(1+\beta\bm{p}^{2})\bm{p}-e\bm{A}\right]^{2}+{\cal O}(\beta^{2})=\hat{H}_{0}+\beta\hat{H}_{1}+{\cal O}(\beta^{2}),$$

(17)

where $e$ is an particle charge and

$$\hat{H}_{0}=\frac{1}{2M}(\bm{p}-2\bm{A})^{2}\hskip 14.22636pt\hat{H}_{1}=\frac{1}{M}\left[(\bm{p}^{2})^{2}-\frac{e}{2}\left\{(\bm{A}\cdot\bm{p})\bm{p}^{2}+\bm{p}^{2}(\bm{p}\cdot\bm{A})\right\}\right].$$

(18)

If we represent the energy eigenvalue in terms of the wave number as $E=E_{0}+\beta E_{1}+{\cal O}(\beta^{2})$ with $E_{0}=\hbar^{2}k^{2}/(2M)$ and $E_{1}=\hbar^{4}k^{4}/M$, it is straightforward to show that the Schrödinger equation $\hat{H}\psi=E\psi$ can be written as

	$\displaystyle(-i\hbar\bm{\nabla}-e\bm{A})^{2}\psi+2\beta\hbar^{4}\left[(\bm{\nabla}^{2})^{2}-\frac{ie}{2\hbar}\left(\bm{A}\cdot\bm{\nabla}\bm{\nabla}^{2}+\bm{\nabla}^{2}\bm{\nabla}\cdot\bm{A}\right)\right]\psi$		(19)
	$\displaystyle\hskip 227.62204pt+{\cal O}(\beta^{2})=(\hbar^{2}k^{2}+2\beta\hbar^{4}k^{4})\psi.$

We assume that there is a thin magnetic flux tube along the $z$-axis, which gives the vector potential in a form:

$$eA_{i}=\frac{\alpha\epsilon_{ij}x_{j}}{r^{2}},$$

(20)

where $\epsilon_{ij}$ is an antisymmetric tensor with $\epsilon_{01}=-\epsilon_{10}=1$. In the usual electromagnetic theory the choice of the vector potential (20) is not unique due to the gauge symmetry. Thus, the choice of Eq. (20) corresponds to the Coulomb gauge $\nabla\cdot{\bf A}=0$. However, the gauge symmetry is modified to Eq. (15) for our case, which contains the particle’s velocity. Thus, our results presented in the paper are valid only for the particular choice of ${\bf A}$ given in Eq. (20). Then, the corresponding magnetic field is $eB=-(\alpha/r)\delta(r)$. Using Eq. (20) explicitly, one can show

$$\bm{\nabla}^{2}\bm{\nabla}\cdot(e\bm{A})=(e\bm{A})\cdot\bm{\nabla}\bm{\nabla}^{2}+\frac{4\alpha}{r^{3}}\left(\frac{\partial}{\partial r}-\frac{1}{r}\right)\frac{\partial}{\partial\phi}.$$

(21)

Then, the Schrödinger equation (19) reduces to

	$\displaystyle\hskip 56.9055pt\left[\frac{\partial^{2}}{\partial r^{2}}+\frac{1}{r}\frac{\partial}{\partial r}+\frac{1}{r^{2}}\left(\frac{\partial}{\partial\phi}+i\alpha^{\prime}\right)^{2}+k^{2}\right]\psi$		(22)
	$\displaystyle-2\beta\hbar^{2}\left[(\bm{\nabla}^{2})^{2}+\frac{i\alpha^{\prime}}{r^{2}}\left(\frac{\partial^{2}}{\partial r^{2}}-\frac{1}{r}\frac{\partial}{\partial r}+\frac{1}{r^{2}}\frac{\partial^{2}}{\partial\phi^{2}}+\frac{2}{r^{2}}\right)\frac{\partial}{\partial\phi}-k^{4}\right]\psi+{\cal O}(\beta^{2})=0$

where $\alpha^{\prime}=\alpha/\hbar$. One can show that the Schrödinger equation (22) is invariant under the simultaneous operations $\alpha\rightarrow-\alpha$, $\phi\rightarrow-\phi$.

If one imposes

$$\psi(\bm{r})=\sum_{m=-\infty}^{\infty}e^{im\phi}f_{m}(r),$$

(23)

the radial equation of (22) can be written as

$$\left(\widehat{S}_{1}-2\beta\hbar^{2}\widehat{S}_{2}\right)f_{m}(r)+{\cal O}(\beta^{2})=0,$$

(24)

where

	$\displaystyle\widehat{S}_{1}=\frac{d^{2}}{dr^{2}}+\frac{1}{r}\frac{d}{dr}-\frac{(m+\alpha^{\prime})^{2}}{r^{2}}+k^{2}$		(25)
	$\displaystyle\widehat{S}_{2}=\left(\frac{d^{2}}{dr^{2}}+\frac{1}{r}\frac{d}{dr}-\frac{m^{2}-\alpha^{\prime}(m+\alpha^{\prime})}{r^{2}}-k^{2}\right)\widehat{S}_{1}$
	$\displaystyle\hskip 56.9055pt-\frac{2\alpha^{\prime}(3m+2\alpha^{\prime})}{r^{2}}\left(\frac{1}{r}\frac{d}{dr}-\frac{1}{r^{2}}+\frac{k^{2}}{2}\right)+\frac{\alpha^{\prime 2}(m+\alpha^{\prime})(2m+\alpha^{\prime})}{r^{4}}.$

The symmetry of the simultaneous operations $\alpha\rightarrow-\alpha$, $\phi\rightarrow-\phi$ is represented in the radial equation as the simultaneous changes $\alpha\rightarrow-\alpha$, $m\rightarrow-m$. If we set

$$f_{m}(r)=f_{0,m}(r)+\beta f_{1,m}(r)+{\cal O}(\beta^{2}),$$

(26)

within ${\cal O}(\beta)$ the radial equation is represented as the following two equations:

	$\displaystyle\widehat{S}_{1}f_{0,m}(r)=0$		(27)
	$\displaystyle\widehat{S}_{1}f_{1,m}(r)=2\hbar^{2}\left[-\frac{2\alpha^{\prime}(3m+2\alpha^{\prime})}{r^{2}}\left(\frac{1}{r}\frac{d}{dr}-\frac{1}{r^{2}}+\frac{k^{2}}{2}\right)+\frac{\alpha^{\prime 2}(m+\alpha^{\prime})(2m+\alpha^{\prime})}{r^{4}}\right]f_{0,m}(r).$

The general solutions of Eq. (27) are

	$\displaystyle f_{0,m}(r)=A_{m}J_{|m+\alpha^{\prime}|}(z)+B_{m}J_{-|m+\alpha^{\prime}|}(z)$		(28)
	$\displaystyle f_{1,m}(r)=C_{m}J_{|m+\alpha^{\prime}|}(z)+D_{m}J_{-|m+\alpha^{\prime}|}(z)+u_{m}(z)J_{|m+\alpha^{\prime}|}(z)+v_{m}(z)J_{-|m+\alpha^{\prime}|}(z)$

where $z=kr$ and $J_{\nu}(z)$ is usual Bessel function of the first kind. In Eq. (28) $u_{m}$ and $v_{m}$ are

	$\displaystyle u_{m}(z)=\frac{\pi\hbar^{2}k^{2}}{\sin(|m+\alpha^{\prime}|\pi)}$		(29)
	$\displaystyle\times\Bigg{[}A_{m}\bigg{\{}2\xi_{m}F_{2}(z;-|m+\alpha^{\prime}|,|m+\alpha^{\prime}|+1)+\xi_{m,-}F_{3}(z;-|m+\alpha^{\prime}|,|m+\alpha^{\prime}|)$
	$\displaystyle\hskip 227.62204pt-\xi_{m}F_{1}(z;-|m+\alpha^{\prime}|,|m+\alpha^{\prime}|)\bigg{\}}$
	$\displaystyle\hskip 22.76228pt+B_{m}\bigg{\{}2\xi_{m}F_{2}(z;-|m+\alpha^{\prime}|,-|m+\alpha^{\prime}|+1)+\xi_{m,+}F_{3}(z;-|m+\alpha^{\prime}|,-|m+\alpha^{\prime}|)$
	$\displaystyle\hskip 227.62204pt-\xi_{m}F_{1}(z;-|m+\alpha^{\prime}|,-|m+\alpha^{\prime}|)\bigg{\}}\Bigg{]}$
	$\displaystyle v_{m}(z)=-\frac{\pi\hbar^{2}k^{2}}{\sin(|m+\alpha^{\prime}|\pi)}$
	$\displaystyle\times\Bigg{[}A_{m}\bigg{\{}2\xi_{m}F_{2}(z;|m+\alpha^{\prime}|,|m+\alpha^{\prime}|+1)+\xi_{m,-}F_{3}(z;|m+\alpha^{\prime}|,|m+\alpha^{\prime}|)$
	$\displaystyle\hskip 227.62204pt-\xi_{m}F_{1}(z;|m+\alpha^{\prime}|,|m+\alpha^{\prime}|)\bigg{\}}$
	$\displaystyle\hskip 22.76228pt+B_{m}\bigg{\{}2\xi_{m}F_{2}(z;|m+\alpha^{\prime}|,-|m+\alpha^{\prime}|+1)+\xi_{m,+}F_{3}(z;|m+\alpha^{\prime}|,-|m+\alpha^{\prime}|)$
	$\displaystyle\hskip 227.62204pt-\xi_{m}F_{1}(z;|m+\alpha^{\prime}|,-|m+\alpha^{\prime}|)\bigg{\}}\Bigg{]}$

where

$$\xi_{m}=\alpha^{\prime}(3m+2\alpha^{\prime})\hskip 28.45274pt\xi_{m,\pm}=\alpha^{\prime 2}(m+\alpha^{\prime})(2m+\alpha^{\prime})+2\xi_{m}(1\pm|m+\alpha^{\prime}|)$$

(30)

and

$$F_{n}(z;\mu,\nu)\equiv\int\frac{J_{\mu}(z)J_{\nu}(z)}{z^{n}}dz.$$

(31)

In order to escape the infinity at $r=0$, we should choose $B_{m}=0$. Therefore, the wave function can be written in a form;

	$\displaystyle\psi({\bm{r}})=\sum_{m=-\infty}^{\infty}e^{im\phi}A_{m}J_{|m+\alpha^{\prime}|}(z)+\beta\sum_{m=-\infty}^{\infty}e^{im\phi}\bigg{(}C_{m}J_{|m+\alpha^{\prime}|}(z)$		(32)
	$\displaystyle\hskip 28.45274pt+D_{m}J_{-|m+\alpha^{\prime}|}(z)+u_{m}(z)J_{|m+\alpha^{\prime}|}(z)+v_{m}(z)J_{-|m+\alpha^{\prime}|}(z)\bigg{)}+{\cal O}(\beta^{2})$

where

	$\displaystyle u_{m}(z)=\frac{A_{m}\pi\hbar^{2}k^{2}}{\sin(|m+\alpha^{\prime}|\pi)}\bigg{\{}2\xi_{m}F_{2}(z;-|m+\alpha^{\prime}|,|m+\alpha^{\prime}|+1)$		(33)
	$\displaystyle\hskip 28.45274pt+\xi_{m,-}F_{3}(z;-|m+\alpha^{\prime}|,|m+\alpha^{\prime}|)-\xi_{m}F_{1}(z;-|m+\alpha^{\prime}|,|m+\alpha^{\prime}|)\bigg{\}}$
	$\displaystyle v_{m}(z)=-\frac{A_{m}\pi\hbar^{2}k^{2}}{\sin(|m+\alpha^{\prime}|\pi)}\bigg{\{}2\xi_{m}F_{2}(z;|m+\alpha^{\prime}|,|m+\alpha^{\prime}|+1)$
	$\displaystyle\hskip 28.45274pt+\xi_{m,-}F_{3}(z;|m+\alpha^{\prime}|,|m+\alpha^{\prime}|)-\xi_{m}F_{1}(z;|m+\alpha^{\prime}|,|m+\alpha^{\prime}|)\bigg{\}}.$

If we choose

$$A_{m}=(-i)^{|m+\alpha^{\prime}|}=e^{-i\pi|m+\alpha^{\prime}|/2},$$

(34)

one can showhagen-91

$$\lim_{r\rightarrow\infty}\sum_{m=-\infty}^{\infty}e^{im\phi}A_{m}J_{|m+\alpha^{\prime}|}(z)=e^{-iz\cos\phi}+\frac{e^{ikr}}{\sqrt{r}}f_{0}(\phi)$$

(35)

where⁴⁴4The incident wave derived by Ref. AB-1 is $e^{-iz\cos\phi-i\alpha\phi}$, which is different from that of Eq. (35). The authors in this reference derived it by solving the appropriate differential equation. It was arguedhagen-90-1 that this discrepancy is originated from the fact that the long-range nature of the vector potential does not allow the interchange of the summation over $m$ with the taking of the $r\rightarrow\infty$ limit in the partial-wave analysis. $f_{0}(\phi)$ is

$$f_{0}(\phi)=\frac{1}{\sqrt{2\pi ik}}\left[-2\pi\delta(\phi-\pi)(1-\cos\pi\alpha^{\prime})-ie^{-iN(\phi-\pi)}\frac{\sin\pi\alpha^{\prime}e^{-i\phi/2}}{\cos\frac{\phi}{2}}\right].$$

(36)

In Eq. (36) we used $\alpha^{\prime}=N+\gamma$, where $N$ is integer and $0\leq\gamma<1$.

Using

$$\frac{1}{z}J_{\nu}(z)=\frac{1}{2\nu}\left[J_{\nu-1}(z)+J_{\nu+1}(z)\right],$$

(37)

it is possible to show

	$\displaystyle F_{2}(z;\mu,\nu)=\frac{1}{2\nu}\left[F_{1}(z;\mu,\nu-1)+F_{1}(z;\mu,\nu+1)\right]$		(38)
	$\displaystyle F_{3}(z;\mu,\nu)=\frac{1}{4\mu\nu}\bigg{[}F_{1}(z;\mu-1,\nu-1)+F_{1}(z;\mu-1,\nu+1)$
	$\displaystyle\hskip 113.81102pt+F_{1}(z;\mu+1,\nu-1)+F_{1}(z;\mu+1,\nu+1)\bigg{]}.$

Then, $u_{m}(z)$ and $v_{m}(z)$ can be expressed as

	$\displaystyle u_{m}(z)=\frac{A_{m}\pi\hbar^{2}k^{2}}{\sin(|m+\alpha^{\prime}|\pi)}\Bigg{[}-\frac{|m+\alpha^{\prime}|\xi_{m}}{|m+\alpha^{\prime}|+1}F_{1}(z;-|m+\alpha^{\prime}|,|m+\alpha^{\prime}|)$		(39)
	$\displaystyle\hskip 170.71652pt+\frac{\xi_{m}}{|m+\alpha^{\prime}|+1}F_{1}(z;-|m+\alpha^{\prime}|,|m+\alpha^{\prime}|+2)$
	$\displaystyle-\frac{\xi_{m,-}}{4|m+\alpha^{\prime}|^{2}}\bigg{\{}F_{1}(z;-|m+\alpha^{\prime}|-1,|m+\alpha^{\prime}|-1)+F_{1}(z;-|m+\alpha^{\prime}|+1,|m+\alpha^{\prime}|+1)$
	$\displaystyle\hskip 28.45274pt+F_{1}(z;-|m+\alpha^{\prime}|-1,|m+\alpha^{\prime}|+1)+F_{1}(z;-|m+\alpha^{\prime}|+1,|m+\alpha^{\prime}|-1)\bigg{\}}\Bigg{]}$
	$\displaystyle v_{m}(z)=-\frac{A_{m}\pi\hbar^{2}k^{2}}{\sin(|m+\alpha^{\prime}|\pi)}\Bigg{[}-\frac{|m+\alpha^{\prime}|\xi_{m}}{|m+\alpha^{\prime}|+1}F_{1}(z;|m+\alpha^{\prime}|,|m+\alpha^{\prime}|)$
	$\displaystyle\hskip 170.71652pt+\frac{\xi_{m}}{|m+\alpha^{\prime}|+1}F_{1}(z;|m+\alpha^{\prime}|,|m+\alpha^{\prime}|+2)$
	$\displaystyle+\frac{\xi_{m,-}}{4|m+\alpha^{\prime}|^{2}}\bigg{\{}F_{1}(z;|m+\alpha^{\prime}|-1,|m+\alpha^{\prime}|-1)+F_{1}(z;|m+\alpha^{\prime}|+1,|m+\alpha^{\prime}|+1)$
	$\displaystyle\hskip 113.81102pt+2F_{1}(z;|m+\alpha^{\prime}|-1,|m+\alpha^{\prime}|+1)\bigg{\}}\Bigg{]}.$

Now, let us examine the behavior of $\psi({\bm{r}})$ around $r\sim 0$. We use the following indefinite integral formula:

$$F_{1}(z;\mu,\nu)=-\frac{z}{\mu^{2}-\nu^{2}}\left[J_{\mu+1}(z)J_{\nu}(z)-J_{\mu}(z)J_{\nu+1}(z)\right]+\frac{1}{\mu+\nu}J_{\mu}(z)J_{\nu}(z).$$

(40)

If we takes $\nu\rightarrow\pm\mu$ limit in Eq. (40), one can also derive

	$\displaystyle F_{1}(z;\mu,\mu)=\frac{2^{-1-2\mu}z^{2\mu}}{\mu\Gamma^{2}(1+\mu)}{{}_{2}F}_{3}\left(\mu,\mu+\frac{1}{2}:1+\mu,1+\mu,1+2\mu:-z^{2}\right)$		(41)
	$\displaystyle F_{1}(z;\mu,-\mu)=-\frac{z^{2}}{4\Gamma(2-\mu)\Gamma(2+\mu)}{{}_{3}F}_{4}\left(1,1,\frac{3}{2}:2-\mu,2+\mu,2,2:-z^{2}\right)+\frac{\ln z}{\Gamma(1-\mu)\Gamma(1+\mu)}$

where $\Gamma(z)$ and ${}_{p}F_{q}(a_{1},\cdots,a_{p}:b_{1},\cdots,b_{q}:z)$ are the usual gamma and generalized hypergeometric functions. Using the limiting form

$$\lim_{z\rightarrow 0}J_{\nu}(z)=\frac{1}{\Gamma(\nu+1)}\left(\frac{z}{2}\right)^{\nu},$$

(42)

one can show

	$\displaystyle\lim_{r\rightarrow 0}F_{1}(z;\mu,\nu)=\frac{1}{(\mu+\nu)\Gamma(\mu+1)\Gamma(\nu+1)}\left(\frac{z}{2}\right)^{\mu+\nu}$		(43)
	$\displaystyle\lim_{r\rightarrow 0}F_{1}(z;\mu,\mu)=\frac{1}{2\mu\Gamma^{2}(1+\mu)}\left(\frac{z}{2}\right)^{2\mu}$
	$\displaystyle\lim_{r\rightarrow 0}F_{1}(z;\mu,-\mu)=\frac{\ln z}{\Gamma(1-\mu)\Gamma(1+\mu)}.$

Then the dominant terms in $u_{m}$ and $v_{m}$ at $r\sim 0$ are

	$\displaystyle\lim_{r\rightarrow 0}u_{m}(z)=-\frac{A_{m}\hbar^{2}k^{2}}{8}\frac{\xi_{m,-}}{|m+\alpha^{\prime}|}\left(\frac{z}{2}\right)^{-2}$		(44)
	$\displaystyle\lim_{r\rightarrow 0}v_{m}(z)=\frac{A_{m}\hbar^{2}k^{2}}{8}\frac{\Gamma(-|m+\alpha^{\prime}|)\xi_{m,-}}{(|m+\alpha^{\prime}|-1)\Gamma(1+|m+\alpha^{\prime}|)}\left(\frac{z}{2}\right)^{2(|m+\alpha^{\prime}|-1)},$

which yield at ${\cal O}(\beta)$

	$\displaystyle\lim_{r\rightarrow 0}\psi({\bm{r}})=\beta\sum_{m=-\infty}^{\infty}e^{im\phi}\Bigg{[}\frac{D_{m}}{\Gamma(1-|m+\alpha^{\prime}|)}\left(\frac{z}{2}\right)^{-|m+\alpha^{\prime}|}$		(45)
	$\displaystyle\hskip 113.81102pt-\frac{A_{m}h^{2}k^{2}}{8}\frac{\xi_{m,-}}{(|m+\alpha^{\prime}|-1)\Gamma(1+|m+\alpha^{\prime}|)}\left(\frac{z}{2}\right)^{|m+\alpha^{\prime}|-2}\Bigg{]}.$

Since we cannot make $\lim_{r\rightarrow 0}\psi({\bm{r}})$ regular by choosing $D_{m}$ appropriately, unlike the AB-scattering in the usual quantum mechanics the AB-like scattering with GUP (5) should allow the irregular solution at the origin.

Now, let us examine the behavior of $\psi({\bm{r}})$ around $r\sim\infty$. Using the limiting behavior of the Bessel function

$$\lim_{r\rightarrow\infty}J_{\nu}(z)=\sqrt{\frac{2}{\pi z}}\cos\left(z-\frac{\nu\pi}{2}-\frac{\pi}{4}\right)=\frac{1}{\sqrt{2\pi z}}\left[\left(-i\right)^{\nu+1/2}e^{iz}+\left(i\right)^{\nu+1/2}e^{-iz}\right],$$

(46)

it is straightforward to show

$$\lim_{r\rightarrow\infty}F_{1}(z;\mu,\nu)=-\frac{1}{2\pi(\mu^{2}-\nu^{2})}\left[\left\{(-i)^{\mu-\nu+1}-(-i)^{\nu-\mu+1}\right\}+\left\{(-i)^{\nu-\mu-1}-(-i)^{\mu-\nu-1}\right\}\right].$$

(47)

When $\nu=\pm\mu$ in Eq. (47), one can derive the following asymptotic formula by making use of Eq. (41):

$$\lim_{r\rightarrow\infty}F_{1}(z;\mu,\mu)=\frac{1}{2\mu}\hskip 14.22636pt\lim_{r\rightarrow\infty}F_{1}(z;\mu,-\mu)=\frac{-\gamma-\psi\left(\frac{1}{2}\right)+\psi(1-\mu)+\psi(1+\mu)}{2\Gamma(1-\mu)\Gamma(1+\mu)},$$

(48)

where $\gamma$ and $\psi(z)$ are Euler number and digamma function. Using $\psi(z+1)=\psi(z)+1/z$ explicitly, one can show

$$\lim_{r\rightarrow\infty}u_{m}(z)=g_{1,m}\hskip 56.9055pt\lim_{r\rightarrow\infty}v_{m}(z)=g_{2,m},$$

(49)

where

	$\displaystyle g_{1,m}=-\frac{A_{m}\hbar^{2}k^{2}}{2(1+|m+\alpha^{\prime}|)}\Bigg{[}\left(\xi_{m}+\frac{\xi_{m,-}}{2|m+\alpha^{\prime}|(1-|m+\alpha^{\prime}|)}\right)\bigg{\{}-\gamma-\psi\left(\frac{1}{2}\right)$		(50)
	$\displaystyle\hskip 256.0748pt+\psi(|m+\alpha^{\prime}|)+\psi(1-|m+\alpha^{\prime}|)\bigg{\}}$
	$\displaystyle\hskip 71.13188pt+\frac{1+2|m+\alpha^{\prime}|}{|m+\alpha^{\prime}|(1+|m+\alpha^{\prime}|)}\xi_{m}-\frac{1-|m+\alpha|-3|m+\alpha^{\prime}|^{2}+|m+\alpha|^{3}}{2|m+\alpha^{\prime}|^{3}(1+|m+\alpha^{\prime}|)(1-|m+\alpha^{\prime}|)^{2}}\xi_{m,-}\Bigg{]}$
	$\displaystyle g_{2,m}=\frac{A_{m}\hbar^{2}k^{2}\Gamma(|m+\alpha^{\prime}|)\Gamma(1-|m+\alpha^{\prime}|)}{2(1+|m+\alpha^{\prime}|)}\left(\xi_{m}+\frac{\xi_{m,-}}{2|m+\alpha^{\prime}|(1-|m+\alpha^{\prime}|)}\right).$

Using Eq. (49) it is straightforward to compute $\lim_{r\rightarrow\infty}f_{1,m}(r)$ explicitly. Since $f_{1.m}(r)$ should be outgoing wave at $r=\infty$, we should impose the coefficient of $e^{-ikr}$ to be zero, which gives

$$C_{m}+g_{1,m}=-e^{-i\pi|m+\alpha^{\prime}|}(D_{m}+g_{2,m}).$$

(51)

Then, $f_{1,m}(r\rightarrow\infty)$ reduces to

$$\lim_{r\rightarrow\infty}f_{1,m}(r)=\frac{e^{ikr}}{\sqrt{2\pi ikr}}e^{i\pi|m+\alpha^{\prime}|/2}\left(1-e^{-2i\pi|m+\alpha^{\prime}|}\right)(D_{m}+g_{2,m}).$$

(52)

Thus, the asymptotic behavior of the wave function given in Eq. (32) can be written as a standard from

$$\lim_{r\rightarrow\infty}\psi({\bm{r}})=e^{-ikr\cos\phi}+\frac{e^{ikr}}{\sqrt{r}}f(\phi),$$

(53)

where the scattering amplitude $f(\phi)$ is

$$f(\phi)=f_{0}(\phi)+\beta f_{1}(\phi)+{\cal O}(\beta^{2}).$$

(54)

In Eq. (54) $f_{0}(\phi)$ is given in Eq. (36) and $f_{1}(\phi)$ is

$$f_{1}(\phi)=\frac{1}{\sqrt{2\pi ik}}\sum_{m=-\infty}^{\infty}e^{im\phi}e^{i|m+\alpha^{\prime}|\pi/2}\left(1-e^{-2i\pi|m+\alpha^{\prime}|}\right)(D_{m}+g_{2,m}).$$

(55)