Scattering theory for the Aharonov-Bohm effect

Transcription

1 Scattering theory for the Aharonov-Bohm effect Takuya MINE Kyoto Institute of Technology 5th September 2017 Tosio Kato Centennial Conference at University of Tokyo Takuya MINE (KIT) Aharonov-Bohm effect Kato Centennial Conference 1 / 60

2 Today s plan 1. Aharonov-Bohm effect by single solenoid on the plane Original result by Aharonov-Bohm Mathematical justification by Ruijsenaars Aharonov-Bohm effect by N solenoids Asymptotics of the scattering amplitude in the large distance limit by Ito-Tamura 2001, Calculation of the two-solenoidal scattering amplitude in the elliptic coordinate by Gu-Qian 1988, and its justification by M Aharonov-Bohm effect by toroidal magnet Experiment by Tonomura et. al Result by Ballesteros-Weder 2009, 2009, Calculation of the scattering amplitude in the spheroidal coordinate (New). All the numerical results are obtained by using Mathematica Takuya MINE (KIT) Aharonov-Bohm effect Kato Centennial Conference 2 / 60

3 Aharonov-Bohm effect In 1959, Y. Aharonov and D. Bohm published a remarkable paper in which they emphasize the significance of the electro-magnetic vector potential in quantum mechanics. They consider an infinitely long, electrically shielded magnetic solenoid, and an electron moving outside the solenoid. They calculate how the incident plane wave is scattered by the solenoid, i.e., they compute the scattering amplitude for this problem, in the limiting case the thickness of the solenoid is 0. Their result is rigorously formulated by Ruijsenaars 1983 in terms of scattering theory. Takuya MINE (KIT) Aharonov-Bohm effect Kato Centennial Conference 3 / 60

4 Aharonov-Bohm Hamiltonian Because of the translational invariance along z-axis, they consider the two-dimensional Hamiltonian ( ) 2 ( ) 1 H = i A α on R 2, = x,, y A α = (A α,x, A α,y ) = α ( y r, x ), r = x 2 r 2 + y 2, 2 with the regular boundary condition at (x, y) = O = (0, 0) (we use the normalization ħ = 1, m = 1/2, and e = 1 for simplicity). Here α is a real constant. In the radial coordinate (x, y) = (r cos θ, r sin θ), the vector potential A α is represented as A α = α θ ( = α arctan y x ). (1) Takuya MINE (KIT) Aharonov-Bohm effect Kato Centennial Conference 4 / 60

5 Aharonov-Bohm Hamiltonian The magnetic field corresponding to A α is given by Then, we have by (1) curl A α = A α,y x curl A α = α curl θ = α A α,x y. ( ) 2 θ x y 2 θ = 0 y x for (x, y) O. However, for the disk D = {x 2 + y 2 = r 2 }, we have by the Stokes formula and (1) curl A α ds = A α dl = α θ dl = 2πα. D D Thus the magnetic field curl A α = 2παδ O, where δ O is the Dirac measure at the origin. Takuya MINE (KIT) Aharonov-Bohm effect Kato Centennial Conference 5 / 60 D

6 Aharonov-Bohm Hamiltonian The representation (1) also implies ( ) ( ) i A α = e iαθ i e iαθ, H = i A α = e iαθ ( )e iαθ. Then, for the stationary Schrödinger equation Hu = λu, we have Hu = λu (e iαθ u) = λe iαθ u v = λv, where v = e iαθ u. Notice that v is a multi-valued function unless α is an integer, because of the phase factor e iαθ. This is the first reason why the magnetic flux shifts the phase of the wave function. Takuya MINE (KIT) Aharonov-Bohm effect Kato Centennial Conference 6 / 60

7 Incoming wave solution In order to study the scattering property, they compute the incoming wave solution φ = W e ix p to the stationary Schrödinger equation Hφ = k 2 φ, in the sense φ e ix p + f (τ θ; k 2 ) eikr r 1/2 (r ), where x = (x, y) = (r cos θ, r sin θ), and p = (k cos τ, k sin τ) is the momentum of the incident wave. The factor f (τ θ; k 2 ) is called the scattering amplitude, and f (τ θ; k 2 ) 2 the differential scattering cross section, which tells us the ratio of the particle coming from the incident direction τ and scattered into the final direction θ. Takuya MINE (KIT) Aharonov-Bohm effect Kato Centennial Conference 7 / 60

8 Incoming wave solution Theorem 1 (Aharonov-Bohm 1959, Ruijsenaars 1983) The incoming wave solution φ is given by φ = m= i m e iδ m(k) J m α (kr)e im(θ τ), where J ν is the Bessel function of order ν, and the scattering phase shift δ m (k) is given by δ m (k) = ( m m α )π/2. The number δ m (k) is determined by the asymptotic formula ( 2 J m α (kr) πkr cos m α π kr π ) 2 4 ( 2 = πkr cos kr m π π ) δ m(k) (r ). Takuya MINE (KIT) Aharonov-Bohm effect Kato Centennial Conference 8 / 60

9 Incoming wave solution φ (k = 1, τ = 0) Figure: The density plot of the real part of the incoming wave solution φ when α = 1/2, k = 1, θ = 0. Takuya MINE (KIT) Aharonov-Bohm effect Kato Centennial Conference 9 / 60

10 Incoming wave solution φ (k = 1, τ = π/4) Figure: The density plot of the real part of the incoming wave solution φ when α = 1/2, k = 1, θ = π/4. Takuya MINE (KIT) Aharonov-Bohm effect Kato Centennial Conference 10 / 60

11 Incoming wave solution φ (k = 1, τ = π/2) Figure: The density plot of the real part of the incoming wave solution φ when α = 1/2, k = 1, θ = π/2. Takuya MINE (KIT) Aharonov-Bohm effect Kato Centennial Conference 11 / 60

12 Formula for the scattering amplitude When we know all the scattering phase shifts δ m (k), we can calculate the scattering amplitude f (τ θ; k 2 ) by the formula f (τ θ; k 2 ) = (2πik) 1/2 m= (e 2iδm(k) 1)e im(θ τ), (2) which is applicable also for the two-dimensional Schrödinger operator with decaying radial scalar potential + V (r). Substituting δ m (k) = ( m m α )π/2 into this formula, they obtain the following formulas. Takuya MINE (KIT) Aharonov-Bohm effect Kato Centennial Conference 12 / 60

13 Scattering amplitude by single solenoid Theorem 2 (Aharonov-Bohm 1959, Ruijsenaars 1983) The scattering amplitude f (τ θ; k 2 ) is given by f (τ θ; k 2 ) = ( ) 1/2 2i sin πα e i[α](θ τ) πk π e i(θ τ) 1 (θ τ), where [x] is the integer part of x. The differential scattering cross section f (τ θ; k 2 ) 2 is given by f (τ θ; k 2 ) 2 = 1 sin 2 (πα) 2πk sin 2 ((θ τ)/2) (θ τ). Ruijsenaars also determines the form of the singularity of f appearing at θ = τ. Takuya MINE (KIT) Aharonov-Bohm effect Kato Centennial Conference 13 / 60

14 Multi solenoidal Aharonov-Bohm effect Next, we consider the case there are N solenoids perpendicular to the plane. The Hamiltonian is given by ( ) 2 1 H = i A N on R 2, N A N = A αj (x x j, y y j ), j=1 where 2πα j is the magnetic flux through the solenoid at P j (x j, y j ) (j = 1,..., N). We impose the regular boundary condition at each P j. Takuya MINE (KIT) Aharonov-Bohm effect Kato Centennial Conference 14 / 60

15 Known results for multi-solenoidal AB effect Šťovíček 1989 Construction of the Green function by path-integral method (developed again in Kocábová-Šťovíček 2008). Nambu 2000 Trial to calculate the scattering amplitude (incomplete). Ito-Tamura 2001, 2003 Asymptotic formula for the scattering amplitude for the multi-solenoidal AB effect in the large distance limit. Gu-Qian 1988 Calculation of the scattering amplitude for the two-solenoidal AB effect, by using the elliptic coordinate (incomplete). M 2015 Justification of Gu-Qian s result. Takuya MINE (KIT) Aharonov-Bohm effect Kato Centennial Conference 15 / 60

16 Ito-Tamura result In Ito-Tamura 2001, they consider the case P 1 (0), P 2 (d) (d R 2 \ {0}), and calculate the asymptotics of the scattering amplitude as d. In this result, we identify an angle θ [0, 2π) with a unit vector ω = (cos θ, sin θ) S 1 (S 1 = {ω R 2 ω = 1}), and we denote f (ω ω; k 2 ) instead of f (θ θ; k 2 ). For x R 2 \ {0} and ω S 1, let γ(x; ω) denotes the azimuth angle of x from the direction ω with 0 γ(x; θ) < 2π. We use the notation { exp(iαγ(x; ω)) (x/ x = ω), exp(iαγ(x; ω)) = (1 + exp(2πiα))/2 (x/ x = ω), exp(iατ(x; ω, ω )) = exp(iαγ(x; ω))exp(iαγ(x; ω )). Takuya MINE (KIT) Aharonov-Bohm effect Kato Centennial Conference 16 / 60

17 Ito-Tamura result Theorem 3 (Ito-Tamura 2001) Let f j (ω ω; k 2 ) be the scattering amplitude by the single solenoid at P j with flux 2πα j (j = 1, 2) (f 2 depends on d), and f d (ω ω; k 2 ) be the scattering amplitude by the two solenoids. Then, for ω ω, f d (ω ω; k 2 ) behaves like f d (ω ω ; k 2 ) = exp(iα 2 τ( d; ω, ω ))f 1 (ω ω ; k 2 ) as d with fixed ˆd = d/ d. + exp(iα 1 τ(d; ω, ω ))f 2 (ω ω ; k 2 ) + o(1) The phase factors in the right hand side are considered to be an appearance of the AB-effect. In Ito-Tamura 2003, they consider the case N 3. Takuya MINE (KIT) Aharonov-Bohm effect Kato Centennial Conference 17 / 60

18 Elliptic coordinate In the case N = 2, P 1 ( a, 0) and P 2 (a, 0), we can analyze the operator H by using the elliptic coordinate defined as follows (this idea is first used in Gu-Qian 1988, and further developed by M 2015). { x = a cosh ξ cos η, ξ 0, π < η π, y = a sinh ξ sin η, where a is a positive constant so that the solenoids are placed at (±a, 0). The Laplacian is represented as ( ) 2 2 = a 2 (cosh 2ξ cos 2η) ξ η 2 Takuya MINE (KIT) Aharonov-Bohm effect Kato Centennial Conference 18 / 60

19 Elliptic coordinate Η 5 6 Η y Η 2 Ξ 1 Η Ξ Η 6 Η Ξ 0 Η 0 2 Η x Η 5 6 Η Η Η 3 Η 6 Figure: Elliptic coordinate for a = 1. The coordinate curves are ellipses and hyperbolas, and the points (±a, 0) are the common foci. Takuya MINE (KIT) Aharonov-Bohm effect Kato Centennial Conference 19 / 60

20 Mathieu function In the elliptic coordinate, the equation u = λu can be solved by separation of variables, and the solutions are u = Ce n (ξ, q)ce n (η, q) (n = 0, 1, 2,...), u = Se n (ξ, q)se n (η, q) (n = 1, 2,...), where q = a 2 λ/4. The functions ce n and se n are called the the Mathieu functions, and Ce n and Se n are called the the modified Mathieu functions. The angular functions ce n and se n are similar to trigonometric functions. In particular ce n and Ce n are even functions, and se n and Se n are odd functions. Takuya MINE (KIT) Aharonov-Bohm effect Kato Centennial Conference 20 / 60

21 Two solenoidal AB Hamiltonian When N = 2, P 1 ( a, 0) and P 2 (a, 0), the AB Hamiltonian H is H = e i(α 1θ 1 +α 2 θ 2 ) ( )e i(α 1θ 1 +α 2 θ 2 ), θ 1 = arg((x + 1) + yi), θ 2 = arg((x 1) + yi). Put v = e i(α 1θ 1 +α 2 θ 2 ) u. Then, Hu = λu is equivalent to v = λv, but v might be a multi-valued function. Analyzing the latter equation, we conclude the eigenequation Hu = λu can be solved by separation of variables if and only if α 1, α 2 (1/2)Z. (3) The condition (3) is called the magnetic quantization condition, since 2π (1/2) = π is equal to the quantum of magnetic flux h/(2e) under our normalization ħ = h/(2π) = 1 and e = 1. Takuya MINE (KIT) Aharonov-Bohm effect Kato Centennial Conference 21 / 60

22 Generalized eigenfunctions When the magnetic quantization condition (3) holds, the solutions u to Hu = λu are u = e i(α 1θ 1 +α 2 θ 2 ) Fe n (ξ, q)ce n (η, q) (n = 0, 1, 2,...), u = e i(α 1θ 1 +α 2 θ 2 ) Ge n (ξ, q)se n (η, q) (n = 1, 2,...), where q = a 2 λ/4. The functions Fe n and Ce n satisfy the same ODE but Fe n has the opposite parity to that of Ce n, and so do Ge n and Se n. Takuya MINE (KIT) Aharonov-Bohm effect Kato Centennial Conference 22 / 60

23 Scattering phase shift We introduce the normalized modified Mathieu functions Jc n, Js n, Jf n, Jg n, which are constant multiples of Ce n, Se n, Fe n, Ge n and behave like 2 Jc n (ξ, q) Js n (ξ, q) (kr πkr cos nπ 2 π ), 4 2 ( Jf n (ξ, q) πkr cos kr nπ 2 π ) 4 + δ n,k, 2 ( Jg n (ξ, q) πkr cos kr nπ 2 π ) 4 + ϵ n,k as ξ, where q = a 2 k 2 /4 and r = a cosh ξ x 2 + y 2. The first formula means Jc n and Js n have the same asymptotics as that of the Bessel function. The constants δ n,k and ϵ n,k are the scattering phase shifts. Takuya MINE (KIT) Aharonov-Bohm effect Kato Centennial Conference 23 / 60

24 Incoming wave solution Theorem 4 (M 2015) Assume N = 2, P 1 ( a, 0), P 2 (a, 0), α 1 = 1/2, α 2 = 1/2 and put x = (a cosh ξ cos η, a sinh ξ sin η), p = (k cos τ, k sin τ). Then, the incoming wave solution φ = W e ix p to Hφ = k 2 φ with momentum p is ( φ = e i( θ 1+θ 2 )/2 i n e iδ n,k Jf n (ξ, q)ce n (η, q)ce n (τ, q) n=0 + n=1 ) i n e iϵ n,k Jg n (ξ, q)se n (η, q)se n (τ, q). Takuya MINE (KIT) Aharonov-Bohm effect Kato Centennial Conference 24 / 60

25 Incoming wave solution (τ = 0, k = 4) The density plot of the real part of the incoming wave solution φ with incident direction τ, with k = 4. The red arrow indicates the incident direction τ, and the red points are the positions of the solenoids (±1, 0). Takuya MINE (KIT) Aharonov-Bohm effect Kato Centennial Conference 25 / 60

26 Incoming wave solution (τ = π/8, k = 4) The density plot of the real part of the incoming wave solution φ with incident direction τ, with k = 4. The red arrow indicates the incident direction τ, and the red points are the positions of the solenoids (±1, 0). Takuya MINE (KIT) Aharonov-Bohm effect Kato Centennial Conference 26 / 60

27 Incoming wave solution (τ = π/4, k = 4) The density plot of the real part of the incoming wave solution φ with incident direction τ, with k = 4. The red arrow indicates the incident direction τ, and the red points are the positions of the solenoids (±1, 0). Takuya MINE (KIT) Aharonov-Bohm effect Kato Centennial Conference 27 / 60

28 Incoming wave solution (τ = 3π/8, k = 4) The density plot of the real part of the incoming wave solution φ with incident direction τ, with k = 4. The red arrow indicates the incident direction τ, and the red points are the positions of the solenoids (±1, 0). Takuya MINE (KIT) Aharonov-Bohm effect Kato Centennial Conference 28 / 60

29 Incoming wave solution (τ = π/2, k = 4) The density plot of the real part of the incoming wave solution φ with incident direction τ, with k = 4. The red arrow indicates the incident direction τ, and the red points are the positions of the solenoids (±1, 0). Takuya MINE (KIT) Aharonov-Bohm effect Kato Centennial Conference 29 / 60

30 Scattering amplitude Theorem 5 (M 2015) Under the assumption of Theorem 4, the scattering amplitude f (τ θ; k 2 ) is given by f (τ θ; k 2 ) = ( 2 (e 2iδ n,k 1)ce n (θ, q)ce n (τ, q) πik n=0 ) + (e 2iϵ n,k 1)se n (θ, q)se n (τ, q), n=1 where q = a 2 k 2 /4. Formally the above formula can be obtained by replacing cos nθ by ce n (θ, q) and sin nθ by se n (θ, q) in the formula (2). Takuya MINE (KIT) Aharonov-Bohm effect Kato Centennial Conference 30 / 60

31 Comparison with Ito-Tamura formula (τ = π/2) Figure: Differential scattering cross section with τ = π/2 by the Ito-Tamura formula. Figure: Differential scattering cross section with τ = π/2 by our formula. Takuya MINE (KIT) Aharonov-Bohm effect Kato Centennial Conference 31 / 60

32 Comparison with Ito-Tamura formula (τ = 0) Figure: Differential scattering cross section with τ = 0 by the Ito-Tamura formula. Figure: Differential scattering cross section with τ = 0 by our formula. Takuya MINE (KIT) Aharonov-Bohm effect Kato Centennial Conference 32 / 60

33 Spectral shift function Figure: The spectral shift function ξ(λ; H, H 0 ) for the pair (H, H 0 ) for H 0 =, computed by using the Birman-Kreĭn formula ξ(λ; H, H 0 ) = log det S(k 2 )/(2πi). The high energy limit ξ(λ) 1/4 is consistent with the result in Tamura Takuya MINE (KIT) Aharonov-Bohm effect Kato Centennial Conference 33 / 60

34 Tonomura experiment In 1986, Tonomura et. al. carried out a famous experiment in order to prove the Aharonov-Bohm effect. They used a toroidal apparatus made from a super-conductive material, in which the magnetic field was enclosed. The observed interference pattern shows the real existence of the Aharonov-Bohm effect. Today, we shall give a rigorous solution of the Schrödinger equation whose interference pattern is similar to the picture by Tonomura et. al. Takuya MINE (KIT) Aharonov-Bohm effect Kato Centennial Conference 34 / 60

35 Known results (3D case) The three dimensional model (the Aharonov-Bohm effect by toroidal magnet) is also analyzed by several authors. M. Ballesteros and R. Weder 2009, 2009, 2011 solve the inverse problem of determining the magnetic flux through the section of the handlebodies (modulo 2π) in the existence of many handlebodies, using the high-velocity limit of the scattering operator. Recently they extend their results for the Klein-Gordon equations (Ballesteros-Weder 16). Moreover, they also justify the Aharonov-Bohm ansatz for the Gaussian wave packet. Iwatsuka-M-Shimada 2009 study the norm-resolvent convergence of the Schrödinger operators in the limit as the thickness of the torus tends to 0. The key tool is the Hardy-type inequality for the magnetic Schrödinger operators. Takuya MINE (KIT) Aharonov-Bohm effect Kato Centennial Conference 35 / 60

36 Idealized Tonomura model Today we consider a ring R of thickness 0 in R 3, R = {x = (x, y, 0) R 3 x = a} in which a quantized magnetic flux is enclosed. This is an idealized model to the experiment by Tonomura et al. Then, as in the twodimensional case, we can explicitly calculate physical quantities under the magnetic quantization condition. Takuya MINE (KIT) Aharonov-Bohm effect Kato Centennial Conference 36 / 60

37 Idealized Tonomura model The corresponding Hamiltonian is H = ( ) 2 1 i A on R 3, where A C (R 3 \ R; R 3 ) is the magnetic vector potential satisfying A = 0 in R 3 \ R, ( A) n ds = A dl = oπ D D (o : odd number) for any small disc D pierced by the ring R, where n is the unit normal vector on D (the direction of n is appropriately fixed). Actually we can take A satisfying the above conditions and the support of A is bounded in R 3 (we assume this condition below). Takuya MINE (KIT) Aharonov-Bohm effect Kato Centennial Conference 37 / 60

38 Oblate spheroidal coordinate The oblate spheroidal coordinate (in Iwanami mathematical formulas III ) is defined as follows. x = a cosh ξ sin η cos ϕ, y = a cosh ξ sin η sin ϕ, z = a sinh ξ cos η, ξ 0, 0 η π, π < ϕ π, where a is a positive constant. Flammer adopts another definition x = d/2 (1 + ξ 2 )(1 η 2 ) cos ϕ, ξ 0, y = d/2 (1 + ξ 2 )(1 η 2 ) sin ϕ, 1 η 1, z = d/2 ξ η, π < ϕ π. Two definitions are connected via ξ = sinh ξ, η = cos η, a = d/2. Takuya MINE (KIT) Aharonov-Bohm effect Kato Centennial Conference 38 / 60

39 Oblate spheroidal coordinate The surface ξ = const. is a flattened ellipsoid, η = const. a hyperboloid of one sheet, ϕ = const. a halfplane. Takuya MINE (KIT) Aharonov-Bohm effect Kato Centennial Conference 39 / 60

40 Laplacian in the oblate spheroidal coordinate The Laplacian is written as follows. u = 1 a 2 cosh ξ sin η(cosh 2 ξ sin 2 η) ( sin η ( cosh ξ u ) + cosh ξ ξ ξ η ) + cosh2 ξ sin 2 η 2 u. cosh ξ sin η ϕ 2 ( sin η u ) η By using this expression, we can reduce the Helmholtz equation u = k 2 u to some ordinary differential equations by the separation of variables u = f (ξ)g(η)h(ϕ). Takuya MINE (KIT) Aharonov-Bohm effect Kato Centennial Conference 40 / 60

41 The generalized eigenfunctions for As a result, the solutions to u = k 2 u are u = S ml ( iak, cos η)r (1) ml ( iak, i sinh ξ) cos mϕ, u = S ml ( iak, cos η)r (1) ml ( iak, i sinh ξ) sin mϕ (m 0), (m = 0, 1, 2,..., l = m, m + 1, m + 2,...), where S ml is called the angular spheroidal wave function, and R (1) ml the radial spheroidal wave function of the first kind. Takuya MINE (KIT) Aharonov-Bohm effect Kato Centennial Conference 41 / 60

42 Idealized Tonomura model Let us return to the ring magnetic field model. Take x 0 R 3 sufficiently large, and put Φ(x) = exp ( x i ) A dl. The function Φ is two-valued, since exp ( i ) A dl = e ioπ = 1 D for any small disc D pierced by R. Moreover, we have x 0 Hu = Φ( )Φ 1 u. Then we can solve Hu = λu similarly to the two-dimensional case. Takuya MINE (KIT) Aharonov-Bohm effect Kato Centennial Conference 42 / 60

43 Idealized Tonomura model As a result, the solutions to Hu = k 2 u are u = Φ S ml ( iak, cos η)r (5) ml ( iak, i sinh ξ) cos mϕ, u = Φ S ml ( iak, cos η)r (5) ml ( iak, i sinh ξ) sin mϕ (m 0), (m = 0, 1, 2,..., l = m, m + 1, m + 2,....) Both functions Φ and S ml ( iak, cos η)r (5) ml ( iak, i sinh ξ) are two-valued with respect to the original coordinate x, and the product of these functions is single-valued. Takuya MINE (KIT) Aharonov-Bohm effect Kato Centennial Conference 43 / 60

44 Scattering phase shift The scattering phase shift δ m l,k R (1) 1 ml (c, z) is defined by the asymptotics of ( cz cos R (5) 1 ml (c, z) cz sin cz l + 1 ) 2 π, ( cz l π + δm l,k as z i. When c = iak and z = i sinh ξ, we have cz = k a sinh ξ kr (r = x ) and z i r. Thus the behavior of R (1) ml (c, z) is similar to that of the spherical Bessel function. ) Takuya MINE (KIT) Aharonov-Bohm effect Kato Centennial Conference 44 / 60

45 Main result 1 Theorem 6 (Incoming wave solution) For x, p R 3, put x = (a cosh ξ sin η cos ϕ, a cosh ξ sin η sin ϕ, a sinh ξ cos η), p = (k sin τ cos ψ, k sin τ sin ψ, cos τ). Then, the incoming wave solution φ = W e ix p with momentum p is φ = Φ m=0 l=m i l cl,me 2 iδm (5) l,k R ml ( iak, i sinh ξ) S ml ( iak, cos η)s ml ( iak, cos τ)cos(m(ϕ ψ)). Takuya MINE (KIT) Aharonov-Bohm effect Kato Centennial Conference 45 / 60

46 Numerical result (the object wave) The real part of the object wave e ip 1 x (p 1 = (0, 0, 8)) in the free system, and that of φ,1 = W e ip 1 x in the ring magnetic field. Figure: The density plot of Re e ip 1 x. Figure: The density plot of Re φ,1. Takuya MINE (KIT) Aharonov-Bohm effect Kato Centennial Conference 46 / 60

47 Numerical result (the reference wave) The real part of the reference wave e ip 2 x (p 2 = (0, 4 3, 4)) in the free system, and that of φ,2 = W e ip 2 x in the ring magnetic field. Figure: The density plot of Re e ip 2 x. Figure: The density plot of Re φ,2. Takuya MINE (KIT) Aharonov-Bohm effect Kato Centennial Conference 47 / 60

48 The interference pattern However, the real part of a wave function is not an observable quantity (it also depends on the choice of the gauge). The experimental interference pattern made by two plane waves e ip 1 x and e ip2 x is calculated as e ip 1 x + e ip 2 x 2 = p 1+p 2 ei x 2 2 e i p 1 p 2 2 = 4 cos 2 ( p1 p 2 2 x + e i p 2 p 1 2 ) x. 2 x Thus the interference pattern consists of parallel planes orthogonal to the vector p 1 p 2, in the free case. We see how the interference pattern is shifted by the ring magnetic field. Takuya MINE (KIT) Aharonov-Bohm effect Kato Centennial Conference 48 / 60

49 Numerical result (the interference pattern) The figures are the density plot of the yz-plane section of the interference pattern e ip 1 x + e ip 2 x 2 and φ,1 + φ,2 2. Figure: The interference pattern in the free system. Figure: Shifted interference pattern by the ring magnetic field. Takuya MINE (KIT) Aharonov-Bohm effect Kato Centennial Conference 49 / 60

50 Shifted interference pattern on the plane z = 3 Figure: Shifted interference pattern. Takuya MINE (KIT) Aharonov-Bohm effect Kato Centennial Conference 50 / 60

51 Shifted interference pattern on the plane z = 2 Figure: Shifted interference pattern. Takuya MINE (KIT) Aharonov-Bohm effect Kato Centennial Conference 51 / 60

52 Shifted interference pattern on the plane z = 1 Figure: Shifted interference pattern. Takuya MINE (KIT) Aharonov-Bohm effect Kato Centennial Conference 52 / 60

53 Shifted interference pattern on the plane z = 0 Figure: Shifted interference pattern. Takuya MINE (KIT) Aharonov-Bohm effect Kato Centennial Conference 53 / 60

54 Shifted interference pattern on the plane z = 1 Figure: Shifted interference pattern. Takuya MINE (KIT) Aharonov-Bohm effect Kato Centennial Conference 54 / 60

55 Shifted interference pattern on the plane z = 2 Figure: Shifted interference pattern. Takuya MINE (KIT) Aharonov-Bohm effect Kato Centennial Conference 55 / 60

56 Main result 2 Theorem 7 (Scattering amplitude for ring magnetic field) We introduce the spherical coordinate (τ, ψ) in S 2 as ω = (sin τ cos ψ, sin τ sin ψ, cos τ), ω = (sin τ cos ψ, sin τ sin ψ, cos τ ). Then, the scattering amplitude with energy k 2 (k > 0) for the quantized ring magnetic field is f (ω ω; k 2 ) = 1 2ik m=0 l=m (e 2iδm l,k 1)c 2 l,m S ml ( iak, cos τ)s ml ( iak, cos τ ) cos(m(ψ ψ )). Takuya MINE (KIT) Aharonov-Bohm effect Kato Centennial Conference 56 / 60

57 Differential scattering cross section Figure: Density plot of the DSCS for τ = 0, ψ = 0, k = 8. Takuya MINE (KIT) Aharonov-Bohm effect Kato Centennial Conference 57 / 60

58 Differential scattering cross section Figure: Density plot of the DSCS for τ = π/3, ψ = 0, k = 8. Takuya MINE (KIT) Aharonov-Bohm effect Kato Centennial Conference 58 / 60

59 Differential scattering cross section Figure: Density plot of the DSCS for τ = π/2, ψ = 0, k = 8. Takuya MINE (KIT) Aharonov-Bohm effect Kato Centennial Conference 59 / 60

60 Remaining problems (1) Rigorous analysis of the scattering amplitude; e.g. the asymptotics as a or a 0, etc. (2) In the case the magnetic flux is not quantized. Asymptotics of the scattering amplitude? Resonance? (3) Efficient technique for the numerical calculation (improvement of Mathematica s built-in functions...). (4) Improvement of the convergence in the thickness 0-limit (Iwatsuka-M-Shimada 2009). Trace class convergence? Takuya MINE (KIT) Aharonov-Bohm effect Kato Centennial Conference 60 / 60