[PPT] - Greens function, wavefunction and Wigner function of the MIC-Kepler PowerPoint Presentation

SLIDE 1

Green’s function, wavefunction and Wigner function

f the MIC-Kepler problem

Tokyo University of Science Graduate School of Science, Department of Mathematics, The Akira Yoshioka Laboratory

Tomoyo Kanazawa

1

SLIDE 2

Outline

1. Hamiltonian description for the MIC-Kepler problem
The reduced Hamiltonian system by an S1 action
The 4-dimensional conformal Kepler problem
2. Green’s function of the MIC-Kepler problem
The infinite series of its wavefunction
3. Wigner function of the MIC-Kepler problem

2

SLIDE 3

1. Hamiltonian description for the MIC-Kepler problem

In 1970, McIntosh and Cisneros studied the dynamical system de- scribing the motion of a charged particle under the magnetic force due to Dirac’s monopole field of strength −µ and the square inverse centrifugal potential force besides the Coulomb’s potential force.

Bµ

= −µ r3 x ∥ Bµ ∥ = |µ| r2 µ is quantized as µ ∈ ℏ 2 Z .

PSfrag repla emen ts O x y z r = k x k _ x Coulom b's for e k r 2 en trifugal for e due to Dira 's eld

2

mr 3 jj r 2 k _ x k magneti for e due to Dira 's eld _ x

B
3

SLIDE 4

The Hamiltonian description for the MIC-Kepler problem is given by

T. Iwai and Y. Uwano (1986) as follows.

The MIC-Kepler problem is the

✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿

reduced Hamiltonian system

f the 4-dimensional

conformal Kepler problem by an S1 action, if the associated momentum mapping ψ(u, ρ) ≡ µ ̸= 0 .

PSfrag repla emen ts O x y z _ R 4 _ R 3 T

_

R 4 T

_

R 3

1

4u 2

1

() 1 4u 2

p

(x ) u u x =

(S

1

u)

(u ;

)

(u ; ) (x ; p) S 1

u

ψ(u , ρ) is invariant under the S1 action, then let ψ−1(µ)⊂ T ∗

u ˙

R4 be a subset s.t. ψ−1(µ) =

{

(u , ρ) ∈ T ∗

u ˙

R4

ψ(u , ρ) = 1

2(−u2ρ1 + u1ρ2 − u4ρ3 + u3ρ4) = µ

}

.

4

SLIDE 5

The MIC-Kepler problem is a triple ( T ∗ ˙ R3, σµ , Hµ ) where σµ = dpx∧dx + dpy∧dy + dpz∧dz− µ r3 ( x dy∧dz + y dz∧dx + z dx∧dy) , Hµ (x, p) = 1 2m(px2 + py2 + pz2) + µ2 2mr2 − k r . Its energy hyper surface : Hµ = E ⇔ Φ(x, p) ≡ r (Hµ − E) = 0 is equal to

(

π∗

µΦ

)

(u, ρ) = 0 where πµ ψ−1(µ) − → T ∗ ˙ R3 , π∗

µΦ = u2 (

H − E

)

,

PSfrag repla emen ts

(u

; ) T

_

R 4 T

_

R 3 1 ()

(x

; p) (u ; ) (x ; p) R

u2 ≡ u12 + u22 + u32 + u42 .

5

SLIDE 6

The conformal Kepler problem is a triple (T ∗ ˙ R4 , dρ∧du , H ) where dρ ∧ du ≡

4

∑

j=1

dρj ∧ duj , H(u , ρ) = 1 2m

(

1 4u2

4

∑

j=1

ρj2

)

− k u2 . Since u2 = r > 0 (invariant under the S1 action), π∗

µΦ = 0 is equal

to H − E = 0 . The energy hyper surface H = E is equivalent to K(u , ρ)= ϵ where K(u, ρ) is the Hamiltonian of 4-dimensional harmonic oscillator: K(u, ρ) = 1 2m

4

∑

j=1

ρj2 + 1 2mω2

4

∑

j=1

uj2

{

m > 0 mass of pendulum ω > 0 angular frequency considering only the case where the real parameter E < 0 and putting both the constant mω2 ≡ −8E and a real parameter ϵ ≡ 4k .

6

SLIDE 7

We solved the harmonic oscillator by means of the Moyal product, which brought the following functions. ♣ Moyal propagator (∗-exponential) e

i ℏ(t+iy′)K

(ui+uf

2

,ρ

)

∗

=

(

cos ωz′ 2

)−4

exp

[

i 2 ℏωK

(ui + uf

2 , ρ

)

tan ωz′ 2

]

where ui = (ui

1, ui 2, ui 3, ui 4) ∈ ˙

R4 and uf = (uf

1, uf 2, uf 3, uf 4) ∈ ˙

R4 denote initial point and final point of motion respectively. ♣ Feynman’s propagator

K (uf, ui ; z′ = t + iy′)

= −m2ω2 4π2ℏ2 1 sin2(ωz′) exp

[

−i mω 2ℏ 1 sin (ωz′)

{

(u2

i + u2 f) cos (ωz′) − 2ui · uf

}]

=

∞

∑

n=−∞

Cneinωt where Cn ≡ ω 2π

∫ π/ω

−π/ω K (uf, ui ; τ + iy′)e−inωτ dτ .

7

SLIDE 8

♣ Green’s function G(uf, ui ; ϵ) = lim

y′ → +0

i ℏ

∫ ∞  

∞

∑

n=−∞

Cn einωt

  e−y′+iϵ

ℏ

(t+iy′)dt

= m2ω2 π2ℏ2 exp

[

−mω 2ℏ (u2

i + u2 f)

]

∞

∑

N=0

∑

l1+l2+l3+l4=N

1 ϵ − (N + 2)ℏω 1 2N l1!l2!l3!l4! Hl1

(√mω

ℏ ui

1

)

Hl1

(√mω

ℏ uf

1

)

Hl2

(√mω

ℏ ui

2

)

Hl2

(√mω

ℏ uf

2

)

Hl3

(√mω

ℏ ui

3

)

Hl3

(√mω

ℏ uf

3

)

Hl4

(√mω

ℏ ui

4

)

Hl4

(√mω

ℏ uf

4

)

where n − 2 ≡ N = l1 + l2 + l3 + l4 (l1, l2, l3, l4 ∈ N ∪ {0}), Hl(X) is the Hermite polynomial : Hl(X) = (−1)l eX2 dl dXl e−X2 .

8

SLIDE 9

Moreover, we denote by ΨN(u) the wave function of 4-dimensional harmonic oscillator on ˙ R4 ΨN(u) = ← − mω πℏ 1

√

2Nl1!l2!l3!l4! exp

[

−mω 2ℏ (u2

1 + u2 2 + u2 3 + u2 4)

]

Hl1

(√mω

ℏ u1

)

Hl2

(√mω

ℏ u2

)

Hl3

(√mω

ℏ u3

)

Hl4

(√mω

ℏ u4

)

satisfying ˆ K ΨN(u) = ϵ ΨN(u) where ˆ K = − ℏ2 2m

 

4

∑

j=1

∂2 ∂u2

j

  + 1

2mω2

4

∑

j=1

u2

j .

Then we verify G(uf, ui ; ϵ) =

∞

∑

N=0

1 ϵ − (N + 2)ℏω ΨN(uf) ΨN(ui) .

9

SLIDE 10 PSfrag repla emen ts H

=

E E <

nformal

1 () [ T

_

R 4 G + (x f ; x i ; E )

r

G

(x

f ; x i ; E ) H = E Harmoni K =

Kepler

T

_

R 3 MIC-Kepler e iz ~ K

K

(u f ; u i ; z )

4k

F 1 L

s illator
=

m! 2

8jE

j G(u f ; u i ; 4k ) y ! +0

=

m m e it ~ K

t

! z = t + iy y > (y 6= 0)

10

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2. Green’s function of the MIC-Kepler problem

We suppose E ̸= −2mk2 ℏ2 (N + 2)2 (N = 0, 1, 2, · · · ) , then reduce the Green’s function of the conformal Kepler problem G(uf, ui ; ϵ ≡ 4k) assumed mω2 ≡ −8E to the Green’s function of the MIC-Kepler problem G+(xf, xi ; E) or G−(xf, xi ; E) by an S1 action. G+(xf, xi ; E) and G−(xf, xi ; E) denote the Green’s functions in the following local coordinates τ+ and τ− respectively. τ+ : π−1(U+) ∋ u − → (π(u), φ+(u)) = (x(r, θ, ϕ), exp(iν/2)) ∈ U+ × S1

    

x = r sin θ cos ϕ y = r sin θ sin ϕ z = r cos θ ,

            

u1 = √r cos θ 2 cos ν + ϕ 2 , u2 = √r cos θ 2 sin ν + ϕ 2 u3 = √r sin θ 2 cos ν − ϕ 2 , u4 = √r sin θ 2 sin ν − ϕ 2 where U+ = ← −

{

x(r, θ, ϕ) ∈ ˙

R3 ; r > 0 , 0 ≤ θ < π , 0 ≤ ϕ < 2π

}

, 0 ≤ ν < 4π .

11

SLIDE 12

τ− : π−1(U−) ∋ u − → (π(u), φ−(u)) = (x(˜ r, ˜ θ, ˜ ϕ), exp(i˜ ν/2)) ∈ U− × S1

    

x = ˜ r sin ˜ θ cos ˜ ϕ y = −˜ r sin ˜ θ sin ˜ ϕ z = −˜ r cos ˜ θ ,

            

u1 = − √ ˜ r sin ˜ θ 2 cos ˜ ν + ˜ ϕ 2 , u2 = − √ ˜ r sin ˜ θ 2 sin ˜ ν + ˜ ϕ 2 u3 = − √ ˜ r cos ˜ θ 2 cos ˜ ν + 3˜ ϕ 2 , u4 = − √ ˜ r cos ˜ θ 2 sin ˜ ν + 3˜ ϕ 2 where U− = ← −

{

x(˜

r, ˜ θ, ˜ ϕ) ∈ ˙ R3 ; ˜ r > 0 , 0 ≤ ˜ θ < π , 0 < ˜ ϕ ≤ 2π

}

, 0 ≤ ˜ ν < 4π . O

x

z x y r = ˜ r θ ˜ θ O

x

z x y r = ˜ r ϕ ˜ ϕ

12

SLIDE 13

Iwai and Uwano also investigated the “quantized” system (1988) : The “quantized” conformal Kepler problem is defined as a pair (L2(R4; 4u2du), ˆ H ) where

              

L2(R4; 4u2du) : The Hilbert space of square integrable complex-valued functions on R4 , ˆ H = − ℏ2 2m

  1

4u2

4

∑

j=1

∂2 ∂u2

j

  − k

u2 : The Hamiltonian operator.

They introduce complex line bundles Ll (l ∈ Z) on which the

linear connection is induced from a connection on the principal fibre bundle π : ˙ R4 → ˙ R3.

By an S1 action, L2(R4; 4u2du) is reduced to the Hilbert space,

denoted by Γl , of square integrable cross sections in Ll over ˙ R3.

13

SLIDE 14

The quantized MIC-Kepler problem is the ✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿ reduced quantum system (Γl, ˆ Hl ) where ∇j stands for the covariant derivation with respect to the linear connection whose curvature gives Dirac’s monopole field of strength −lℏ/2 , ˆ Hl = − ℏ2 2m

3

∑

j=1

∇2

j + (lℏ/2)2

2mr2 − k r .

Cross section in Ll corresponds uniquely to an eigenfunction of

the momentum operator ˆ N ˆ N = iℏ 2

(

−u2 ∂ ∂u1 + u1 ∂ ∂u2 − u4 ∂ ∂u3 + u3 ∂ ∂u4

)

, ˆ N Ψ(u) = − l 2 ℏ Ψ(u) Ψ(u) ∈ L2(R4; 4u2du) , l ∈ Z .

Accordingly, the introduction of Ll is understood as a geometric

consequence of the conservation of the angular momentum associated with the U(1) ≃ S1 action.

14

SLIDE 15

We can obtain the wave function of the quantized MIC-Kepler problem, denoted by ΨN(x)∈ Γl , as the following cross section with either of the local coordinates. i) ∀x ∈ U+ , Ψ+

N(x) ≡

√π 2 e−i l ν/2 ΨN, l (u) ii) ∀x ∈ U− , Ψ−

N(x) ≡

√π 2 e−i l ˜

ν/2 ΨN, l (u)

where L2(R4; 4u2du) ∋ ΨN, l(u) satisfies

        

ˆ H Ψ(u) = E Ψ(u) ⇔ ˆ K Ψ(u) = ϵ Ψ(u) s.t. ϵ ≡ 4k and mω2 ≡ −8E ˆ N Ψ(u) = − l 2 ℏ Ψ(u) . Finally, we can calculate the Green’s function of the MIC-Kepler problem by the following infinite series consists of ΨN(x) with either of the local coordinates. G (xf, xi; E = −mω2/8) =

∞

∑

N=0

1 4k − (N + 2)ℏω ΨN(xf) ΨN(xi)

15

SLIDE 16

Proposition 1-1. [The Green’s function of the MIC-Kepler problem] (i) When xi , xf ∈ U+ , G+(xf, xi; E = −mω2/8) = m2ω2 4πℏ2 e−mω

2ℏ (ri+rf)

∞

∑

N=0

1 4k − (N + 2)ℏω

(mω

ℏ

)N

1 k1!k2!k3!k4!

( √rirf cos θi

2 cos θf 2

)k1+k3 ( √rirf sin θi

2 sin θf 2

)k2+k4

P

(

ri cos2θi 2 , ri sin2θi 2

)

P

(

rf cos2θf 2 , rf sin2θf 2

)

ei(k1−k2−k3+k4)(ϕi−ϕf)/2 where k1, k2, k3, k4 ∈ N ∪ {0} s.t.

{

k1 + k2 + k3 + k4 = N k1 + k2 − k3 − k4 = −l (Z ∋ l = 2µ/ℏ) ,

P (X , Y ) is the following polynomial. P (X , Y ) =

k1

∑

j=0 k2

∑

s=0

j!s!

(

− ℏ mω

)j+s

k1Cj · k3Cj · k2Cs · k4Cs X−j Y −s

16

SLIDE 17

Proposition 1-2. [The Green’s function of the MIC-Kepler problem] (ii) When xi , xf ∈ U− , G−(xf, xi; E = −mω2/8) =m2ω2 4πℏ2 e−mω

2ℏ (˜

ri+˜ rf) ∞

∑

N=0

1 4k − (N + 2)ℏω

(mω

ℏ

)N

1 k1!k2!k3!k4!

(√

˜ ri˜ rf sin ˜ θi 2 sin ˜ θf 2

)k1+k3 (√

˜ ri˜ rf cos ˜ θi 2 cos ˜ θf 2

)k2+k4

P

(

˜ ri sin2˜ θi 2 , ˜ ri cos2˜ θi 2

)

P

(

˜ rf sin2˜ θf 2 , ˜ rf cos2˜ θf 2

)

ei(k1+3k2−k3−3k4)(˜

ϕi−˜ ϕf)/2 .

(iii) When xi , xf ∈ U+ ∩ U− , the following correlation of G− with G+ is shown by ˜ r = r , ˜ θ = π − θ and ˜ ϕ = 2π − ϕ. G−(xf, xi; E) = G+(xf, xi; E) ei l (ϕi−ϕf) Incidentally, we can also find the following proposition.

17

SLIDE 18

Proposition 2. [The wave function of the MIC-Kepler problem] (i) When x ∈ U+ , Ψ+

N (x) = mω

2√πℏ

(√mω

ℏ

)N P (

rcos2 θ

2 , rsin2 θ 2

)

√k1!k2!k3!k4! e−mω

2ℏ r

(√r cos θ

2 )k1+k3 (√r sin θ

2 )k2+k4

exp

[

−i(k1 − k2 − k3 + k4)ϕ 2

]

. (ii) When x ∈ U− , Ψ−

N (x) = mω

2√πℏ

(

−

√mω

ℏ

)N P (

˜ rsin2 ˜

θ 2 , ˜

rcos2 ˜

θ 2

)

√k1!k2!k3!k4! e−mω

2ℏ ˜

r

(√

˜ r sin ˜ θ 2

)k1+k3 (√

˜ r cos ˜ θ 2

)k2+k4

exp

[

−i(k1 + 3k2 − k3 − 3k4) ˜ ϕ 2

]

. (iii) When x ∈ U+ ∩ U− , Ψ−

N (x) = Ψ+ N (x) e−i l ϕ

where N = 0, 1, 2, · · · , the combination of non-negative integers (k1, k2, k3, k4) and the polynomial P are the same as those shown in Proposition 1.

18

SLIDE 19

3. Wigner function of the MIC-Kepler problem

We showed the energy-eigenspace of the MIC-Kepler problem in our proceeding as follows. Theorem [The eigenspace of the MIC-Kepler problem] Its eigenspace associated with the negative energy EN = −2mk2 ℏ2 (N + 2)2 (N = 0, 1, 2, · · · ) is spanned by fN(u, ρ) = (−1)N (πℏ)4 e−2(N+2) Lna(4b+

3 b3)Lnb(4b+ 1 b1)Lnc(4b+ 2 b2)Lnd(4b+ 4 b4)

where na, nb, nc, nd ∈ N ∪ {0} , l ∈ Z s.t.

    

2(na + nd) ≡ N + l 2(nb + nc) ≡ N − l i.e.

    

| l | ≤ N N and l are simultaneously even or odd. The dimension is

(

N + l 2 + 1

)(

N − l 2 + 1

)

= (N + l + 2)(N − l + 2) 4 .

19

SLIDE 20

In the above-mentioned theorem, Ln(X) denotes the Laguerre polynomial of degree n s.t. Ln(X) =

n

∑

α=0

(−1)α n! (α!)2(n − α)! Xα ,

∞

∑

n=0

Ln(X) ξn = 1 1 − ξ exp

(

− ξ 1 − ξ X

)

. Further,

                                    

4b+

3 b3 = mω

ℏ (u12 + u22) + 1 mℏω(ρ12 + ρ22) + 2 ℏ(u1ρ2 − u2ρ1) 4b+

1 b1 = mω

ℏ (u12 + u22) + 1 mℏω(ρ12 + ρ22) − 2 ℏ(u1ρ2 − u2ρ1) 4b+

2 b2 = mω

ℏ (u32 + u42) + 1 mℏω(ρ32 + ρ42) − 2 ℏ(u3ρ4 − u4ρ3) 4b+

4 b4 = mω

ℏ (u32 + u42) + 1 mℏω(ρ32 + ρ42) + 2 ℏ(u3ρ4 − u4ρ3) . We reduce the eigenfunction fN(u, ρ) on T ∗ ˙ R4 to that on T ∗ ˙ R3 with the above-mentioned local polar coordinates.

20

SLIDE 21

Proposition 3-1. [The Wigner function of the MIC-Kepler problem] Suppose x ∈ U+ ∩ U− , (i) fN(r, θ, ϕ, pr, pθ, pϕ) =(−1)N (πℏ)4 e−2(N+2) Lna

(

N + 2 2mk

[

A2 + C2 r(1 + cos θ)

])

Lnb

(

N + 2 2mk

[

A2 + D2 r(1 + cos θ)

])

Lnc

(

N + 2 2mk

[

B2 + E2 r(1 − cos θ)

])

Lnd

(

N + 2 2mk

[

B2 + F2 r(1 − cos θ)

])

(ii) fN(˜ r, ˜ θ, ˜ ϕ, p˜

r, p˜ θ, p˜ ϕ)

=(−1)N (πℏ)4 e−2(N+2) Lna

(

N + 2 2mk

[

˜ A2 + ˜ C2 ˜ r(1 − cos ˜ θ)

])

Lnb

(

N + 2 2mk

[

˜ A2 + ˜ D2 ˜ r(1 − cos ˜ θ)

])

Lnc

(

N + 2 2mk

[

˜ B2 + ˜ E2 ˜ r(1 + cos ˜ θ)

])

Lnd

(

N + 2 2mk

[

˜ B2 + ˜ F2 ˜ r(1 + cos ˜ θ)

])

where px dx + py dy + pz dz = pr dr + pθ dθ + pϕ dϕ = p˜

r d˜

r + p˜

θ d˜

θ + p˜

ϕ d˜

ϕ .

21

SLIDE 22

Proposition 3-2. [The Wigner function of the MIC-Kepler problem] Functions A, B, C, D, E and F on T ∗(U+ ∩ U−) are as follows. A(r, θ, ϕ, pr, pθ, pϕ) ≡ pr

√

r(1 + cos θ) − pθ

√

1 − cos θ r B(r, θ, ϕ, pr, pθ, pϕ) ≡ pr

√

r(1 − cos θ) + pθ

√

1 + cos θ r C(r, θ, ϕ, pr, pθ, pϕ) ≡ pϕ + r(1 + cos θ)

{

2mk ℏ(N + 2) + µ r

}

D(r, θ, ϕ, pr, pθ, pϕ) ≡ pϕ − r(1 + cos θ)

{

2mk ℏ(N + 2) − µ r

}

E(r, θ, ϕ, pr, pθ, pϕ) ≡ pϕ + r(1 − cos θ)

{

2mk ℏ(N + 2) − µ r

}

F(r, θ, ϕ, pr, pθ, pϕ) ≡ pϕ − r(1 − cos θ)

{

2mk ℏ(N + 2) + µ r

}

22

SLIDE 23

Proposition 3-3. [The Wigner function of the MIC-Kepler problem] Functions ˜ A, ˜ B, ˜ C, ˜ D, ˜ E and ˜ F on T ∗(U+ ∩ U−) are as follows. ˜ A(˜ r, ˜ θ, ˜ ϕ, p˜

r, p˜ θ, p˜ ϕ) ≡ p˜ r

√

˜ r(1 − cos ˜ θ) + p˜

θ

√

1 + cos ˜ θ ˜ r ˜ B(˜ r, ˜ θ, ˜ ϕ, p˜

r, p˜ θ, p˜ ϕ) ≡ p˜ r

√

˜ r(1 + cos ˜ θ) − p˜

θ

√

1 − cos ˜ θ ˜ r ˜ C(˜ r, ˜ θ, ˜ ϕ, p˜

r, p˜ θ, p˜ ϕ) ≡ p˜ ϕ − ˜

r(1 − cos ˜ θ)

{

2mk ℏ(N + 2) + µ ˜ r

}

˜ D(˜ r, ˜ θ, ˜ ϕ, p˜

r, p˜ θ, p˜ ϕ) ≡ p˜ ϕ + ˜

r(1 − cos ˜ θ)

{

2mk ℏ(N + 2) − µ ˜ r

}

˜ E(˜ r, ˜ θ, ˜ ϕ, p˜

r, p˜ θ, p˜ ϕ) ≡ p˜ ϕ − ˜

r(1 + cos ˜ θ)

{

2mk ℏ(N + 2) − µ ˜ r

}

˜ F(˜ r, ˜ θ, ˜ ϕ, p˜

r, p˜ θ, p˜ ϕ) ≡ p˜ ϕ + ˜

r(1 + cos ˜ θ)

{

2mk ℏ(N + 2) + µ ˜ r

}

23

SLIDE 24

Proposition 3-4. [The Wigner function of the MIC-Kepler problem] Using the following equivalences : ˜ r = r , cos ˜ θ = − cos θ , p˜

r = pr , p˜ θ = −pθ , p˜ ϕ = −pϕ

we verify ˜ A = A , ˜ B = B ˜ C = −C , ˜ D = −D , ˜ E = −E , ˜ F = −F . Then we have fN(r, θ, ϕ, pr, pθ, pϕ) = fN(˜ r, ˜ θ, ˜ ϕ, p˜

r, p˜ θ, p˜ ϕ) .

References

[1] Bayen F., Flato M., Fronsdal C., Lichnerowicz A. and Sternheimer D., Deformation Theory and Quantization. II. Physical Applications, Ann.Phys.111 (1978) 111–151.

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SLIDE 25

[2] Fujii K. and Funahashi K., 一次元調和振動子のFeynman核の再考,

横浜市立大学論叢, 自然科学系列第50巻, 第1.2合併号(1999),

pp 77–92. [3] Gracia-Bond ´ ıa J., Hydrogen Atom in the Phase-Space Formulation of Quantum Mechanics, Phys.Rev.A 30 (1984)

no. 2, 691–697.

[4] Hoang L., On the Green Function for the MIC-Kepler Problem, Izv.Akad.Nauk Belarusi, Ser.Fiz.-Mat.Nauk (1992) no. 2, 76–80. [5] Hoang L., Komarov L. and Romanova T., On the Coulomb Green Function, J.Phys.A: Math.Gen. 22 (1989) 1543–1552. [6] Iwai T., On a “Conformal” Kepler Problem and its Reduction, J.Math.Phys. 22 (1981) no. 8, 1633–1639. [7] Iwai T. and Uwano Y., The Four-dimensional Conformal Kepler Problem Reduces to the Three-dimensional Kepler Problem with a Centrifugal Potential and Dirac’s Monopole Field. Classical Theory, J.Math.Phys. 27 (1986) no. 6, 1523–1529.

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SLIDE 26

[8] Iwai T. and Uwano Y., The Quantised MIC-Kepler Problem and Its Symmetry Group for Negative Energies, J.Phys.A: Math.Gen. 21 (1988) 4083–4104. [9] Kanazawa T. and Yoshioka A., Star Product and its Application to the MIC-Kepler Problem, Journal of Geometry and Symmetry in Physics, 25 (2012) 57–75. [10] Kanazawa T., A Specific Illustration of Section derived from ∗-unitary Evolution Function, RIMS–Research Institute for Mathematical Sciences– Kˆ

kyˆ

uroku 1774 (2012) January, pp 172–191. [11] Moreno C. and Silva J., Star-Products, Spectral Analysis, and Hyperfunctions, In: Conf´ erence Mosh´ e Flato 1999, vol. 2, G.Dito and D.Sternheimer (Eds), Kluwer Academic Publishers, Netherlands 2000, pp 211–224. [12] Zachos C., Fairlie D. and Curtright T., Quantum Mechanics in Phase Space, World Scientific, Singapore 2005.

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