Levinson theorem with the nonlocal Aharonov-Bohm effect

Levinson theorem with the nonlocal Aharonov-Bohm effect

De-Hone Lin*

Department of Applied Mathematics, National Chiao Tung University, Hsinchu 30043, Taiwan

共Received 28 April 2003; published 5 November 2003兲

Levinson theorem for a charged particle moving in an arbitrary short-range potential and the field of the Aharonov-Bohm magnetic flux is established. The theorem constructs the relation␦_␣(0)⫽n_␣␲ between the phase shift␦_␣(k) of scattering state at zero momentum and the total number n_␣of bound states for the␣th angular-momentum channel, where␣⫽兩m⫹␮0兩 is a real number (m⫽integer, and␮0⫽⫺⌽/⌽0with⌽ being

the magnetic flux and⌽0⫽hc/e the fundamental flux quantum兲. The relation means that the phase shift at the threshold of zero momentum can serve as a counter for the bound states in the general angular-momentum channel.

DOI: 10.1103/PhysRevA.68.052705 PACS number共s兲: 34.10.⫹x, 34.90.⫹q, 03.65.Vf

I. INTRODUCTION

In 1949, Levinson discovered one of the most beautiful theorems in quantum mechanics 关1兴. Well known as the Levinson theorem, it clarifies the relation between the phase shifts of a quantum particle scattered by a short-range poten-tial and the number of bound states therein. In three-dimensional space, the theorem can be described as

␦l共0兲⫽nl␲, l⫽1,2, . . . ,

where␦l(0) denotes the phase shift of scattered state with a linear momentum k at the threshold of zero momentum, i.e., k⫽ 0, in the angular-momentum channel l, and n_lis the total number of bound states in the angular-momentum channel l allowed by the short-range potential. When the angular mo-mentum l⫽0, the theorem must be modified to

␦0共0兲⫽共n0⫹1/2兲␲

due to the existence of a zero-energy resonance 共a half-bound state兲. Later some authors devoted Levinson’s verifi-cation to discuss the elegant theorem by way of the different manners, or generalized it to the more general cases 关2–8兴. Slightly different from the three-dimensional case, Levinson theorem in two dimensions can be expressed as关8兴

␦m共0兲⫽nm␲, m⫽0,1,2, . . . , 共1兲 where␦m(0) is the phase shift of scattering state at threshold of angular-momentum channel m, and n_mis the total number of the bound states in the same channel. The total number of bound states in the angular-momentum channel ⫺m is the same as that of m. This is due to the fact that the phase shift and the number of bound states just relate to the angular momentum m via its absolute value兩m兩 in cylindrically sym-metric system. Ten years after Levinson’s work, Aharonov and Bohm found that a charged particle can be influenced by the magnetic field even if the particle is nowhere in the re-gion of nonzero field strength关9兴. The phenomenon is some-what counterintuitive and represents a nonlocal and

topologi-cal effect in quantum mechanics. The term ‘‘nonlotopologi-cal’’ means that it exists even when the charged particle passes through a field-free region and is only associated with the entire closed curve. It is ‘‘topological’’ in the sense that the phase interference is unaffected when the particle path of closed curve is deformed within the field-free region. Forty years later, Aharonov-Bohm共AB兲 effect had great impact on our comprehension of the foundation of quantum theory 关10兴, and helped in the understanding of the quantum Hall effect关11,12兴, superconductivity 关12,13兴, and so forth.

In this paper we shall generalize the Levinson theorem for a charged particle moving in an arbitrary short-range poten-tial to include the field of the nonlocal AB effect. This paper is organized as follows. In the following section we establish the partial-wave method for scattering theory in two dimen-sions for a short-range potential and the nonlocal AB effect. The asymptotic behavior of phase shifts at threshold is dis-cussed in Sec. III. In Sec. IV, the Levinson theorem is gen-eralized to charged particles moving in the potential V(␳), which is less singular than ␳⫺2 when ␳⭐a and V(␳)⫽0 when ␳⭓a, and in the field of the nonlocal AB effect using Green’s-function method. The number of bound states n_␣for a given general angular momentum␣⫽兩m⫹␮0兩 is related to

the phase shifts␦_␣(0) of zero-momentum scattering states as follows:

␦␣共0兲⫽n␣␲, ␣⫽兩m⫹␮0兩, 共2兲

where m is an integer, and ␮0⫽⫺⌽/⌽0 with ⌽ being the

AB flux and ⌽0⫽hc/e the fundamental flux quantum. Our

discussions are summarized in Sec. V.

II. PARTIAL-WAVE METHOD FOR A SHORT-RANGE POTENTIAL AND AN AB MAGNETIC FLUX The fixed-energy Green’s function G0_(r,r

_⬘

_{;E) for a}

charged particle with mass␮ propagating from r

⬘

to r satis-fies the Schro¨dinger equation

冋

E⫺H0

冉

r,

ប i ⵜ

冊册

G

0_共r,r

_⬘

_;E兲⫽␦2_共r⫺r

_⬘

_{兲. 共3兲}

Here the Hamiltonian of the system is given by H0

⫽⫺ប2_ⵜ2_{/2␮⫹V(r) and r represents the two-dimensional}

position vector. In the cylindrically symmetric system, the angular decomposition of the Green’s function can be written as G0共r,r

⬘

;E兲⫽

兺

m⫽⫺⬁ ⬁ G_m0共␳,␳

⬘

;E兲e im(␸⫺␸⬘) 2␲ , 共4兲 with (␳,␸) being the polar coordinates in two-dimensional space and G_m0(␳,␳

⬘

;E) the radial Green’s function. As a re-sult, the left-hand side共lhs兲 of Eq. 共3兲 can be brought to the following form:

兺

m_⫽⫺⬁ ⬁

再

E⫹

冋

ប 2 2␮

冉

d2 d␳2⫹ 1 ␳ d d␳⫺ m2 ␳2

冊

册

⫺V共␳兲

冎

⫻Gm 0_共 ␳,␳

⬘

;E兲e im(␸⫺␸⬘) 2␲ . 共5兲

For a charged particle affected by a magnetic field, Green’s function G(r,r

⬘

;E) is different from G0(r,r

⬘

;E) by a glo-bally nonintegrable phase factor关14,15兴:

G共r,r

⬘

;E兲⫽G0共r,r

⬘

;E兲exp

再

ie បc

冕

r⬘

r

A共r˜兲•dr˜

冎

. 共6兲 Here we have used the vector potential A(r˜) to represent the magnetic field. For an infinitely thin tube of finite magnetic flux along the z-direction under consideration, the vector po-tential can be described by

A共r兲⫽2g⫺yeˆx⫹xeˆy

x2⫹y2 , 共7兲

where eˆx,eˆy stand for the unit vector along the x,y axes, respectively. Introducing the azimuthal angle around the AB tube,

␸共r兲⫽tan⫺1共y/x兲, 共8兲 the components of the vector potential can be expressed as

Ai⫽2g⳵i␸共r兲. 共9兲

The associated magnetic-field lines are confined to an infi-nitely thin tube along the z axis:

B3⫽2g⑀3i j⳵i⳵j␸共r兲⫽4␲g␦共r⬜兲, 共10兲 where r_⬜ stands for the transverse vector r_⬜⬅(x,y). Note that the derivatives in front of ␸(r) commute everywhere, except at the origin where Stokes’ theorem yields

冕

d2x共⳵x⳵y⫺⳵y⳵x兲␸共r兲⫽

冖

d␸⫽2␲. 共11兲 Since the magnetic flux through the tube is defined by the integral

⌽⫽

冕

d2_xB

3, 共12兲

the coupling constant g is related to the magnetic flux by g ⫽⌽/4␲. Inserting Ai⫽2g⳵i␸ into the nonintegrable phase factor in Eq. 共6兲, the magnetic interaction takes the form exp关⫺i␮0兰␶d␶

⬘

␸˙ (␶

⬘

)兴, where ␸˙⫽d␸/d␶ and ␮0

⫽⫺2eg/បc⫽⫺⌽/⌽0 is a dimensionless number. The

mi-nus sign is a matter of convention. According to the discus-sion in Refs. 关14–17兴, only phase factors with closed-loop contour are considered where the description of the electro-magnetic phenomenon is complete关18兴. Hence, we have

m⫽ 1 2␲

冕

␶

d␶

⬘

␸˙共␶

⬘

兲, 共13兲 with integer values m corresponding to the winding number. The magnetic interaction is therefore purely nonlocal, and topological. The nonintegrable phase factor now becomes exp兵⫺i␮0(2␲m⫹␸⫺␸

⬘

)其. It can be included with the help of

Poisson’s summation formula 共e.g., p. 469 of Ref. 关19兴兲

兺

k⫽⫺⬁ ⬁ f共k兲⫽

冕

⫺⬁ ⬁ dx

兺

n⫽⫺⬁ ⬁ e2␲nxif共x兲. 共14兲 So expression共5兲 can be written as

冕

dz

兺

m⫽⫺⬁ ⬁

再

E⫹

冋

ប 2 2␮

冉

d2 d␳2⫹ 1 ␳ d d␳⫺ z2 ␳2

冊

册

⫺V共␳兲

冎

⫻Gz共␳,␳

⬘

;E兲 ei(z⫺␮0)(␸⫹2m␲⫺␸⬘) 2␲ , 共15兲

where the superscript 0 in G_m0 has been suppressed to reflect the inclusion of the AB effect. The summation over all indi-ces m forindi-ces z⫽␮0 modulo an arbitrary integer number.

Thus, we have

兺

m⫽⫺⬁ ⬁

再

E⫹

冋

ប 2 2␮

冉

d2 d␳2⫹ 1 ␳ d d␳⫺ 兩m⫹␮0兩 2 ␳2

冊

册

⫺V共␳兲

冎

⫻G兩m⫹␮0兩共␳,␳

⬘

;E兲 eim␸ 2␲ . 共16兲

In what follows, we shall denote兩m⫹␮0兩⫽␣briefly. We see

that the influence of the AB effect to the radial Green’s func-tion is to replace the integer quantum number m with a real one ␣ which depends on the magnitude of magnetic flux. Applying the Fourier expansion of␦ function,

␦共␸⫺␸

⬘

兲⫽

兺

m_⫽⫺⬁ ⬁ 1 2␲e im(␸⫺␸⬘)_{, 共17兲}

to the right-hand side共rhs兲 of Eq. 共3兲, one can show that the radial Green’s function satisfies

再

E⫹

冋

ប 2 2␮

冉

d2 d␳2⫹ 1 ␳ d d␳⫺ ␣2 ␳2

冊

册

⫺V共␳兲

冎

G␣共␳,␳

⬘

;E兲 ⫽␦共␳⫺␳

⬘

兲. 共18兲

As a result, the corresponding radial wave equation reads

再

E⫹

冋

ប 2 2␮

冉

d2 d␳2⫹ 1 ␳ d d␳⫺ ␣2 ␳2

冊

册

⫺V共␳兲

冎

R␣k共␳兲⫽0, 共19兲 where the subscript set (␣,k) with k⬅

冑

2␮E/ប denotes the state of scattering particle.

For short-range potential, i.e., V(␳) vanishes for ␳⬎a, the domain of the variable␳ is divided into an internal region (␳⬍a) and an external region (␳⬎a). The normalized ex-terior solution is the linear combination of Bessel functions J_␣(k␳) and N_␣(k␳) of the first and second kind, and may be given by

R_␣k共␳兲⫽

冑

k关cos␦_␣共k兲J_␣共k␳兲⫺sin␦_␣共k兲N_␣共k␳兲兴, 共20兲 where ␦_␣(k) is the phase shift of the scattered radial wave function which is used to measure the interaction in poten-tial. The general solution of a scattering particle ⌿k(r) is given by superposition of the partial wave ⌿_␣k(r) ⫽R␣k(␳)eim␸, and reads ⌿k共r兲⫽

兺

m⫽⫺⬁ ⬁

冑

k关cos␦_␣共k兲J_␣共k␳兲 ⫺sin␦␣共k兲N␣共k␳兲兴eim␸. 共21兲

Because it must describe both incident wave and scattered wave at large distance, we naturally expect it to become

⌿k共r兲 ——→

兩r兩→⬁

Fasymp

冉

exp兵ik•r其exp

再

ie បc

冕

r⬘

r

A共r˜兲•dr˜

冎

冊

⫹ f共␸兲

冑

_␳iexp兵ik␳其, 共22兲 where exp(ik•r) describes the incident plane wave of a charged particle with momentum p⫽␮k and Fasymp(•) stands for its asymptotic form. The phase modulation of the nonintegrable phase factor comes from the fact that the field

A(r˜) of AB magnetic flux affects the charged particle

glo-bally. To find the amplitude f (␸) we note that the plane wave in Eq. 共22兲 can be in terms of the expansion of the partial waves in polar coordinates:

eik•r⫽

兺

m_⫽⫺⬁

⬁

imJm共k␳兲eim␸. 共23兲 Using the same procedure as in Eqs.共14兲–共16兲, the nonlocal flux effect can be combined into the partial-wave expansion, and yields exp共ik•r兲exp

冉

ie បc

冕

r⬘ r A共r˜兲•dr˜

冊

⫽

兺

m_⫽⫺⬁ ⬁ i␣J_␣共k␳兲eim␸. 共24兲 Inserting the result into Eq. 共22兲, making use of the asymptotic approximations of Bessel functions, and then comparing both asymptotic forms of Eqs. 共21兲 and 共22兲, we find the scattering amplitude in terms of phase shifts:

f共␸兲⫽ 1

冑

2␲k m

兺

⫽⫺⬁

⬁

ei(␦␣⫺␲/4)_{2i sin}␦_␣eim␸_{. 共25兲} It is worth noting that if the flux is quantized, i.e., ␮0 is an

integer, the result reduces to the free of flux case. In most cases, one concern is the total cross section, which is defined by

␴t⫽

冕

⫺␲ ␲

兩 f共␸兲兩2_{d␸. 共26兲}

Accordingly, the partial-wave representation of the total cross section for a charged particle scattered by a short-range potential and the nonlocal AB effect is given by

␴t⫽ 4 k m

兺

⫽⫺⬁ ⬁ sin2_␦ ␣. 共27兲

We see that the cross section is completely determined by the scattered phase shifts which are concluded by the potential of different types and the magnetic flux. On the other hand, the potential also determines the number of bound states. The relation between the phase shifts and the number of bound states was first clarified by Levinson 关1兴. Here, due to the nonlocal AB magnetic flux existence, the phase shifts are affected globally, and so are the number of bound states关20兴. In next two sections it will be showed that the relation be-tween the phase shift at threshold of the scattered wave func-tion and the number of bound states for the corresponding angular-momentum channel is connected by a general Levinson theorem.

III. PHASE SHIFTS NEAR␳\0 AT THRESHOLD Since the behavior of phase shifts near␳→0 at threshold is useful in the procedure of proof, the asymptotic behavior is discussed in that follows. According to Eq. 共19兲, when a potential V(␳) is less singular than␳⫺2, the solution has the power dependence on␳ near␳⫽0,

R_␣共␳,k兲 ⬃

␳→0

␳␣_{, 共28兲}

where we have used R_␣(␳,k) to denote the solution of Eq. 共19兲 which satisfies the boundary condition Eq. 共28兲. On the other hand, the external solution is given by Eq. 共20兲. The boundary conditions at␳⫽a require that the logarithmic de-rivative be continuous,

1 R_␣ dR_␣ d␳

冏

_␳⫽a⫺⫽ 1 R_␣k dR_␣k d␳

冏

_␳⫽a⫹, 共29兲 and thus yield the formula for the phase shift,

tan␦_␣⫽kaJ␣

⬘

共ka兲⫺␥␣J␣共ka兲

kaN_␣

⬘

共ka兲⫺␥_␣N_␣共ka兲. 共30兲 Here we define J_␣

⬘

(␳)⫽dJ_␣(␳)/d␳ and ␥_␣ ⫽adR␣/R␣d␳兩␳⫽a⫺. Note that Eq.共28兲 is independent of k, and Eq.共18兲 depends on k only through k2. Therefore either R_␣ or dR_␣/d␳ must be an integral function of k, and hence are the even function of k. Accordingly, ␥_␣ can only have one of the following forms 关8兴:

␥␣→b␣⫹共ka兲2l␣ ⫹ , ␥␣→b␣⫺共ka兲⫺2l␣ ⫺ , ␥␣→c␣⫹b␣共ka兲2l␣,

where b_␣⫾, and c_␣, and b_␣are nonzero constants, and l_␣⫾, l_␣ are the natural numbers. Using the asymptotic forms of Bessel functions at ␳⬃0, it is easy to find that the leading term of Eq. 共30兲 at threshold in any case is given by

tan␦_␣→d_␣共ka兲2␣˜, 共31兲 where d_␣⫽0 and ␣˜ is a nonzero positive real number. The result will be useful in the following proof of the Levinson method.

IV. THE LEVINSON THEOREM WITH THE NONLOCAL AB EFFECT

Using the spectrum representation of the radial Green’s function of Eq.共18兲, G_␣共␳,␳

⬘

;E兲⫽

兺

␬ u_␣␬共␳兲u_␣␬*_共␳

⬘

_兲

冑

␳␳

⬘

共E⫺E␣␬⫹i⑀兲 , 共32兲

the Green’s function G(r,r

⬘

;E) is G共r,r

⬘

;E兲⫽

兺

m_⫽⫺⬁ ⬁ G_␣共␳,␳

⬘

;E兲e im(␸⫺␸⬘) 2␲ ⫽

兺

m⫽⫺⬁ ⬁

兺

_␬ u␣␬共␳兲u␣␬*共␳

⬘

兲

冑

␳␳

⬘

共E⫺E␣␬⫹i⑀兲 eim(␸⫺␸⬘) 2␲ , 共33兲 where⑀⫽0⫹has been defined for the retarded Green’s func-tion and we use␬ to denote the discretized energy levels of a system moving in an attractive field. It is easy to see that by making use of R_␣␬⫽u_␣␬/

冑

␳ in Eq.共19兲 the wave func-tion u_␣␬(␳) solves d2u_␣␬ d␳2 ⫹

冋

2␮ ប2关E␣␬⫺V共␳兲兴⫺ ␣2_⫺1/4 ␳2

册

u␣␬⫽0. 共34兲

The wave function u_␣␬(␳) satisfies normalization condition

具

u_␣␬,u_␣␬⬘

典

⫽

冕

0

⬁

d␳u_␣␬* _共␳_{兲u␣␬⬘共}␳_兲⫽␦_␬␬⬘. 共35兲 So we find the following trace:

冕

d␳␳G_␣共␳,␳;E兲⫽

兺

␬

1

共E⫺E␣␬⫹i⑀兲. 共36兲

With the help of the formula 1 x⫹i⑀⫽P

1

x⫺i␲␦共x兲, 共37兲

the imaginary part of integral in Eq. 共36兲 reads

Im

冕

d␳␳G_␣共␳,␳;E兲⫽⫺␲

兺

␬ ␦共E⫺E␣␬兲. 共38兲

Thus the total number of bound states can be read off by integrating the formula over E from ⫺⬁ to 0⫺, and yields

Im

冕

⫺⬁

0⫺

dE

冕

d␳␳G_␣共␳,␳;E兲⫽⫺n_␣⫺␲, 共39兲 where n_␣⫺is the number of bound states with negative ener-gies corresponding to the channel of a general angular mo-mentum ␣ប. We have performed the integral over E up to 0⫺ instead of 0 for avoiding ambiguity. The possible exis-tence of a bound state with zero energy will be considered in Sec. V. We point out that the degeneracy between m and ⫺m in Eq. 共1兲 is broken in general due to the existence of magnetic flux of the nonlocal AB effect. A similar discussion as above can be applied to the free charged particle moving in the field of the AB effect and gives

Im

冕

⫺⬁

0⫺

dE

冕

d␳␳G_␣0共␳,␳;E兲⫽0. 共40兲 Combining both Eqs. 共39兲 and 共40兲, we find

Im

冕

⫺⬁

0⫺

dE

冕

d␳␳关G_␣共␳,␳;E兲⫺G_␣0共␳,␳;E兲兴⫽⫺n_␣⫺␲. 共41兲 It is useful to discuss this result by the Dyson equation

G共r,r

⬘

;E兲⫽G_AB0 共r,r

⬘

;E兲

⫹

冕

dr

⬙

G_AB0 共r,r

⬙

;E兲V共r

⬙

兲G共r

⬙

,r

⬘

;E兲, 共42兲 where we have used G_AB0 (r,r

⬘

;E) to represent the Green’s function G(r,r

⬘

;E) in the case of V(␳)⫽0. With the help of

Eq. 共33兲, the integration of angular part can be carried out and turns the equation into the single-dimensional one with a general quantum number␣ 关16兴

G_␣共␳,␳

⬘

;E兲⫽G_␣0共␳,␳

⬘

;E兲

⫹

冕

d␳

⬙

G_␣0共␳,␳

⬙

;E兲V共␳

⬙

兲G_␣共␳

⬙

,␳

⬘

;E兲. 共43兲 Here G_␣0(␳,␳

⬘

;E) is the radial Green’s function with V(␳) ⫽0. Its spectrum representation can be in terms of discrete sum G_␣0共␳,␳

⬘

;E兲⫽

兺

␬ u_␣␬0 共␳兲u_␣␬0*_共␳

⬘

_兲

冑

␳␳

⬘

共E⫺E␣␬⫹i⑀兲 . 共44兲

This can be achieved by requiring the wave functions to vanish at a sufficiently large radius R when aⰆR for a short-range potential. Taking the same trace with respect to Eq. 共43兲 as Eq. 共36兲, we obtain

冕

d␳␳关G_␣共␳,␳;E兲⫺G_␣0共␳,␳;E兲兴

⫽

冕

d␳␳

再

冕

d␳

⬙

G_␣0共␳,␳

⬙

;E兲V共␳

⬙

兲G_␣共␳

⬙

,␳;E兲

冎

. 共45兲 With the help of Eqs.共32兲 and 共44兲, the equation becomes

冕

d␳␳关G_␣共␳,␳;E兲⫺G_␣0共␳,␳;E兲兴 ⫽

兺

␬␬⬘

具

u_␣␬⬘,u_␣␬0

典具

u_␣␬0 兩V兩u_␣␬⬘

典

共E⫺E_␣␬0 _{⫹i⑀兲共E⫺E}

␣␬_⬘⫹i⑀兲

. 共46兲

The matrix element

具

u_␣␬0 兩V兩u_␣␬⬘

典

can explicitly carry out

具

u_␣␬0 兩V兩u_␣␬⬘

典

⫽

冕

0 ⬁ d␳u_␣␬0*_{共␳兲V共␳兲u␣␬⬘共␳兲} ⫽

冕

0 ⬁ d␳u_␣␬0*_共␳_兲共H˜_⫺H˜ 0兲u␣␬⬘共␳兲 ⫽共E␣␬_⬘⫺E␣␬0 兲

具

u␣␬0 ,u␣␬_⬘

典

, 共47兲

where H˜ is the Hamiltonian in Eq.共34兲,

H ˜ u_␣␬⬅

冋

⫺ ប 2 2␮ d2 d␳2⫹

冉

V共␳兲⫹ 共␣2_⫺1/4兲ប2 2␮␳2

冊

册

u␣␬ ⫽E␣␬u␣␬, 共48兲

and H˜₀ is the Hamiltonian with V(␳)⫽0. Substituting Eq. 共47兲 into Eq. 共46兲 and taking the imaginary part, we obtain

Im

冕

d␳␳关G_␣共␳,␳;E兲⫺G_␣0共␳,␳;E兲兴 ⫽␲

兺

␬␬⬘关␦共E⫺E␣␬ 0 _{兲⫺␦共E⫺E} ␣␬_⬘兲兴兩

具

u␣␬0 ,u␣␬_⬘

典

兩2. 共49兲 Integrating this equation over E from⫺⬁ to ⬁ gives

Im

冕

⫺⬁ ⬁

dE

冕

d␳␳关G_␣共␳,␳;E兲⫺G_␣0共␳,␳;E兲兴⫽0. 共50兲 The equation indicates that the total number of states in a specific angular-momentum channel is not changed by an attractive field, except that some scattering states are pulled down into the bound-state region. Comparing Eqs. 共41兲 and 共50兲, we obtain the result

Im

冕

0⫺

⬁

dE

冕

d␳␳关G_␣共␳,␳;E兲⫺G_␣0共␳,␳;E兲兴⫽n_␣⫺␲. 共51兲 Arriving here we complete the proof of rhs of the Levinson theorem with the nonlocal AB effect in Eq. 共2兲 by discretiz-ing the energy spectrum of continuous part. In the followdiscretiz-ing we shall prove the lhs of the Levinson theorem by directly treating the continuous part of energy spectrum which will gives the phase-shift expression of the total number of bound states at threshold. Including the continuous spectrum, Eq. 共32兲 takes the expression

G_␣共␳,␳

⬘

;E兲⫽

兺

␬ u_␣␬共␳兲u_␣␬* _共␳

⬘

_兲

冑

␳␳

⬘

共E⫺E␣␬⫹i⑀兲 ⫹

冕

dk u␣k共␳兲u␣k *_共␳

⬘

_兲

冑

␳␳

⬘

共E⫺E␣k⫹i⑀兲 , 共52兲

where we have used␬ and k to denote the discrete and con-tinuous spectrum, respectively. Using Eqs.共35兲 and 共37兲, we have

Im

冕

d␳␳G_␣共␳,␳;E兲 ⫽⫺␲

兺

␬ ␦共E⫺E␣␬兲

⫺␲

冕

dk␦共E⫺E_␣k兲

具

u_␣k,u_␣k

典

. 共53兲 Note that E_␣k may be zero energy, and the wave functions corresponding to the continuous spectrum has the normaliza-tion condinormaliza-tion

具

u_␣k,u_␣k⬘

典

⫽

冕

0

⬁

d␳u_␣k*_{共␳兲u␣k⬘共␳兲⫽}␦_共k⫺k

⬘

_{兲. 共54兲} Integrating Eq.共53兲 over E from 0⫺to⬁, one finds that

Im

冕

0⫺

⬁

dE

冕

d␳␳G_␣共␳,␳;E兲⫽⫺␲

冕

dk

具

u_␣k,u_␣k

典

, 共55兲 which is divergent due to

具

u_␣k,u_␣k

典

⫽␦(0)⫽⬁. The same treatment for G_␣0(␳,␳

⬘

;E) gives

Im

冕

0⫺

⬁

dE

冕

d␳␳G_␣0共␳,␳;E兲⫽⫺␲

冕

dk

具

u_␣k0 ,u_␣k0

典

, 共56兲 which is also infinity due to

具

u_␣k0 ,u_␣k0

典

⫽␦(0). But both in-finities are of different order which leads to the Levinson theory. To see this, let us first evaluate the difference

具

u_␣k,u_␣k⬘

典

_␳ 0⫺

具

u␣k 0 _,u ␣k_⬘ 0

典

␳0 ⫽

冕

0 ␳0 d␳u_␣k*_{共␳兲u␣k⬘共␳兲⫺}

冕

0 ␳0 d␳u_␣k0*_{共␳兲u␣k} ⬘ 0 共␳兲, 共57兲 and then take the limit k

⬘

→k and␳0→⬁. Here␳0is a large

but finite radius. Employing Eq.共34兲 and the boundary con-ditions

u_␣k共0兲⫽0, u_␣k0 共0兲⫽0, 共58兲 it is easy to find the expression

共k2_⫺k

_⬘

2_兲

_具

_u ␣k,u␣k_⬘

典

␳0 ⫽u_␣k*_共␳0兲 du_␣k⬘共␳兲 d␳

冏

_␳ 0 ⫺u␣k_⬘共␳0兲 du_␣k*_共␳_兲 d␳

冏

_␳ 0 . 共59兲 Since ␳0 is a large radius, the asymptotic form of Eq. 共20兲

can be used to evaluate the equality. With the help of asymptotic behavior of the Bessel functions it can be found

u_␣k⫽

冑

␳R_␣k ⬃ ␳→⬁

冑

2 ␲cos

冋

k␳⫺ ␣␲ 2 ⫺ ␲ 4⫹␦␣共k兲

册

, 共60兲 which in the limit k

⬘

→k leads to

具

u_␣k,u_␣k⬘

典

_␳ 0⫽ ␳0 ␲⫹ 1 ␲ d␦_␣共k兲 dk ⫺ 1 2␲k ⫻兵cos␣␲cos关2k␳0⫹2␦␣共k兲兴 ⫹sin␣␲sin关2k␳₀⫹2␦_␣共k兲兴其. 共61兲 The same procedure for u_␣k0 gives

u_␣k0 ⫽

冑

␳R_␣k ⬃ ␳→⬁

冑

2 ␲cos

冋

k␳⫺ ␣␲ 2 ⫺ ␲ 4

册

共62兲 and

具

u_␣k0 ,u_␣k ⬘ 0

典

␳0⫽ ␳0 ␲⫺ 1 2␲k兵cos␣␲cos 2k␳0 ⫹sin␣␲sin 2k␳0其. 共63兲 So we obtain

具

u_␣k,u_␣k⬘

典

_␳ 0⫺

具

u␣k 0 ,u_␣k0 _⬘

典

_␳ 0 ⫽_␲1 d␦␣共k兲 dk ⫹ 1 2␦共k兲cos␣␲sin 2␦␣共k兲 ⫹␦共k兲sin␣␲sin2␦_␣共k兲 ⫹cos_␲␣␲ k cos共2k␳0兲sin 2_␦ ␣共k兲 ⫺sin␣␲ 2␲k cos共2k␳0兲sin 2␦␣共k兲, 共64兲 where we have used the well-known formula

lim

␳0→⬁

sin2k␳0

␲k ⫽␦共k兲. 共65兲

Since Eq.共31兲 is valid, two terms containing␦(k) in Eq.共64兲 vanish. So from Eqs.共55兲 and 共56兲 we find

Im

冕

0⫺ ⬁ dE

冕

d␳␳关G_␣共␳,␳;E兲⫺G_␣0共␳,␳;E兲兴 ⫽␦␣共0兲⫺␦␣共⬁兲⫺cos␣␲ lim ␳0→⬁ ⫻

冕

0 ⬁ dkcos共2k␳0兲 k sin 2_␦ ␣共k兲⫹ sin␣␲ 2 _␳lim 0→⬁ ⫻

冕

0 ⬁ dkcos共2k␳0兲 k sin 2␦␣共k兲. 共66兲 The integrals can be divided into two regions. The first from 0 to 0⫹ vanishes on account of Eq. 共31兲, while the second from 0⫹ to⬁ also vanishes in the limit␳0→⬁ because the factor cos(2k␳0) oscillates very rapidly. Thus we have

Im

冕

0⫺

⬁

dE

冕

d␳␳关G_␣共␳,␳;E兲⫺G_␣0共␳,␳;E兲兴

⫽␦␣共0兲⫺␦␣共⬁兲. 共67兲

Combining Eqs.共51兲 and 共67兲, we obtain the Levinson theo-rem with the nonlocal AB effect:

␦␣共0兲⫺␦␣共⬁兲⫽n␣⫺␲, ␣⫽兩m⫹␮0兩,

V. DISCUSSION

A. On the existence of a zero-energy bound state As an explanation, let us consider a potential well with radius a and depth V(␳)⫽⫺V0 for ␳⬍a; V(␳)⫽0 for

␳⬎a. Using Eq. 共19兲, it is not difficult to find that the energy spectrum is determined by

␤J␣⫺1_J 共␤a兲

␣共␤a兲 ⫽i␭

H_␣⫺1(1) 共i␭a兲

H_␣(1)共i␭a兲 , 共69兲 where␤⫽

冑

2␮(V₀⫺兩E兩)/ប, ␭⫽

冑

2␮兩E兩/ប, and H_␣(1) is the Hankel function of the first kind. So a zero-energy bound state in this case is determined by J_␣⫺1(k0a)⫽0 with k0

⫽

冑

2␮V0/ប for ␣⬎1 共see below兲. The existence of a

zero-energy bound state would not change the result in Eqs.共39兲– 共41兲, and thus Eq. 共51兲. But Eq. 共55兲 will receive an addi-tional␲ to become Im

冕

0⫺ ⬁ dE

冕

d␳␳G_␣共␳,␳;E兲⫽⫺␲⫺␲

冕

dk

具

u_␣k,u_␣k

典

. 共70兲 Hence Eq.共67兲 gets an additional␲ and turns into

Im

冕

0⫺

⬁

dE

冕

d␳␳关G_␣共␳,␳;E兲⫺G_␣0共␳,␳;E兲兴

⫽␦␣共0兲⫺␦␣共⬁兲⫺␲. 共71兲

Therefore when a system contains a zero-energy bound state, the Levinson theorem reads

␦␣共0兲⫺␦␣共⬁兲⫽共n␣⫺⫹1兲␲⫽n␣␲, 共72兲

with n_␣⬅(n_␣⫺⫹1). Here only when␣⬎1 the bound state is a real zero-energy bound state. To see this recall Eq. 共34兲. When E_␣0⫽0, the exterior solution (␳⬎0) satisfies

d2u_␣0 d␳2 ⫺ ␣2_⫺1/4 ␳2 u␣0⫽0. 共73兲 Explicitly, u_␣0 is given by u_␣0⬃␳⫺␣⫹1/2, 共74兲 which leads to the fact that the wave function ⌿_␣0 ⬃u␣0/

冑

␳⫽1/␳␣ cannot be normalized when␣⭐1. On the

flip side, as ␣⬎1, ␦_␣(0) obtains an additional ␲ if a zero-energy solution actually exists. For this case, Levinson theo-rem becomes Eq.共72兲.

B. The phase shifts at high energies

Let us investigate the behavior of the phase shifts when␣ is fixed but k→⬁. For this purpose, we consider the scatter-ing by two potentials V(␳) and V˜ (␳). The corresponding radial equations read

d2u_␣k d␳2 ⫹

冋

2␮ ប2 关E␣k⫺V共␳兲兴⫺ ␣2_⫺1/4 ␳2

册

u␣k⫽0, 共75兲 d2˜u_␣k d␳2 ⫹

冋

2␮ ប2 关E␣k⫺V˜共␳兲兴⫺ ␣2_⫺1/4 ␳2

册

u ˜ ␣k⫽0. 共76兲

With the boundary conditions of Eq.共58兲 and the asymptotic form of radial function u_␣kin Eq.共60兲, it is easy to find that

sin关␦_␣共k兲⫺˜␦_␣共k兲兴 ⫽⫺␲␮ ប2_k

冕

₀ ⬁ d␳关V共␳兲⫺V˜共␳兲兴u_␣k共␳兲u˜_␣k共␳兲. 共77兲 When V˜ (␳)⫽0 we deduce the integral representation

sin␦_␣共k兲⫽⫺␲␮ ប2_k

冕

₀

⬁

d␳V共␳兲u_␣k共␳兲u_␣k0 共␳兲. 共78兲 In the case k⫽

冑

2␮E_␣k/ប→⬁ we expect that the potential will become vanishingly small since the potentials V(␳) and V

˜ (␳) should not be more singular than r⫺2 at the origin and well behaved elsewhere as assumed. So the radial function u_␣k will be very close to the corresponding free wave, i.e., u_␣k(␳) can be replaced with u_␣k0 . Thus with the help of the asymptotic expression of Eq. 共62兲 we deduce that

sin␦_␣共k兲⫽⫺2␮ ប2_k

冕

₀ ⬁ d␳V共␳兲cos2

冋

k␳⫺␣␲ 2 ⫺ ␲ 4

册

. 共79兲 The square of cosine function can be replaced with its mean value 1/2 since a very large k value leads to very rapid os-cillations. So we have sin␦_␣共k兲 → k→⬁ ⫺ ␮ ប2_k

冕

₀ ⬁ d␳V共␳兲. 共80兲

Hence we see that the phase shifts ␦_␣(k) tend to zero 共modulo ␲) as k→⬁ provided that the integral exists. This suggests that a reasonable absolute definition of the phase shift may be given by requiring that

lim k→⬁

␦␣共k兲⫽0. 共81兲

The definition is physically reasonable since we require that ␦␣(k)⫽0 when the particle is effectively free. With this

con-vention the Levinson theorem is given by

␦␣共0兲⫽n␣␲, ␣⫽兩m⫹␮0兩, m⫽0,⫾1,⫾2, . . . ,

共82兲 a result given in Eq.共2兲. It means that the phase shift at the threshold serves as a counter for the bound states in a general angular-momentum channel.

C. The effects of magnetic flux

Several interesting effects caused by the nonlocal influ-ence of the magnetic flux are concluded as follows.

共a兲 When the flux is quantized, i.e., ⌽⫽m⌽0the multiple

of a fundamental flux quantum hc/e, the Levinson theorem will reduce to the free of flux case as in关8兴. In this case the total number of bound states for the quantum number m and ⫺m are the same except m⫽0, and thus have the same phase shifts.

共b兲 When the flux satisfies ⌽/⌽0⫽ half-odd integer, there

are two different m corresponding to the same total number of the bound states, so are the phase shifts at threshold. These are such that the number pairs (m,m)⫽(1,⫺2),(2, ⫺3) for ⌽/⌽0⫽⫺1/2.

共c兲 In general, when ⌽/⌽0⫽ integer, and half-odd

inte-ger, the total number of bound states for ⫾m are no longer identical, and the phase shifts will be different from each other.

D. Extension of the potential to a more general case Although in the procedure of our proof we assume that the potential must be less singular than ␳⫺2 in Eq.共28兲 and V(␳)⫽0 for␳⬎a, we do not specify the radius a beyond which V(␳)⫽0. Hence we expect that the Levinson theorem

given in the paper should be valid for a very general poten-tial as long as the potenpoten-tial decreases rapidly enough when r→⬁ such that the total number of bound states in a general angular-momentum channel is finite.

E. A possible experimental test

In Ref.关21兴, a general fractional 共nonquantized兲 magnetic flux is observed in the superconducting film. Because of the inevitable pinning of flux in superconductor, the flux finally attaches to the defect or impurity which may carry the charge. A thin film can be viewed as a two-dimensional sys-tem and because the screen effect exists in solid, the electric interaction becomes a finite range interaction as mentioned in the preceding paragraph. If a charged particle moves near the impurity which may be captured by the impurity and forms a bound-state system during a period, the system scat-tered by the other low-energy charged particle can be the test ground of the phase shift and the number of bound states for a general angular-momentum channel.

ACKNOWLEDGMENT

The author would like to thank Dr. Jang-Yu Hsu for read-ing the manuscript.

关1兴 N. Levinson, K. Dan. Vidensk. Selsk. Mat. Fys. Medd. 25, 9 共1949兲.

关2兴 J.M. Jauch, Helv. Phys. Acta 30, 143 共1957兲. 关3兴 A. Martin, Nuovo Cimento 7, 607 共1958兲.

关4兴 R.G. Newton, J. Math. Phys. 18, 1348 共1977兲; 18, 1582 共1977兲; Scattering Theory of Waves and Particles

共McGraw-Hill, New York, 1966兲.

关5兴 G.J. Ni, Phys. Energ. Fort. Phys. Nucl. 3, 432 共1979兲. 关6兴 Z.Q. Ma and G.J. Ni, Phys. Rev. D 31, 1482 共1985兲. 关7兴 N. Poliatzky, Phys. Rev. Lett. 70, 2507 共1993兲; Helv. Phys.

Acta 66, 241共1993兲.

关8兴 Q.G. Lin, Phys. Rev. A 56, 1938 共1997兲; 57, 3478 共1998兲. 关9兴 Y. Aharonov and D. Bohm, Phys. Rev. 115, 485 共1959兲. 关10兴 Foundations of Quantum Mechanics, edited by S. Nakajima, Y.

Murayama, and A. Tonomura 共World Scientific, Singapore, 1996兲.

关11兴 Z.F. Ezawa, Quantum Hall Effects 共World Scientific,

Sin-gapore, 2000兲.

关12兴 F. Wilczek, Fractional Statistics and Anyon Superconductivity

共World Scientific, Singapore, 1990兲.

关13兴 H. Kleinert, Gauge Fields in Condensed Matter 共World

Scien-tific, Singapore, 1989兲, Vol. I.

关14兴 T.T. Wu and C.N. Yang, Phys. Rev. D 12, 3845 共1975兲. 关15兴 C.N. Yang, Phys. Rev. Lett. 33, 445 共1974兲.

关16兴 D.H. Lin, J. Phys. A 31, 4785 共1998兲; 32, 6783 共1999兲. 关17兴 D.H. Lin, J. Math. Phys. 41, 2723 共2000兲; W.F. Kao, P.G.

Luan, and D.H. Lin, Phys. Rev. A 65, 052108共2002兲.

关18兴 C.N. Yang, in Foundations of Quantum Mechanics, edited by

S. Nakajima, Y. Murayama, and A. Tonomura 共World Scien-tific, Singapore, 1996兲.

关19兴 W. Magnus, F. Oberhettinger, and R.P. Soni, Formulas and

Theorems for the Special Functions of Mathematical Physics

共Springer, Berlin, 1966兲.

关20兴 D.S. Chuu and D.H. Lin, J. Phys. A 34, 2561 共2001兲. 关21兴 A.K. Geim, S.V. Dubonos, I.V. Grigorieva, K.S. Novoselov,

F.M. Peeters, and V.A. Schweigert, Nature共London兲 407, 55