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Aharonov-Bohm effect in resonances for scattering by three solenoids.

1. Introduction. In quantum mechanics, a vector potential is said to have a direct significance to particles moving in a magnetic field. This quantum effect is known as the Aharonov-Bohm effect (AB effect) ([3]). We study this effect through resonances in scattering systems by three solenoids the centers of which are placed almost in line. The resonances are shown to be generated near the real axis by the trajectories trapped between these centers, when the centers are largely separated from one another. The location of the resonances depends on the ratio of the distances between the centers as well as on the magnetic fluxes. We also discuss the case of four solenoids.

We begin by fixing the basic notation to formulate the obtained results. We work in the two dimensional space [R.sup.2] with generic point x = ([x.sub.1], [x.sub.2]) and write

H(A) = [2.summation over (j=1)][(-i[[partial derivative].sub.j] - [a.sub.j]).sup.2], [[partial derivative].sub.j] = [partial derivative]/[partial derivative][x.sub.j],

for the Schrodinger operator with A =([a.sub.1], [a.sub.2]) : [R.sup.2] [right arrow] [R.sup.2] as a vector potential. The magnetic field b associated with A is defined as

b = [nabla] x A = [[partial derivative].sub.1][a.sub.2] - [[partial derivative].sub.2][a.sub.1] : [R.sup.2] [right arrow] R

and the magnetic flux is defined by the integral [(2[pi]).sup.-1] [integra] b(x)dx, where the integral with no domain attached is taken over the whole space.

We now take A(x) to be

(1) A(x) = (-[x.sup.2]/[[absolute value of (x)].sup.2], [x.sub.1]/[[absolute value of (x)].sup.2]) = (-[[partial derivative].sub.2] log [absolute value of (x)], [[partial derivative].sub.1] log [absolute value of (x)])

which generates the solenoidal field

[nabla] x A = ([[partial derivative].sup.2.sub.1] + [[partial derivative].sup.2.sub.2]) log [absolute value of (x)] = 2[pi][delta](x)

with center at the origin. This vector potential is often called the Aharonov-Bohm potential in physics literatures. The scattering system by one solenoid is known as one of exactly solvable models in quantum mechanics ([1,3,6]). We consider the operator H = H([alpha]A) associated with the solenoid 2[pi][alpha][delta](x), [alpha] being a magnetic flux. The operator formally defined is symmetric over [C.sup.[infinity].sub.0]([R.sup.2]\{0}), but it is not necessarily essentially self adjoint in [L.sup.2] = [L.sup.2]([R.sup.2]) because of the strong singularities at the origin. The Friedrichs extension is realized by imposing the boundary condition

(2) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

at the center. We use the same notation H to denote this self adjoint operator. The operator admits the partial wave expansion. We denote by f([omega] [right arrow] [theta]; E) the amplitude for scattering from the incident direction [omega] [member of] [S.sup.1] to the final one [theta] at energy E > 0. The amplitude is explicitly calculated as

f([omega] [right arrow] [theta]; E) = [c.sub.0](E)sin([alpha]][pi])[e.sup.i[a]([theta]-[omega])] [e.sup.i([theta] - [omega])]/1 - [e.sup.i([theta] - [omega]])

with [c.sub.0](E) = [(2/[pi]).sup.1/2][[e.sup.i[pi]]/.sup.4][E.sup.-1/4], where the Gauss notation [[alpha]] denotes the greatest integer not exceeding [alpha] and the coordinates [theta], [omega] over the unit circle [S.sup.1] are identified with the azimuth angles from the positive [x.sub.1] axis. In particular, the backward amplitude takes the form

(3) f([omega] [right arrow] -[omega]; E) = [(2[pi]).sup.-1/2][[e.sup.[pi]/.sup.4][(-1).sup.[[alpha]]+1] sin([alpha][pi])[E.sup.-1/4]

independent of [omega]. We note that the amplitude vanishes for integer flux a and that the forward amplitude f([omega] [right arrow] [omega]; E) diverges for non integer flux [alpha].

We define the Hamiltonian associated with the three centers

(4) [d.sub.[- or +]] = ([- or +][[kappa].sub.[- or +]]d, 0), [d.sub.0] = (0, [kappa][d.sup.1/2])

labelled by the large parameter d [much greater than] 1, where [[kappa].sub.[+ or -]] > 0 and [[kappa].sub.-] + [[kappa].sub.+] = 1. By assumption, it follows that [absolute value of [d.sub.+] - [d.sub.-]] = d for the distance between [d.sub.-] and [d.sub.+]. Let [A.sub.d](x) be the potential defined by

(5) [A.sub.d](x) = [[alpha].sub.-]A(x - [d.sub.-])] + [[alpha].sub.0]A(x - [d.sub.0]) + [[alpha].sub.+]A(x - [d.sub.+]),

which generates the three solenoids 2[pi][[alpha].sub.[+ or -]][delta](x - [d.sub.[+ or -]]) and 2[pi][[alpha].sub.0][delta] (x - [d.sub.0]). We consider the self adjoint operator

(6) [H.sub.d] = H([A.sub.d])

under the boundary conditions like (2) at each center. It is known that [H.sub.d] has no positive eigenvalues and the continuous spectrum occupied by [0, [infinity]) is absolutely continuous. We can show that the resolvent

R([zeta]; [H.sub.d]) = [([H.sub.d] - [zeta]).sup.-1] : [L.sup.2] [right arrow] [L.sup.2],

with Re [zeta] > 0 and Im [zeta] > 0 is meromorphically continued from the upper half plane of the complex plane to the lower half plane {[zeta] [member of] C : Re [zeta] > 0, Im [zeta] < 0} across the positive real axis where the continuous spectrum of [H.sub.d] is located. Then R([zeta]; [H.sub.d]) with Im [zeta] [less than or equal to] 0 is well defined as an operator from [L.sup.2.sub.comp]([R.sup.2]) to [L.sup.2.sub.loc]([R.sup.2]) in the sense that [chi]R([zeta]; [H.sub.d])[chi] : [L.sup.2] [right arrow] [L.sup.2] is bounded for every [chi] [member of] [C.sup.[infinity].sub.0]([R.sup.2]), where [L.sup.2.sub.comp]([R.sup.2]) and [L.sup.2.sub.loc]([R.sup.2]) denote the spaces of square integrable functions with compact support and of locally square integrable functions, respectively. The resonances of [H.sub.d] are defined as the poles of R([zeta]; [H.sub.d]) in the lower half plane (the second sheet or the unphysical plane).

The argument here is restricted only to a neighborhood of the positive axis. We fix [E.sub.0] > 0 and take a complex neighborhood

(7) [D.sub.d] = {[zeta] : [absolute value of Re[zeta] - [E.sub.0]] < [[delta].sub.0][E.sub.0], [absolute value of Im [zeta]] < (1 + 2[[delta].sub.0])[E.sup.1/2.sub.0]((log d)/d)}

for [[delta].sub.0] , 0 < [[delta].sub.0] [much less than] 1, small enough. We denote by [f.sub.[+ or -]]([omega] [right arrow] [theta]; E) the scattering amplitude by 2[pi][[alpha].sub.[+ or -]][delta](x) and set

(8) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

for [[omega].sub.1] = (1,0) (see (3)), where the branch [kappa] = [[zeta].sup.1/2] is taken in such a way that Re [kappa] > 0 for Re [zeta] > 0. Let [[kappa].sub.[+ or -]] and [kappa] be as in (4). We define the integral I([zeta]) by

(9) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

with [tau]([zeta]) = [kappa][(1/[[kappa].sub.-] + 1/[[kappa].sub.+]).sup.1/2][[zeta].sup.1/4], and the two terms [[pi].sub.[+ or -]]([zeta]) by

(10) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

where the contour is taken to be the segment from 0 to [tau] and

[P.sub.0]([zeta]) = (1 + I([zeta]))/2.

We further define

h([zeta]; d) = ([e.sup.2ikd]/d)[f.sub.0]([zeta])[[pi].sub.- ]([zeta])[[pi].sub.+]([zeta])

over [D.sub.d]. If the fluxes [[alpha].sub.-] and [[alpha].sub.+] are not an integer and if [[pi].sub.[+ or -]]([E.sub.0]) [not equal to] 0, then [f.sub.0]([zeta]) [not equal to] 0 and [[pi].sub.[+ or -]]([zeta]) [not equal to] 0 over the complex neighborhood [D.sub.d] of [E.sub.0]. We can show ([7, Lemma 4.6]) that the equation

(11) h([zeta]; d) = 1

has the solutions

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

such that [[zeta].sub.j](d) behaves like

Im [[zeta].sub.j](d) ~ -[E.sup.1/2.sub.0](log d)/d, Re([[zeta].sub.j+1] (d) - [[zeta].sub.j] (d)) ~ 2[pi][E.sup.1/2.sub.0]/d

for d [much greater than] 1. Then the first result is formulated as the theorem below, and the second one is obtained as a consequence of this theorem.

Theorem 1. Let the notation be as above. Assume that [[alpha].sub.-] and [[alpha].sub.+] are not an integer and [[pi].sub.[+ or -]]([E.sub.0]) does not vanish. Then we can take [[delta].sub.0] > 0 so small that the neighborhood [D.sub.d] defined by (7) has the following property: For any [epsilon] > 0 small enough, there exists [d.sub.[epsilon]] [much greater than] 1 such that for d > [d.sub.[epsilon]], [H.sub.d] has the resonances {[[zeta].sub.res,j](d)}, [[zeta].sub.res,j] [member of] [D.sub.d], with

Re[[zeta].sub.res,1](d) < ... < Re[[zeat].sub.res], [N.sub.d](d)

in the neighborhood {[zeta] [member of] C : [absolute value of [zeta] - [[zeta].sub.j](d)] < [epsilon]/d}, and R([zeta]; [H.sub.d]) is analytic as a function with values in operators from [L.sup.2.sub.comp]([R.sup.2]) to [L.sup.2.sub.loc]([R.sup.2]) over

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Corollary 2. Let I = [a, b] with 0 < a < b < [infinity] and let

[D.sub.d](I) = {[zeta] [member of] C : Re [zeta] [member of] I, [absolute value of Im[zeta]] [less than or equal to] (1 + 2[[delta].sub.0])[(Re[zeta]).sup.1/2]((log d)/d)}

for [[delta].sub.0] > 0 small enough. Denote by [N.sub.d](I) the number of resonances in [D.sub.d](I). Assume that [[alpha].sub.-] and [[alpha].sub.+] are not an integer and [[pi].sub.[+ or -]](E) does not vanish over the interval I. Then one can take [[delta].sub.0] > 0 so small that [N.sub.d](I) obeys the asymptotic formula

[N.sub.d](I) = (([b.sup.1/2] - [a.sup.1/2])/[pi])d + O([d.sup.1/2])

as d [right arrow] [infinity].

The rigorous proof of the theorem is long. We are going to discuss the details elsewhere ([9]).

2. Heuristic arguments. We see from an intuitive point of view how reasonable (11) is as an approximate relation to determine the location of the resonances. For brevity, we consider the special case [kappa] = 0 ([d.sub.0] = (0,0)). In this case, the three centers are exactly placed in line and the integral I([zeta]) defined by (9) vanishes, and hence h([zeta]; d) is explicitly represented as

h([zeta]; d) = ([e.sup.2ikd]/d)[f.sub.0]([zeta])[cos.sup.2]([[alpha].sub.0][pi]).

We denote by [[phi].sub.0](x; [omega], E)= exp(i[E.sup.1/2] x [omega]) the plane wave with [omega] as an incident direction at energy E > 0 and write [x.sub.[+ or -]] for [x.sub.[+ or -]] = x - [d.sub.[+ or -]]. The incident plane wave [[phi].sub.0]([x.sub.-]; -[[omega].sub.1], E) takes the form

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

after scattered into the direction a1 by the solenoid 2[pi][[alpha].sub.-][delta]([x.sub.-]), and the scattered wave hits the other solenoid 2[pi][[alpha].sub.+][delta]([x.sub.+]). Since [absolute value of [x.sub.-]] behaves like

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

around [d.sub.+], the scattered wave behaves like the plane wave

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

when it arrives at [d.sub.+], provided that there is not the third solenoid 2[pi][[alpha].sub.0][delta](x) between [d.sub.-] and [d.sub.+]. If the potential [[alpha].sub.0]A(x) associated with it is present, then the wave function undergoes a change of the phase factor by the AB effect. We consider particles moving from [d.sub.-] to [d.sub.+] and distinguish between the trajectories passing over [x.sub.2] > 0 and [x.sub.2] < 0 to denote the former and latter trajectories by [l.sub.+] and [l.sub.-], respectively. The vector potential A(x) defined by (1) satisfies the relation A(x) = [[nabla].sub.[gamma]](x) for the azimuth angle [gamma](x) from the positive [x.sub.1] axis. The change in the phase factor of the wave function is given by the line integral

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

along [l.sub.[+ or -]]. The contribution from [l.sub.+] and [l.sub.-] is fifty- fifty, and hence cos([[alpha].sub.0][pi]) arises as the sum of the phase factors exp([- or +]i[[alpha].sub.0][pi]). Thus the scattered wave takes

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

as an approximate form, when it hits the center [d.sub.+]. A similar argument applies to the plane wave [[phi].sub.0]([x.sub.+]; [[omega].sub.1], E) after scattered into the direction --a1 by the solenoid 2[pi][[alpha].sub.+][delta]([x.sub.+]), so that it again returns to the center [d.sub.-], taking the approximate form h(E; d)[[phi].sub.0]([x.sub.-]; -[[omega].sub.1], E). Hence the trapping phenomenon between [d.sub.-] and [d.sub.+] is described by the series

([[infinity].summation over (n=1)]h[(E; d).sup.n])[[phi].sub.0]([x.sub.-]; - [[omega].sub.1], E).

For example, the term with h[(E; d).sup.n] describes the contribution from the trajectory oscillating n times. This is the reason why the resonances are approximately determined by (11) and why the magnetic flux [[alpha].sub.0] is related to their locations through the AB effect. If [kappa] [not equal to] 0, then the contribution from [l.sub.[+ or -]] is not necessarily fifty-fifty, but it depends on the ratio of the distances between the centers, as is seen from (10). The coefficient [[pi].sub.[+ or -]]([zeta]) describes the AB effect term along the trajectory from [d.sub.[- or +]] to [d.sub.[+ or -]].

3. Strategy. The proof of the theorem is done by constructing the resolvent kernel R([zeta]; [H.sub.d])(x,y) with the spectral parameter [zeta] [member of] [D.sub.d]. Here we mention only the basic strategy briefly. The idea is based on the results obtained by [4] and [8].

As already stated, the scattering system by one solenoid is known to be exactly solvable. We make a full use of the information from such a system. Let K = H([[alpha].sub.0]A(x - [d.sub.0])) with flux [[alpha].sub.0] in (5). We know that the Hamiltonian with one solenoid has no resonances in C\{0}. The first step is to analyze the behavior as [absolute value of x - y] [right arrow] [infinity] of the resolvent kernel R([zeta]; K)(x,y). The kernel is represented in terms of a complex integral and admits the decomposition

R([zeta]; K)(x,y) = [R.sub.fr]([zeta]; K)(x,y) + [R.sub.sc]([zeta]; K)(x,y),

where [R.sub.fr]([zeta]; K)(x,y) corresponds to the free trajectory which goes directly from y to x without being scattered by the solenoid 2[pi][[alpha].sub.0][delta](x - [d.sub.0]), while [R.sub.sc]([zeta]; K)(x,y) corresponds to the scattered trajectory which goes from y to x after scattered at the center [d.sub.0]. The first term on the right side behaves like

(12) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

with [??] = x/[absolute value of (x)] as [absolute value of x - y] [right arrow] [infinity], where [gamma]([theta]; [omega]), 0 [less than or equal to] [gamma]([theta]; [omega]) < 2[pi], denotes the azimuth angle from [omega] to [theta] and we skip some numerical constants. The second term takes the asymptotic form

(13) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

where [f.sub.0]([omega] [right arrow] [theta]; E) denotes the scattering amplitude by 2[pi][[alpha].sub.0][delta](x).

The second step is to construct the resolvent kernel with two solenoids for the pair ([d.sub.-], [d.sub.0]) by composing two resolvent kernels with one solenoid. In doing this, a difficulty comes from the exponential growth of the resolvent kernel. For magnetic fields compactly supported, the corresponding vector potentials can not be expected to fall off rapidly at infinity, because of the topological feature of the two dimensional space that [R.sup.2]\{0} is not simply connected. This is the case where the magnetic fluxes do not vanish. In fact, it is seen from (1) that vector potentials have the long range property. Thus the vector potentials can not be expected to be well separated, even if the supports of the two magnetic fields are largely separated from each other. In other words, cut off functions used to separate the two centers do not have bounded supports. As is seen from (12) and (13), the resolvent kernels with spectral parameters in the lower half plane grow exponentially at infinity, and hence the composition of the resolvent kernels can not be controlled simply by integration by parts using oscillatory properties. We make use of gauge transformations and of a complex scaling method to construct the resolvent kernels for two solenoids. Then the asymptotic form (13) plays an important role in construction. We have already constructed resolvent kernels for two solenoids in [4]. The complex scaling method in the resonance problem has been initiated by [2]. We refer to the book [5] and literatures there for details and further developments.

The third step is to construct the resolvent kernel of the operator [H.sub.d] in question from the two kernels corresponding to the two centers ([d.sub.-], [d.sub.0]) and to one center [d.sub.+]. The construction is again based on the complex scaling method. Then the asymptotic analysis on the behavior as d = [absolute value of [d.sub.+] - [d.sub.-]] [right arrow] [infinity] of R([zeta]; K)([d.sub.[+ or -]], [d.sub.[- or +]) is crucial. We note from (12) and (13) that [R.sub.fr]([zeta]; K)(x,y) and [R.sub.sc](([zeta] K)(x,y) are singular along the forward direction [??] = -[??]. In fact,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

is not necessarily continuous along the direction [??] = -[??], and the forward amplitude [f.sub.0](-[??] [right arrow] -[??]; [zeta]) is divergent for [R.sub.sc]([zeta]; K)(x,y). However, these singularities are canceled, and we have

R([zeta]; K)([d.sub.[+ or -]], [d.sub.[- or +]]) ~ ([e.sub.ikd]/[d.sup.1/2])[[pi].sub.[- or +]([zeta]).

The AB effect term [[pi].sub.[- or +]([zeta]) is obtained through this asymptotic form.

4. Four solenoids. We move to the case of four solenoids. Assume that the four centers are located at [d.sub.[+ or -]] and at

[d.sub.1] = (-[[kappa].sub.0]d, [[kappa].sub.1][d.sub.1/2]), [d.sub.2] = ([[kappa].sub.0]d, [[kappa].sub.2] [d.sup.1/2]),

where [d.sub.[+ or -]] are as in (4), and [[kappa].sub.0] > 0 satisfies [[kappa].sub.0] < min([[kappa].sub.-], [[kappa].sub.+]). We use the same notation

[A.sub.d](x) = [[alpha].sub.-]A(x - [d.sub.-]) + [[alpha].sub.1]A(x - [d.sub.1]) + [[alpha].sub.2]A(x - [d.sub.2]) + [[alpha].sub.+]A(x - [d.sub.+])

to denote the vector potential associated with these centers (see (5)). We also write [H.sub.d] = H([A.sub.d]) for the self adjoint realization obtained by imposing the condition like (2) at each center. For the operator [H.sub.d] with four solenoids, we can obtain a result similar to Theorem 1 in the two special cases: (1) [[kappa].sub.1] = [[kappa].sub.2] = 0; (2) [[kappa].sub.0] = 0. The first case means that all the centers are horizontally placed along an identical direction, while the second one means that the two centers [d.sub.1] and [d.sub.2] are vertically placed to the trajectory trapped between [d.sub.-] and [d.sub.+].

We discuss case (1). We define the angle [[psi].sub.0], 0 < [[psi].sub.0] < [pi]/2, through the relation

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

and set

[[pi].sub.0] = (1 - [[psi].sub.0]/[pi]] cos([[beta].sub.+] [pi]) + ([psi].sub.0]/[pi]) cos([[beta].sub.-] [pi])

with [[beta].sub.+] = [[alpha].sub.2] + [[alpha].sub.1] and [[beta].sub.-] = [[alpha].sub.2] - [[alpha].sub.1]. The constant [[pi].sub.0] describes the AB effect arising from the trajectories from [d.sub.[- or +]] to [d.sub.[+ or -]]. Let [f.sub.0]([zeta]) be as in (8). We further define

[h.sub.1]([zeta]; d) = ([e.sup.2ikd]/d)[f.sub.0]([zeta])[[pi].sup.2.sub.0]

over the neighborhood [D.sub.d] defined by (7). If [[alpha].sub.[+ or -]] is not an integer and [[pi].sub.0] [not equal to] 0, then we can show that the location of the resonances in [D.sub.d] of [H.sub.d] is approximately determined by the solutions to the equation [h.sub.1]([zeta]; d) = 1 as in Theorem 1.

Next we consider case (2). For brevity, we assume that [[kappa].sub.1] < [[kappa].sub.2]. We define the integrals [I.sub.j]([zeta]), j = 1, 2, by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

with [[pi].sub.j]([zeta]) = [[kappa].sub.j][(1/[[kappa].sub.-] + 1/[[kappa].sub.+]).sup.1/2][[zeta].sup.1/4], and we set

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

with [[beta].sub.[+ or -]] as above, where

[p.sub.1]([zeta]) = (1 + [I.sub.1]([zeta]))/2, [p.sub.2]([zeta]) = (1 - [I.sub.2]([zeta]))/2

and [p.sub.3]([zeta]) = 1 - [p.sub.1]([zeta]) - [p.sub.2]([zeta]). The term [[rho].sub.-]([zeta]) describes the AB effect arising from the trajectory from [d.sub.-] to [d.sub.+], and [[rho].sub.+]([zeta]) corresponds to the trajectory from [d.sub.+] to [d.sub.-]. We further define

[h.sub.2]([zeta]; d) = ([e.sup.2ikd]/d)[f.sub.0]([zeta])[[rho].sub.- ]([zeta])[[rho].sub.+]([zeta])

for [zeta] [member of] [D.sub.d]. Assume that [[alpha].sub.[+ or -]] is not an integer and [[rho].sub.[+ or -]]([E.sub.0]) [not equal to] 0. Then we can show that the location of the resonances in [D.sub.d] of [H.sub.d] is specified by the equation [h.sub.2]([zeta]; d) = 1 as in Theorem 1. To illustrate the vertical case, we end the note by discussing the particular case when the sum of the two fluxes vanishes ([[alpha].sub.1] + [[alpha].sub.2] = 0), and [[kappa].sub.1] = - [kappa] and [k.sub.2] = [kappa] with [kappa] > 0. We set [[alpha].sub.1] = -[alpha] and [[alpha].sub.2] = [alpha]. Then we have [I.sub.1]([zeta]) = -[I.sub.2]([zeta]) = -I([zeta]C) for the integral I([zeta]) defined by (9), and [[rho].sub.[- or +]]([zeta]) takes the form

[[rho].sub.[- or +]]([zeta]) = (1 - I([zeta]))+ I([zeta])[e.sup.[+ or - ]2i[alpha][pi]].

We add a brief comment. Loosely speaking, the AB effect is not observed, provided that 0 < [kappa] [much less than] 1 or [kappa] [much greater than] 1. If the width 2[kappa][d.sup.1/2] between the two centers [d.sub.1] = (0, -[kappa][d.sup.1/2]) and [d.sub.2] = (0, [kappa][d.sup.1/2]) is small in comparison with the distance d = [absolute value of [d.sub.+] - [d.sub.-]], then a large contribution comes from the closed trajectories enclosing the two centers [d.sub.1] and [d.sub.2], and the phase factor of the wave function along such trajectories is not changed. In fact, the integral I([zeta]) goes to zero as the interval [0, [tau]] shrinks ([kappa] [right arrow] 0), and hence [[rho].sub.[- or +]]([zeta]) = 1. If, conversely, the width is large, then a large contribution comes from the closed trajectories passing between the two centers. In this case, the integral interval [0, [tau]] expands to [0, [infinity]) ([kappa] [right arrow] [infinity]), and [[rho].sub.[- or +]] is calculated as [[rho].sub.[- or +]]([zeta]) = [e.sup.[+ or -]2i[alpha][pi]] by making use of the formula [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. As a result, [[rho].sub.-]([zeta])[[rho].sub.+]([zeta]) = 1, and hence the AB effect term disappears.

doi: 10.3792/pjaa.91.45

References

[1] G. N. Afanasiev, Topological effects in quantum mechanics, Fundamental Theories of Physics, 107, Kluwer Acad. Publ., Dordrecht, 1999.

[2] J. Aguilar and J. M. Combes, A class of analytic perturbations for one-body Schrodinger Hamiltonians, Comm. Math. Phys. 22 (1971), 269-279.

[3] Y. Aharonov and D. Bohm, Significance of electromagnetic potentials in the quantum theory, Phys. Rev. (2) 115 (1959), 485-491.

[4] I. Alexandrova and H. Tamura, Resonances in scattering by two magnetic fields at large separation and a complex scaling method, Adv. Math. 256 (2014), 398-448.

[5] P. D. Hislop and I. M. Sigal, Introduction to spectral theory, with applications to Schrodinger operators, Applied Mathematical Sciences, 113, Springer, New York, 1996.

[6] S. N. M. Ruijsenaars, The Aharonov Bohm effect and scattering theory, Ann. Physics 146 (1983), no. 1, 1-34.

[7] H. Tamura, Aharonov Bohm effect in resonances of magnetic Schrodinger operators in two dimensions, Kyoto J. Math. 52 (2012), no. 3, 557-595.

[8] H. Tamura, Asymptotic properties in forward directions of resolvent kernels of magnetic Schrodinger operators in two dimensions. (to be published in Math. J. Okayama Univ. 58 (2016)).

[9] H. Tamura, Aharonov Bohm effect in resonances for scattering by three solenoids at large separation. (in preparation).

By Hideo TAMURA

Department of Mathematics, Okayama University, 3-1-1 Tsushima-naka, Kita-ku, Tsushimanaka, Okayama 700-8530, Japan

(Communicated by Kenji FUKAYA, M.J.A., March 12, 2015)
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