数理解析研究所講究録 1260 巻 2002 年 77-104

77
Boundary conditions at singular configurations of
three-body systems

岩井敏洋、 山岡英孝、 廣瀬亨
京都大学情報学研究科

Toshihiro Iwai, Hidetaka Yamaoka, and Toru Hirose
Department of Applied Mathematics and Physics
Kyoto University
Abstract
For some time, singular configurations have not been taken into account indealing with the center-0f-mass system of many particles. This is because thecenter-0f-mass system can be made into afiber bundle, provided the configurations of particles are restricted to non-singular ones. This article is an attempt
to analyzing the center-0f-mass system with configurations including the singularones, and we present the condition for which awave function must satisfy, should
the particle configuration become singular in finite time. This article should giveasimilar result to [2], but from acompletely different view point.

Contents
1Introduction
1.1 Historical background .
1.2 Organization of the article

2
2
3

2Review of the center-0f-mass system
2.1 Set-up of the problem
2.2 Avariety of configurations
2.3 Restriction to regular configurations.

4
4
4
5

3Taking singular configurations into account
3.1 Construction of colliding free particles
3.2 The total angular momentum of singular configuration
3.3 Fourier Analysis .
3.4 Projection and Transition Operators

7
7
9
10
11

78
4Wavefunctions around singular configurations
4.1 Two worked examples
4.2 Three particles in the nbd of alnear configuration
4.3 Remark

125Collision Singularity
5.1 Configuration in nbd of the collsion .
5.2 Decomposition of space of homogeneous polynomials
5.3 Implication of the decomposition
5.4 Example

19
20
22
23
23

13
15
19

1Introduction
It is well known that reducing adynamical system is closely related to asymmetry. One
famous example is the reduction of quantum system. One may consider an eigenvalue
problem for angular momentum, and would find that the eigenstate is aproduct of theradial and spherical harmonic functions. Then the original Schrodinger’s equation is
reduced to the one for finding the unknown radial function. That way, the system’s
degrees of freedom is reduced to alower value.
The above mentioned reduction is afine example of the heart of this subject.
The tw0-body angular momentum problem happens to be aparticular example of
the Fourier analysis on SO(3) for the three-body one.

1.1

Historical background

The idea of viewing many-particle center-0f-mass system as aprincipal fiber bundle isin fact fairly new. However, there had been many painful attempts to separate rotationfrom vibrations. It was Guichardet, in 1984 who first defined rotational and vibarational vectors, and showed that the rotation can not be separated from vibrations,using the connection theory, provided that the particles do not collde nor align in astraight line. As aconsequence, it was shown that any rotation angle of the particles
were possible, purely by vibrations. In 1987, one of the authors (T.I) showed that if thesystem is restricted to regular configurations, it can be described in the bundle picture,in the sense that the behavior of the system can be discussed on associate vector bundle2

79of the configuration space. Further, in 1990 and 1991, Montgomery defined the falling
cat problem, and showed the cat constructed from point particles, can land on her feetwhen she is launched in air. This was done by viewing the system from abundle picturewith non-holonomic constraints. However one must realize that all above articles dealt
with center-0f-mass systems with the free action of 50(3). The consequence of thisautomatically excludes the cases when particles collide at the origin, or are aligned inaline. These mentioned cases are called the singular configurations, and the questions
on how the mechanics is set up on such configurations are still unsolved. This article
deals with such aquestion.

1.2

Organization of the article

This article is organized as follows. After this chapter, in section 2we give abrief reviewof the center-0f-mass system. The definition of regular and singular configurations
is given in this chapter, and gives abrief idea as to what can be done with regularconfigurations. The most of the facts mentioned in the chapter are already investigatedand cooked by past researchers. For abeginner to this subject, we suggest it would beof great help to start from reading [3]. One may wonder whether it would be possible
for asystem to come to asingular configuration from aregular one in finite time.
Demonstration of this is given in section 3. It further gives us amotivation to see thebehavior of aquantum system instead of classical ones. Also awavefunction can be
written in atidy form by introducing something called the projection and transition
operators. These operators have remarkable properties, and they are described in thissection, too. In section 4, we work through two examples. The main theme in thissection is to write wavefunction in terms of local coordinates, given that it is analytic
and that can be expanded into power series around asingular configuration. One of
the main results of the article is also describen in this section. Section 5deals with the
expansion of the wavefunction when triple collision takes place, stating the other main
result.

2Review of the center-0f-mass system
As is widely discussed, the center-0f-mass system has been very well studied by pastmathematicians and physicists. In this section we shall concentrate on some of the we 1

3

80
known facts.

2.1

Set-up of the problem

Out of particles, let
be the position ofparticle, andbe its respective
mass. Then we define the configuration space of the-mass system to be$n$$x_{j}\in \mathbb{R}^{3}$$j\mathrm{t}\mathrm{h}$$m_{j}$$\mathrm{c}\mathrm{e}\mathrm{n}\mathrm{t}\mathrm{e}\mathrm{r}-\mathrm{o}\mathrm{f}$$X_{0}=\{x=(x_{1}, \ldots, ox_{n})|x_{j}\in \mathbb{R}^{3}$,$\sum_{j=1}^{n}m_{j}x_{j}=0\}$.

(1)

The translational motion of particles are deliberately removed, and the consequence of
this appears in the constraint(2)$\sum_{j=1}^{j=n}m_{j}ox_{j}=0$

in Xo, in order to fix the center of mass at the origin. This configuration space alowsagroup action of SO{3), and this is given by$SO(3)\cross X_{0}arrow X_{0}$2.2,(g, $x)\mapsto gx=(\mathrm{g}$, \ldots gxn).

(3)

Avariety of configurations

When n particles are placed in , we may consider how the particles are arranged inthat space. Therefore, we define the spread of particles to be$\mathbb{R}^{3}$.

(4)$\dim F_{x}=0$:all particles at the origin(5)$\dim F_{x}=1$:particles aligned in aline(6)$\dim F_{x}=2$:particles on aplane(7)$\dim F_{x}=3$:particles scattered in space.

(8)$x_{n}\}$\ldots ,$F_{x}=\mathrm{s}\mathrm{p}\mathrm{m}\{x_{1},$,x$\in X_{0}$ThenProm this, we see that there are different configurations for different dimensions of4$F_{x}$.

81$\dim F_{x}\bullet=0$$[$

dirn $F_{x}=1$$\dim F_{x}=3$

Figure 1: The diagram shows the interpretations of the dimensions of . Prom left,
corresponds to 1) all particles at the origin, 2) particles aligned in aline, 3) particles
on aplane, 4) particles scattered in space.
$F_{x}$

This suggests us to break the configuration space$X_{0}$

up into subspaces$Q\mathrm{o}$,$Q_{1}$, and$\dot{Q}$;(9)dirn $F_{x}=0$ } $=\{0\}$$Q_{0}$$=${$Q_{1}$$=$$\{x\in X_{0}|\dim F_{x}=1\}$(10)$\dot{Q}$

$=$

{

}.

(11)

$x\in X_{0}|$$x\in X_{0}|$dirn$F_{x}\geq 2$

are the configurations for colliding particles and aligned
and
The members ofparticles respectively, and such configurations are called the singular configurations.
$Q_{1}$$Q_{0}$

Restriction to regular configurations

2.3of 50(3) is the subgroup of SO (3) which
For afixed $x\in X_{0}$ , an isotropy subgroup, we havemakes invariant under the action of $g\in G_{x}$ . For different members ofdifferent isotropy subgroups;$G_{x}$$X_{0}$$x$(12)$G_{x}\cong\{\begin{array}{l}SO(3)(x\in Q_{0})SO(2)\{e\}(x\in Q_{1})(x\in\dot{Q})\end{array}$, $G_{x}\cong\{e\}$ , so action 50(3) is free. Therefore $\dot{Q}=\{x\in X|\dim F_{x}\geq 2\}$ isForbundle through the equivalence relation
made in to an$x\in\dot{Q}$$\sim$$5\mathrm{O}(3)$$y\sim x\Leftrightarrow\exists g\in SO(3)$

for ,
$x$

$y\in\dot{X}$

$s.t$

.

$y=gx$(13), with the quotient map$\pi$: $\dot{Q}arrow M:=\dot{Q}/SO(3)--\{[x]|x\in X_{0}\}$ .

5

(14)

82
Then we shall define open set in $M$ by the following statement. Asubset of $M$ isan open subset of $M$ for the quotient topology if and only if its inverse image
$U$

$U$

$\pi^{-1}(U)=\{x\in X_{0}|[x]\in U\}$(15)

is an open subset of X$. For agiven $U\subset M$ , its inverse image has alocal structure$\pi^{-1}(U)\cong U\cross G$ . We define
section in $\pi^{-1}(U)$ to be the continuous map$\sigma$$\sigma$: $M\supset Uarrow X_{0}$ .

(16)Then further, we have an associated bundles$E_{l}=\dot{Q}\mathrm{x}_{l}H^{l}:=\dot{Q}\cross H^{l}/\sim$(17)$(x, z)\sim(gx, D^{l}(g)z)\in\dot{Q}\cross H^{l}$(18)

defined throughwhere$\prime H^{l}$is the representation space of SO(3) with$\dim H^{l}=2l+1$

.

(19)

Like before, one can define section of the associated vector bundle$\sigma_{l}$:$\dot{Q}/SO(3)arrow\dot{Q}\mathrm{x}_{l}\mathcal{H}^{l}$

,

(20)

being interpreted as areduced state, because the reduced state is in one to one correspondence with .
$\sigma_{l}$3Taking singular configurations into account
The previous argument of vector bundles in section 2was all good provided that theaction of SO(3) was free. However this is not the case for the singular configurations.
Such configurations have
. In this section we shall consider configurations
including the singular ones.
Before we start, one might be led to anatural question whether or not it would bepossible for anatural dynamical system to come to asingular configuration in finite
time. To answer the question, we shall manulally construct acolinear configuration for
the free particles$G_{x}\not\cong\{e\}$

6

83
3.1

Construction of colliding free particles

For simplicity, we shall only consider three free particles in space, that is, for each,is linear in time . Since the configuration space can be, $j=1,2,3$ ,identified with set of Jacobi vectors, from now on we shall work in terms of Jacobi
vectors. The Jacobi vectors are given by the following formulae$x_{j}\in \mathbb{R}^{3}$$x_{j}$$t$$x_{j}$(21)$r_{1}$$=$$\sqrt{\frac{m_{1}m_{2}}{m_{1}+m_{2}}}(x_{2}-x_{1})$$r_{2}$$=$$\sqrt\frac{m_{1}m_{2}(m_{1}+m_{2})}{m_{1}+m_{2}+m_{3}}(x_{3}-\frac{m_{1}x_{1}+m_{2}x_{2}}{m_{1}+m_{2}})$(22), so we see that they
. Jacobi vectors are just alinear combination ofare also linear with respect to . This gives the form of Jacobi vectors;for$r_{j}\in \mathbb{R}^{3}$$x_{j}’ \mathrm{s}$$t$$r_{1}$$=$$a_{1}+b_{1}t$(23)$r_{2}$$=$$a_{2}+b_{2}t$(24). Here assume that neither, and,for constant vectors
vanishes. Then when particles form acolinear shape, we have$a_{j}$$t\in \mathrm{R}$$b_{j}\in \mathbb{R}^{3}$for some$r_{1}\cross r_{2}=0$$r_{1}$

nor$r_{2}$$t_{c}\in \mathbb{R}$

(25)

.

26)

or equivalently$r_{1}=\lambda r_{2}$for some$t_{c}$,A$\in \mathbb{R}$

Here we can consider two cases. Case one is when $r_{1}\cross r_{2}=0$ and case two when
. If it is the former case, then it is trivial to see that if the plane spanned
do not coincide, then we do have acolinear
by , ,and that by a2) ,. Hereconfiguration. Therefore we shall consider the latter case, that is,
is in the plane spanned
we note that if there is acolinear shape, then the loci of. We shall call this plane . Similarly, the loci of should be in theby , ,. Again we shall call this plane .
plane spanned by , ,We are now in position to construct acolinear configuration manually. In this$b_{1}=0$ ,so that they satisfy
example, we letandcoincide. Chooseand$r_{2}\cross r_{2}\neq 0$$a_{1}$$O$$O$$r_{1}(t)$$r_{2}(t)$$r_{1}\cross r_{2}\neq 0$$r_{1}$$a_{1}$$O$$r_{1}(t_{c})$$P_{1}$$a_{2}$$P_{1}$$O$$r_{1}$$P_{2}$$r_{2}(t_{c})$$P_{2}$$a_{1}$

7$a_{2}$$a_{1}\cdot$

84

Figure 2: The lines show the loci of two Jacobi vectors with respect to time. The
diagram on the left describes how two skewing loci can come to aconfiguration which
gives parallel Jacobi vectors. Similarly the diagram on the right shows it in the three
dimensional manner. Planes drawn are and . This shows that the intersection of
the planes must be in the direction of parallel Jacobi vectors.
$P_{1}$$P_{2}$

and a2 $b_{2}=0$ . This means that we may assume thatandare perpendicular,
without loss of generality. For given
and , the plane defined by 01, , O is$a_{j}$$a_{1}$$a_{2}$$a_{2}$$r$ $=\alpha_{1}a_{1}+\alpha_{2}a_{2}$So for chosen
andlie on the plane(s)$a_{1}$$a_{2}$, we then choose$\hat{b}_{1}$.

$(a_{1}\cross a_{2})$$b_{j}$$b_{1}$$=0$$\alpha_{1}$and$\hat{b}_{2}$.

,$b_{2}$$\alpha_{2}\in \mathrm{R}$

.

(27)

such that the loci$r_{1}(t)$, and$r_{2}(t)$(28)$(a_{1}\cross a_{2})=0$

If we set
, then at the linear configuration, we must have $r_{1}(t_{c})=r_{2}(t_{c})$ , whereis the time at which colinear configuration takes place. Now note that we may change
the time scales of loci of Jacobi vecotors in the folowing way$\mathrm{A}=1$$t_{c}$$r_{1}=a_{1}+|b_{1}|\hat{b}_{1}t=a_{1}+\hat{b}_{1}t_{1}$(29)$r_{2}=a_{2}+|b_{2}|\hat{b}_{2}t=a_{2}+\hat{b}_{2}t_{2}$(30)Then, now we solve for the time at which these loci intersect. If$a_{1}+\hat{b}_{1}t_{1}^{*}=a_{2}+\hat{b}_{2}t_{2}^{*}$

8.

$t^{*}$

is such time, then
(31)

85
Now if we set$t^{*}$

such that it satisfies,(32)$\frac{|\hat{b}_{1}t_{1}|}{|\hat{b}_{2}t_{2}|}=\frac{\mathrm{s}\mathrm{p}\mathrm{e}\mathrm{e}\mathrm{d}\mathrm{o}\mathrm{f}r_{1}}{\mathrm{s}\mathrm{p}\mathrm{e}\mathrm{e}\mathrm{d}\mathrm{o}\mathrm{f}r_{2}}=\frac{t_{1}^{*}}{t_{2}}*=\frac{|b_{1}|}{|b_{2}|}$then we have particles aligned in straight line. If we further set
then the Jacobi vectors in this case are$r_{1}$$=$$r_{2}$$=$$b_{1}$

to be aunit vector,(33)$a_{1}+\hat{b}_{1}t$$a_{2}+ \frac{t_{2}^{*}}{t_{1}}*\hat{b}_{2}t$

.

(34)

The configurations corresponding to Jacobi vectors (33-34) are regular except for when$t$$=t_{1}^{*}$.

3.2

The total angular momentum of singular configuration

For free particles, the total angular momentum of the system is given by
$L$

$=$

$\sum_{j=1}^{2}r_{j}\cross\frac{dr_{j}}{dt}$

(35)

–

$(a_{1}+b_{1}t)\cross b_{1}+(a_{2}+b_{2}t)\cross b_{2}$

(36)

$=$

$a_{1}\cross b_{1}+a_{2}\mathrm{x}$(37)$b_{2}$

constant.

$=$

(38)

When singular configuraion undergoes, the directions ofThen the angular momentum about the colinear axis is$n\cdot L$$r_{1}(t_{c})$and$r_{2}(t_{c})$coincide.

$=$$n$ $\cdot(a_{1}\cross b_{1}+a_{2}\cross b_{2})$

(39)

$=$

$n\cdot(a_{1}\cross b_{1})+n\cdot(a_{2}\cross b_{2})$

(40)

$=$

$r_{1}(t_{c})\cdot(a_{1}\cross b_{1})+\lambda r_{2}(t_{c})\cdot(a_{2}+b_{2})$

(41)

$=$

$0$

,

(42)

where
$n=r_{1}(t_{c})$

9

.

(43)

86
This indicates that any free particles, at its singular configuration would have azero
angular momentum about its axis. It then makes us wonder if this result would be thesame for the quantum center-0f-mass system.
These particles under no external force may travel in straight lines in Xo. The
corresponding motions in $X_{0}/SO(3)$ can be represented in local coordinates,
where$(\eta_{1}, \eta_{2}, \eta_{3})$$\eta_{1}$y72$=r_{1}^{2}-r_{2}^{2}=(a_{1}^{2}-a_{2}^{2})+(b_{1}^{2}-b_{2}^{2})t^{2}$(44)$=2r_{1}\cdot$ $r_{2}=2(a_{1}\cdot a_{2}+b_{1}\cdot b_{2}t^{2})$

(45)

$=$

$\eta_{3}$$2|r_{1}\mathrm{x}r_{2}|=2[a_{1}\cross a_{2}+b_{1}t\cross a_{2}+a_{1}\cross b_{2}t+b_{1}\cross b_{2}t^{2}]$The singular configuration, as one can see, corresponds to the plane

3.3$\eta_{3}=0$

.

(46)

.

Fourier Analysis

From what we know we have the group action(47)$SO(3)\cross X_{0}arrow X_{0}$We shall consider aquantum system with wave functions inresentation : $SO(3)\cross L^{2}(X_{0})arrow L^{2}(X_{0})$ , defined by$L^{2}(X_{0})$, and unitary rep-$U$$(g, f)\mapsto U(g)f$;$(U(g)f)(x)=f(g^{-1}x)$

.

(48)

As what the unitary representation show, is defined on the configuration space Xq.
However, suppose that $x\in X_{0}$ is fixed, and let us define$f$$f_{x}(g):=f(gx)$ ,(49)

then we can see that is afunction on 50(3). Later if we require it to be that of ,then we can simply put $g=e$ , and we revert to the function evaluated at the point$x\in X_{0}$ .
Now, since f can be seen as afunction on(3), we can apply the Peter-Weyltheorem, which allows us to expand functions on compact Lie groups. Then the Fourier
expansion of is expressed as$f_{x}$$X_{0}$$f$$f$$f(gx)= \sum_{l=0}^{\infty}(2l+1)\sum_{|m|,|n|\leq \mathrm{t}}\prime D_{mn}^{l}(g)\langle D_{mn}^{l}, f_{x}\rangle_{SO(3)}$

10

,

(50)

87
where (,$\rangle_{SO(3)}$denotes the inner product for functions on 50(3)$\langle D_{mn}^{l}, f_{x}\rangle_{SO(3)}=\int_{SO(3)}\overline{D}_{mn}^{l}(k)f(kx)d\mu(k)$

.

(51)

Above expansion holds for all $g\in SO(3)$ , so for $g=e$ in particular, we have$f(x)= \sum_{l=0}^{\infty}(2l+1)\sum_{|m|\leq l}\int_{SO(3)}\overline{D}_{mm}^{l}(h)f(hx)d\mu(h)$The $d\mu(h)$ is the invariant measure for
local coordinates, we have

$5\mathrm{O}(3)$

where

3.4

$(\theta, \phi, \psi)$(52). When it is expressed in terns of thewith$d\mu(h)=\sin\theta d\theta d\phi d\psi/2\pi^{2}$.

$\int_{SO(3)}d\mu(h)=1$

,

(53)

are the Euler angles.

Projection and Transition Operators

Here, in order to write equation (52) in aneater manner, we introduce operators$(P_{n}^{l}f)(x)$$=$$(2l+1) \int_{SO(3)}D_{nn}^{l}(h)f(h^{-1}x)d\mu(h)$$(P_{mn}^{l}f)(x)$$=$$(2l+1) \int_{SO(3)}D_{mn}^{l}(h)f(h^{-1}x)d\mu(h)$$P^{l}$

(54)

.

(55)

Equations (54-55) have the following properties;$(P_{n}^{l})^{\uparrow}=P_{n}^{l}$$P_{nm}^{l}(P_{nm}^{l})^{\mathrm{t}}=P_{n}^{l}$

,

,

(56)

$P_{n}^{l}P_{m}^{l}=\delta_{nm}P_{n}^{l}$$P_{nm}^{l}P_{m}^{l}=P_{n}^{l}P_{nm}^{l}$,$P_{nm}^{l}$:${\rm Im} P_{m}^{l}arrow{\rm Im} P_{n}^{l}$(57)

The most striking properties are the last equations from (56) and (57). Equation (56)twice has the same effect as applying it only once, whereas
says that applying
to that of . For the above
shifts apoint inequation (57) says the map
reasons, we shall from here call the operators in (54) and (55) projection and transition
operators respectively. Further, these operators can be written in aslightly different$P_{n}^{l}$${\rm Im} P_{m}^{l}$$P_{nm}^{l}$$n$manner since$\int_{SO(3)}D_{mn}^{l}(h)f(h^{-1}x)d\mu(h)=\int_{SO(3)}\overline{D}_{nm}^{l}(k)f(kx)d\mu(k)$

11

.

(58)

88
by change of variables $h=k^{-1}$ . Then we have$(P_{mn}^{l}f)(x)=(2l+1) \int_{SO(3)}\overline{D}_{nm}(k)f(kx)d\mu(k)$So Fourier series expansions (52) in

$x$

is;

$f(x)= \sum_{l=0}^{\infty}\sum_{|m|\leq l}(P_{m}^{l}f)(x)$Now define map$E_{m}^{l}$

(59)

.

(60)

: $L^{2}(X_{0})arrow H^{l}\otimes L^{2}(X_{0})$ ;$E_{m}^{l}f= \frac{1}{\sqrt{2l+1}}$(61)$(\begin{array}{l}P_{l-1m}^{l}fP_{lm}^{l}f\vdots P_{-lm}^{l}f\end{array})$

Satisfies the condition$(E_{m}^{l}f)(hx)=D^{l}(h)(E_{m}^{l}f)(x)$,$h\in SO(3)$

.

(62)

This means that the -valued functions
are -equivariant functions. Here we
did not specify nor say anything about form of or the point , and therefore thiscondition is true for any configuration including the singular ones.
$H^{l}$$E_{m}^{l}f$$\rho^{l}$

$f$

$x$

4Wavefunctions around singular configurations
In this section of the article, we shall try to write (60) and hence (52) in terms of local
sections of Xo. However, for the simplicity, we shall consider 3-body system.
We have already seen that the space of center-0f-mass system is identified with thespace of coulpled Jacobi vectors (21-22) in terms of vector spaces$X_{0}\cong\{(r_{1}, r_{2})|r_{j}\in \mathbb{R}^{3},j=1,2\}$At singular configuration, anddependent. Further, we define$r_{1}$$r_{2}$.

(63)

become parallel to each other, and they are linearly$M:=X_{0}/SO(3)\cong\{ox \in \mathbb{R}^{3}|x_{3}\geq 0\}$

12

.

(64)

89
includes the singular configu. The whole ofthe internal space of the whole ofrations. As already stated, $M$ fails to be amanifold. However, we see that
$X_{0}$$X_{0}$$\pi(\partial\dot{X})=\partial\pi(\dot{X})$

(65)

,

and so the boundary gets mapped to the boundary. We say that $M$ is amanifold withboundary.
Away from the boundary, there are three coordinate variables, and we define themto besuch that$(r_{1}, r_{2}, \varphi)\in M$$r_{1}=|r_{1}|$,$r_{2}=|r_{2}|$,$r_{1}\cdot r_{2}=r_{1}r_{2}\cos\varphi$

.

(66)

The boundary of $M$ is identified with $\varphi=0$ . Further, choose local section$Marrow X_{0}$ :$(r_{1}, r_{2}, \varphi)\mapsto\zeta=(r_{1}e_{3}, r_{2}e^{\varphi R(e_{2})}e_{3})$,$R:\mathbb{R}^{3}arrow so(3)$then any point in $\pi^{-1}(U)$ is expressed as $x=g\zeta$ , and expansion of$f$,$P_{nm}^{l}$$(E_{m}^{l}f)(g\zeta)=D^{l}(g)(E_{m}^{l}f)(\zeta)$

4.1.

$\zeta$$U\subset$

(67)

(68)
is

(69)

$f(g()= \sum_{l=0}^{\infty}\sum_{|m|,|n|\leq l}D_{mn}^{l}(g)(P_{nm}^{l}f)(\zeta)$

The equivariance condition (62) which is equivalent for the:

takes the form:$f(g \zeta)=\sum_{l=0}^{\infty}(2l+1)\sum_{|m|,|n|\leq l}D_{mn}^{l}(g)\langle D_{mn}^{l}, f_{\zeta}\rangle_{SO(3)}$where $f_{\zeta}(k)=f(k\zeta)$ for $k\in SO(3)$ . Further, this in terms of$\sigma$

expansion is(70)

Two worked examples

It comes very natural for us to investigate the behavior of wavefunctions at the singularand near-singular configurations. Here we shall choose sections to represent these
configurations. Of the two examples, first, we shall look into the case when we have acolinear singular configuration.

13

90

Figure 3: Section for the cohnear singular configuration (left) and that for the nearsingular configuration (right). Notice the decrease in dimension in the singular configuration.
The boundary of M can be represented by local coordinates
consider the following section$(r_{1}, r_{2})$$\partial M\ni(r_{1}, r_{2})\mapsto\zeta_{0}=(r_{1}e_{3}, r_{2}e_{3})\in X_{0}$and here we note that the isotropy subgroup of SO(3) atg$=e^{tR(e_{3})}$,$D_{nm}^{l}(g)=e^{-\dot{\iota}tn}\delta_{nm}$,$\zeta_{0}$t, and we shall(71)

and it’s representation are$\in \mathrm{R}$(72)respectively. Now, as already mentioned, the equivariance condition (70) holds for any
g $\in SO(3)$ . So in particular, for g, the action of
on f is$P_{nm}^{l}$$\in G_{\zeta_{0}}$$(P_{nm}^{l}f)(\zeta_{0})$

$=$

(73)

$(P_{nm}^{l}f)(g\zeta_{0})$$\sum_{|n|\leq \mathrm{t}},D_{nn’}^{l}(g)(P_{nm}^{l},f)(\zeta_{0})$(74)$\sum_{|n|\leq \mathrm{t}},e^{-\dot{|}nt}\delta_{nn’}(P_{n’m}f)(\zeta_{0})$

(75)

$=$

$=$

$=$

$e^{-\dot{|}nt}(P_{nm}^{l}f)(\zeta_{0})$

.

(76)

We used two facts here. One is that Co is invariant under. The other is theequivariance. Furthermore the bove equations (73-76) is just the calculation of (70)component wise, since the matrix
is diagonal in this case. The consequence of thecondition is that$g\in G_{\zeta_{0}}$$D^{l}$$(P_{nm}^{l}f)(\zeta_{0})=0$

14

if n

$\neq 0$

.

(77)

91
We have the physical interpretation on the result (77) as follows: For acolinear configuration with agiven angular momentum around the axis of alignment, the wave
function must vanish for this configuration, unless the angular momentum is zero. In
other words, we have no probablity of finding colinear configuration if there is anon$n$zero angular momentum round that axis.
Using this result, we may further calculate (52) by letting$\zeta$tend to$\zeta_{0}$. Then we

obtain$f(g\zeta_{0})$$=$$\sum_{l=0}^{\infty}\sum_{|m|,|n|\leq l}D_{mn}^{l}(g)(P_{nm}^{l}f)(\zeta_{0})$

(78)

$=$

$\sum_{l=0}^{\infty}\sum_{|m|\leq l}D_{m0}^{l}(g)(P_{0m}^{l}f)(\zeta_{0})$

(79)

$=$

$\sum_{l=0}^{\infty}\sum_{|m|\leq l}\sqrt{\frac{4\pi}{2l+1}}\overline{\mathrm{Y}}_{lm}(ge_{3})(P_{0m}^{l}f)(\zeta_{0})$

(80)

.

This implies that the wave functions for linear triatomic molecues can be described
determined by rank
, in the subspace ofin terms of local coordinates,(81) $r_{2})=1$ . As aside note, recall that tw0-body system can be characterized by$X_{0}$$(\theta, \phi, r_{1}, r_{2})$considering the section(81)$r\mapsto\zeta_{0}=re_{3}$Then Fourier expansion is$f(g \zeta_{0})=\sum_{l=0}^{\infty}\sum_{|m|\leq l}\sqrt{\frac{4\pi}{2l+1}}\overline{\mathrm{Y}}_{lm}(ge_{3})(P_{0m}^{l}f)(\zeta_{0})$

.

(82)

is dependent only on . The system is then reduced to wavefuncActuallytion for the radial component.
$(P_{0m}^{l}f)(\zeta_{0})$

4.2

$r$

Three particles in the nbd of alinear configuration

As before, we shall choose asection for configurations very close to the singular one.
.
For point which lies in the interior of $M$ , we may write in coordinates
Then we may choose the section to be$(r_{1}, r_{2}, \varphi)$

$p$

$p$

$(r_{1}, r_{2}, \varphi)\mapsto\zeta=(r_{1}e_{3}, r_{2}e^{\varphi R(e_{2})}e_{3})\in X_{0}$

15

.

(83)

92
Because we are trying to express in terms of the local coordinates, we shall write out
explicitly $k\in SO(3)$ in Euler angles$k=e^{\phi’R(e_{3})}e^{\theta’R(e_{2})}e^{\psi’R(e_{3})}:=k_{0}e^{\psi’R(e\mathrm{a})}$

.

(85)

Here we split $k\in SO(3)$ into two parts–namely, because we are
andconsidering an expansion about theaxis. Then as before, we may write $x\in\pi^{-1}(U)$$e^{\psi’R(e\mathrm{a})}$$k_{0}$$e_{3}$

as$x=k\zeta=k_{0}(r_{1}e_{3}, r_{2}e^{\psi’R(e_{3})}e^{\varphi R(e_{2})}e_{3})$

,

(85)

and recall that.

(86)

$D_{mn}^{l}(k)=e^{-\dot{|}m\phi’}d_{mn}^{l}( \oint)e^{-\cdot n\psi’}$and we shall attempt to write (60) using those introduced variables. First we shall look
into the calculation ofthat appears in (60).
$P_{nm}^{l}f$$(P_{nm}^{l}f)(\zeta)$$=$$(2l+1) \int_{SO(3)}\overline{D}_{mn}^{l}(k)f(k\zeta)d\mu(k)$$=(2l+1) \frac{1}{\pi}\int_{S^{2}}e^{\dot{|}m\phi’}d_{mn}^{l}(\theta)\sin\Psi d\Psi d\phi’$$\cross\frac{1}{2\pi}\int_{0}^{2\pi}e^{\dot{|}n\psi}f(r_{1}k_{0}e_{3}, r_{2}k_{0}e^{\mathrm{V}^{lR(e_{3})}}e^{\varphi R(e_{2})}e_{3})d\psi’$Observe that the last integral in (87) is aFourier coefficient of$U(k_{0}^{-1})f$

(87)

;

$c_{-n}(k_{0;}r_{1}e_{3}, r_{2}e^{\varphi R(e_{2})}e_{3})$$= \frac{1}{2\pi}\int_{0}^{2\pi}e^{\dot{|}n\psi’}(U(k_{0}^{-1})f)(r_{1}e_{3}, r_{2}e^{\psi’R(e_{3})}e^{\varphi R(e_{2})}e_{3})d\psi’$

.

(85)

Then (87) comes down to$(P_{nm}^{l}f)(\zeta)$$=$$(2l+1) \int_{\mathrm{S}O(3)}\overline{D}_{mn}^{\mathrm{t}}(k)f(k\zeta)d\mu(k)$

(89)

$=$

$\frac{1}{\pi}\int_{S^{2}}e^{\dot{|}m\phi’}d_{mn}^{l}(\theta)c_{-n}(k_{0};r_{1}e_{3}, r_{2}e^{\varphi R(e_{2})}e_{3})\sin\theta’d\theta’d\phi’$(60)

We would like to take acloser look at these Fourier coefficients. To do so, we takeadifferent set of Local coordinates, by setting$(r_{1}, r_{2}, \psi’, \varphi)\mapsto(r_{1},\xi_{1},\xi_{2},\xi_{3})$

16

93

Figure 4: Changing the local coordinates. , , are changed into cartesian coordias thenates in the standard way. Further change of variable gives as viewing
on the plane.
complex plane, and as the modulus of$r_{2}$$\phi$$\psi’$$\xi_{1}-\xi_{2}$$z$$r_{2}$$\rho$$\xi_{1}=r_{2}\sin\varphi\cos\psi’$,$\xi_{2}=r_{2}\sin\varphi\sin\psi’$,$\xi_{3}=r_{2}\cos\varphi$

,

(91)

and we further put$z=\rho e^{i\psi’}$,$\rho=r_{2}\sin\varphi$

.

(92)

The geographical interpretation for the lately introduced variables in (92) is that isplane, and is the complex variable when
the length of when projected onto
is identified as C.
the plane spanned byat colinear singular conand
Now, assume the wave function is analytic in
into power series in the following
figuration $\xi_{1}=\xi_{2}=0$ . We may expand$\rho$$z$$\xi_{1}-\xi_{2}$$r_{2}$$(\xi_{1}, \xi_{2})$$\xi_{1}$$\xi_{2}$$U(k_{0}^{-1})f$way$(U(k_{0}^{-1})f)(r_{1}e_{3}, r_{2}e^{\psi’R(e_{3})}e^{\varphi R(e_{2})}e_{3})$(93)$= \sum_{p,q\geq 0}^{\infty}c_{\mathrm{p}q}(k_{0;}r_{1}, \xi_{3})z^{p}\overline{z}^{q}$(94)$= \sum_{p,q\geq 0}^{\infty}c_{pq}(k_{0;}r_{1}, \xi_{3})\rho^{\mathrm{p}+q}e^{i(p-q)\psi’}$$= \sum_{n’=-\infty}^{\infty}e^{in’\psi’}\sum_{j=|n’|}^{\infty}\dot{\oint}c_{L_{\frac{+n’}{2}}L_{\frac{-n’}{2}}},(k_{0;}r_{1}, \xi_{3})$where$c_{pq}$,(95)

are the coefficients of the power series. This implies that$c_{n’}(k_{0;}r_{1}e_{3}, r_{2}e^{\varphi R(e_{2})}e_{3})= \sum_{j=|n’|}^{\infty}j_{\mathrm{C}_{L}}+_{\frac{n’}{2}2arrow-n’},\cdot(k_{0;}r_{1}, \xi_{3})$

17

(96)

94
with $n’=-n$ . So
lowest order ;$(P_{nm}^{l}f)(\zeta)$

turns out to be apower series of$\rho$

having terms of the$|n|$$(P_{nm}^{l}f)(\zeta)$

$=$

$=$

$\frac{2l+1}{\pi}\int_{S^{2}}e^{\dot{|}m\phi’}d_{mn}^{l}(\theta’)$(97)$\cross c_{-n}(k_{0;}r_{1}e_{3}, r_{2}e^{\varphi R(e_{2})}e_{3})\sin\theta d\emptyset d\phi’$(98)$\frac{2l+1}{\pi}\int_{S^{2}}e^{\dot{|}m\phi’}d_{mn}^{l}(\Psi)$(99)$\cross\sum_{j=|n|}^{\infty}d\mathrm{C}.\llcorner^{\underline{n}i_{\frac{-n}{2}(k_{0;}r_{1},\xi_{3})\sin\nu d\Psi d\phi’}}+2$

’

$=$

The

$\tilde{c}_{mn}$(100)$\frac{2l+1}{\pi}\sum_{j=|n|}^{\infty}j_{\tilde{\mathrm{C}}_{mn_{22}^{arrow-np\pm\underline{n}}}},\cdot,(r_{1}, \xi_{3})$are the outcome of the integration of,over$c_{i_{\frac{+n}{2}}i_{\frac{-n}{2}}}$$\theta’$and$\phi’$. Hence theexpansion of $f$ is;$f(g\zeta)$$=$$\sum_{l=0}^{\infty}\sum_{|m|,|n|\leq l}D_{mn}^{l}(g)(P_{nm}^{l}f)(\zeta)$$=$$\frac{2l+1}{\pi}\sum_{l=0}^{\infty}\sum_{|m|,|n|\leq l}D_{mn}^{l}(g)$

(101)

(102)

$\cross\sum_{j=|n|}^{\infty}j\tilde{c}_{mn^{i_{\frac{-n}{2}}i_{\frac{+n}{2}}}},,(r_{1},\xi_{3})$This gives aboundary condition on wave function in the sense that when expanded
into power series, the function starts with the power. As aside note, it must be
stressed that an increase of
by one corresponds to adoulbe increase in , which isin consistent with the result given in the reference [2].
The associated vector bundle$\rho^{|n|}$

$|n|$

$j$

$E_{l}=\dot{X}_{0}\mathrm{x}_{l}H^{l}$(103)

defined through (18) has alocal structure. We recall that the equivariant
functions defined in (62) is in one to one correspondence with local sections in theassociated vector bundle. Prom (62), we know that at the colinear configuration, theall but one components ofvanishes. This means the sections for the corresponding$U\cross H^{l}$$E_{m}^{l}f$

18

95

Figure 5: Sketch graphs show components of alocal section in the associated fiber
bundle. In the open set containing the boundary of , all but one components of thesection tend to zero as they approach the boundary. For different components, thedecay speed is dependent on its corresponding angular momentum value.
$\dot{X}$equivariant functions must approach zero, because they are continuous. In other words,
towards the singular configuration, all (apart from the one which corresponds to $n=0$ )components of the section must somehow decay to zero. How they decay can be seenfrom the power series expansion in (100). The bigger the , the faster the decay.
$|n|$

4.3

Remark(corresponds toIf we letterms of the right-hand side of$\zetaarrow\zeta_{0}$$\varphiarrow 0$$U(k_{0}^{-1})f$or ), thenvanish if $n’\neq 0$ :$\xi_{3}arrow r_{2}$

$\pi$

and

$\rhoarrow 0$, so that the$(U(k_{0}^{-1})f)(r_{1}e_{3}, r_{2}e^{\psi R(e_{3})}e^{\varphi R(\mathrm{e}_{2})}e_{3})$$= \sum_{n’=-\infty}^{\infty}e^{in’\psi’}\sum_{j=|n’|}^{\infty}j_{\mathrm{C}_{arrow+n’arrow-n’}},(k_{0};r_{1}, \xi_{3})22$$arrow \mathrm{c}_{0,0}(k_{0;}r_{1}, \xi_{3})$

.

(104)
(105)

This implies further that;if$(P_{nm}^{l}f)(\zeta_{0})=0$$n\neq 0$

,

(106)

which gives the result that is consistent with the one derived in (77).

5Collision Singularity
In section 4, we discussed one type of singularity, that is when the particles are aligned
in astraight line. Here we consider the case when all three particles collide at the origin.

19

96
Let $(r_{1}, r_{2})=(0,$ 0). Then the corresponding end of section is $\zeta_{0}=(0,$ 0). We recall
that (70) holds for any and any g $\in 50(3)$ . Therefore we have for g $\in G_{\zeta 0}\cong 50(3)$$\zeta$$(E_{m}^{l}f)(0)=D^{l}(g)(E_{m}^{l}f)(0)$

.

(107)

This in terms of the local coordinates gives$(P_{nm}^{l}f)(0)=e^{-\dot{|}n\phi’} \sum_{|n|\leq l},d_{nn’}^{l}(\Psi)e^{-\dot{|}n’\psi}(P_{nm}^{l}f)(0)$and this holds for all ,invariant. Noting that$\Psi$$\psi’$$D_{m}^{l}$

,

,

(108)

. (107) says that the linear subspace spanned byis airreducible representation, we conclude that$\phi’$$(E_{m}^{l}f)(0)=0$,$l\neq 0$.

$E_{m}^{l}f$

is

(109)

For $l=0$ we simply have the identity equation since $D^{l}=1$ . The implication of thisis that in general we can not say that the wavefunction is zero, and therefore as aconsequence of this, there is apossible collision after all. The whole purpose of this is
to try to express the wavefunction as aFourier expansion. The wavefunction which is
expanded at the point 0is$f(0)$

5.1$=$$\sum_{l=0}^{\infty}\sum_{|m|,|n|\leq l}D_{mn}^{l}(g)(P_{nm}^{l}f)(0)$–$(P_{0,0}^{0}f)(0)$

.

(110)

(107)

Configuration in nbd of the collision

Since the center of mass of the system is fixed at the origin, we must have $ox_{1}=$$x_{2}=x_{3}=0$ when the collision takes place. In terms of Jacobi vectors (21-22), thiscorresponds to $r_{1}=r_{2}=0$ . The colliding configuration corresponds to$\eta_{3}=0$ in terms of the internal local
coordinates. There is no reason why we can notexpand the wavefunction in terms ofas before. The boundary of the shape space $M$
is two dimensional, as opposed to three for the interior. However it is not always wise
to think that there is adrop in the number of variables. Instead, here we simply see
the triple colision situation as the special case–the case that some variables become
insignificant. As already discussed, the configuration space for the triatomic system
is six dimensional, and we revert our discussion back on the wavefunctionon$\eta_{1}=\eta_{2}=$$D^{l}$$f(\alpha, \beta)$20

97, where and really are nothing other than the Jacobi vectors (21-22). Eachand have components , and , $j=1,2,3$ respectively. Now, if this wavefunction
is analytic at the origin, and therefore analytic when triple collision takes place, has$\mathbb{R}^{3}\cross \mathbb{R}^{3}$$\alpha$$\beta$$\alpha$$\beta$$\beta_{j}$$\alpha_{j}$$f$the expansion of the form$f( \alpha, \beta)=\sum_{I,J}c_{IJ}\alpha^{I}\beta^{J}$

(112)

,

where we have used the notations$I=(i_{1}, i_{2}, i_{3})$,$\mathcal{J}=(j_{1}, j_{2}, j_{3})$,$\alpha^{I}=\alpha_{1}^{i_{1}}\alpha_{2}^{i_{2}}\alpha_{3}^{i_{3}}$,$\beta^{J}=\beta_{1}^{j_{1}}\beta_{2}^{j_{2}}\beta_{3}^{j_{3}}$

.

(113)

Here we emphasize once again that, for given we would like to expand this series intoaFourier series in terms of the $D$-functions. First we consider asection just as we didfor the colinear case. We have already discussed that
$f$

(114)

$M:=X_{0}/SO(3)\cong \mathbb{R}^{3}\cross \mathbb{R}^{3}/SO(3)$. For
is the shape space which is homeomorphic with(14) and for an open set $U\subset M$ , we may express any $(\alpha, \beta)\in\pi^{-1}(U)$ as$\{x\in \mathbb{R}^{3}|x_{3}\geq 0\}$$(\alpha, \beta)=g\sigma$ $=(g\sigma_{1}(q), \sigma_{2}(q))$

,

$\pi$

defined in(115)$q\in U$are the components of
where are the local coordinates in $M$ , $g\in SO(3)$ , anddefined in (16).
be the space of homogeneous polynomials of degree in , .
Let
isThen we see that it is invariant under the SO(3) action, which means thatwill be decom. This implies that the space
represented in
posed into irreducible subspaces with respect to the SO(3) action. In each irreducible
subspace, the basis polynomial transformation is subject to$\sigma$$q$$\sigma_{j}$$n$$P^{n}(\mathbb{R}^{3}\cross \mathbb{R}^{3})$$\alpha_{i}$$\beta_{j}$$5\mathrm{O}(3)$$P^{n}(\mathbb{R}^{3}\cross \mathbb{R}^{3})$$P^{n}(\mathbb{R}^{3}\cross \mathbb{R}^{3})$$p_{m}(g^{-1} \alpha, g^{-1}\beta)=\sum_{|n|\leq l}p_{n}(\alpha, \beta)D_{nm}^{l}(g)$

.

(116)

Hence if we substitute (115) into (112), we expect to be able to put
series in terms of D-functions$f(g \sigma_{1}(q), g\sigma_{2}(q))=\sum_{l=0}^{\infty}\sum_{|m|,|n|\leq l}D_{mn}^{l}(g^{-1})c_{mn}^{(l)}(q)$

21

.

$f$

into aFourier

(117)

98
5.2

Decomposition of space of homogeneous polynomials

It is of our interest to investigate this Fourier series in detail. Let
denote thespace of homogeneous polynomials in $Uj,j=1,2,3$ . Thenis decomposed into$P^{n}(\mathbb{R}^{3})$$P^{n}(\mathbb{R}^{3})$$P^{n}(\mathbb{R}^{3})=H^{n}(\mathbb{R}^{3})\oplus r^{2}H^{n-2}(\mathbb{R}^{3})\oplus\cdots\oplus\{$if
if$r^{n-1}H^{1}(\mathbb{R}^{3})r^{n}H^{0}(\mathbb{R}^{3})$

$n$
$n$

is evenis odd

(118)

where
is the space of solid harmonics of degree . Here we point out that it isisomorphic with the $(2n+1)$ -dimensional spacefor unitary irreducible representations of 50(3). In addition to that is invariant under $g\in SO(3)$ , the decomposition(118) is reduced to$H^{n}(\mathbb{R}^{3})$$n$$H^{n}$$r^{2}$

if
if$H^{0}$$P^{n}(\mathbb{R}^{3})\cong H^{n}\oplus H^{n-1}\oplus\cdots\oplus\{$$H^{1}$

Therefore this decomposition applied to$P^{l}(\mathbb{R}^{3}\cross \mathbb{R}^{3})$$P^{l}(\mathbb{R}^{3}\cross \mathbb{R}^{3})$

$n$
$n$

is evenis odd

(119)

gives

$=$
$\sum_{n+m=l}H^{n}(\mathbb{R}_{\alpha}^{3})\otimes H^{m}(\mathbb{R}_{\beta}^{3})$$\oplus$$\sum_{n+m=l}H^{n}(\mathbb{R}_{\alpha}^{3})\otimes|\beta|^{2}H^{m-2}(\mathbb{R}_{\beta}^{3})$(120)$\oplus$$\sum_{n+m=l}|\alpha|^{2}H^{n-2}(\mathbb{R}_{a}^{3})\otimes H^{m}(\mathbb{R}_{\beta}^{3})\oplus\cdots$

where we have used the identity$P^{l}( \mathbb{R}^{3}\cross \mathbb{R}^{3})=\sum_{n+m=l}P^{n}(\mathbb{R}^{3})\otimes P^{m}(\mathbb{R}^{3})$

,

(120)

and the greek subscripts are placed so that one may not confuse the twoshould further be put in terms of$P^{l}(\mathbb{R}^{3}\cross \mathbb{R}^{3})$$H^{n}$$\mathbb{R}^{3}$’s. This’s;$\cong$$\sum_{n+m=l}H^{n}\otimes H^{m}$$\oplus$$\sum_{n+m=l}7\{^{n}\otimes H^{m-2}$$\oplus$$\sum_{n+m=l}H^{n-2}\otimes \mathcal{H}^{m}\oplus\cdots$

22

.

(122)

99
We can again decompose (122) further, if we apply the Clebsch-Gordan decomposition
formula for SO (3),(123)$H^{p}\otimes H^{q}\cong H^{|p-q|}\oplus H^{|p-q|+1}\oplus\cdots\oplus H^{p+q}$

and we finally obtain the decomposition$P^{l}(\mathbb{R}^{3}\cross \mathbb{R}^{3})\cong H^{l}\oplus \mathcal{H}^{l-1}\oplus 2H^{l-2}\oplus\cdots$

5.3

(124)

.

Implication of the decomposition

We may deduce the main result of the article from the decomposition (124). The
with $m\leq l$
includes representations spaces
decomposition implies thatwith $n\geq l$ . Basisarise fromonly. Therefore the representation space
are subject to the transformation (116).
polynomials in
Now suppose that the wavefunction is analytic at the origin. Given that we haveatriatomic system, and if is an eigenstate associated with the eigenvalue $l(l+1)$ of,, must be alinear combination of
the total angular momentum operator
coefficients of which are expressed as power series in the local coordinates . This
must have theexpressed as apower series in ,implies that the eigenstatewith $|I|+|J|=l$ , where $|I|=i_{i}+i_{2}+i_{3}$ , andlowest order terms of the form
$H^{m}$$P^{l}(\mathbb{R}^{3}\cross \mathbb{R}^{3})$$H^{n}$$P^{n}(\mathbb{R}^{3}\cross \mathbb{R}^{3})$$H^{n}$$f$$f$$L^{2}$$D_{mn}^{l}(g)$

$f$

$q$

$f$

$\alpha_{i}$$\beta_{j}$$\alpha^{I}\beta^{J}$$|J|=j_{1}+j_{2}+j_{3}$5.4.

ExampleIn order to justify the main result obtained in the previous subsection, we present an
example here. Here we pay particular attention to the fact that the representation
are isomorphic, and we consider the tensor product, andspaces of 50(3),$P^{1}(\mathbb{R}^{3})$$H^{1}$.

$P^{1}( \mathbb{R}^{3})\cross P^{1}(\mathbb{R}^{3})=\{\mathrm{t}\mathrm{r}(C^{T}\alpha\beta T)=\sum_{i,j}C_{ij}\alpha_{i}\beta_{j}|C\in \mathbb{C}^{3\mathrm{x}3}\}$(125)

The Clebsch-Gordan formula applied on this gives$\mathcal{H}^{1}\otimes H^{1}\cong H^{0}\oplus \mathcal{H}^{1}\oplus H^{2}$

23

,

(126)

100
and here we aim to identify each component in the right hand side of (126). Just as
we considered the basis of tensor product like the one in (116), here we consider thetransformation of the basis. This transformation is subject to$\alpha\beta^{T}$(127)$\alpha\beta^{T}\mapsto g(\alpha\beta^{T})g^{-1}$then one has
subject to$\mathrm{t}\mathrm{r}(C^{T})g(\alpha\beta^{T})g^{-1})=\mathrm{t}\mathrm{r}(gCg^{-1}\alpha\beta^{T})$. So the transformation of C is

C $\mapsto A_{g}:=gCg^{-1}$ .

(127)

Here we note that the symmetric and anti-symmetric matrices are invariant under theadjoint action of , and since, the representation of SO(3) inis reducible down to three subspaces$\mathrm{t}\mathrm{r}(g^{-1}Cg)=\mathrm{t}\mathrm{r}(C)$$g$$\mathbb{C}_{0}^{3\mathrm{x}3}$$\mathbb{C}_{1}^{3\mathrm{x}3}$$\mathbb{C}_{2}^{3\mathrm{x}3}$$\mathbb{C}^{3\mathrm{x}3}$

$:=$

{

$:=$

$\{C\in \mathbb{C}^{3\mathrm{x}3}|C=-C^{T}\}$$:=$$\{C\in \mathbb{C}^{3\mathrm{x}3}|C=C^{T},\mathrm{t}\mathrm{r}(C)=0\}$$\lambda I_{3}|$A$\in \mathbb{C}$

}

(129)

It is trivial to observe that,, so we may identify
with
.
For ,
, define the inner product$\dim \mathbb{C}_{0}^{3\mathrm{x}3}=1$$\oplus_{j=0}^{3}\mathbb{C}_{j}^{3\mathrm{x}3}$$C_{1}$$\dim \mathbb{C}_{1}^{3\mathrm{x}3}=3$

(130)

,

.

(130)

$\dim \mathbb{C}_{2}^{3\mathrm{x}3}=5$, and$\mathbb{C}^{3\mathrm{x}3}=$$H^{j}$$\mathbb{C}_{j}^{3\mathrm{x}3}$$C_{2}\in \mathbb{C}^{3\mathrm{x}3}$$\langle C_{1}, C_{2}\rangle=\mathrm{t}\mathrm{r}(C_{1}^{*}C_{2})$

,

(132)

and under this inner product, it can be shown easily that the adjoint operatorisunitary. Recall thatare invariant subspaces ofunder , we observe thatthe adjoint operator restricted todomain is a $U(2j+1)$ operator. Further, notethatis an orthogonal direct sum. In particular, if we chooseandsuch that-Ci,, we have$A_{g}$$\mathbb{C}^{3\mathrm{x}3}$$\mathbb{C}_{j}^{3\mathrm{x}3}$$A_{\mathit{9}}$$\mathbb{C}_{j}^{3\mathrm{x}3}$$\oplus_{j=0}^{3}\mathbb{C}_{j}^{3\mathrm{x}3}$$C_{1}^{T}=$$C_{1}$$C_{2}$$C_{2}^{T}=C_{2}$$\langle C_{1}, C_{2}\rangle=0$(133)

and this implies thatandare orthogonal to each other. In addition, if
we particularly choose
andwith, we again have (133), implying
andare orthogonal to each other.
$\mathbb{C}_{0}^{3\mathrm{x}3}\oplus \mathbb{C}_{2}^{3\mathrm{x}3}$$C_{1}=\lambda I_{3}$$\mathbb{C}_{0}^{3\mathrm{x}3}$$\mathbb{C}_{1}^{3\mathrm{x}3}$$C_{2}$$\mathrm{t}\mathrm{r}(C_{2})=0$$\mathbb{C}_{2}^{3\mathrm{x}3}$

24

101
respectively. The
and the components ofareandThe bases of. However, these bases,,
arebases of
, $l=1,2$
should be transformed into suitable ones in order to get unitary matrices
as transformation matrices such that (116) holds.
andFor the case $l=1$ , let $7=\alpha\cross\beta$ . If and are transformed torespectively, then 7transforms subject to $7\mapsto\# 7$ , and we have$7\{^{0}$$H^{2}$$\mathcal{H}^{1}$$\alpha\cross\beta$$\alpha\cdot\beta$$\alpha_{2}\beta_{2}-\alpha_{3}\beta_{3}$$\alpha_{1}\beta_{1}-\alpha_{2}\beta_{2}$$\alpha_{i}\beta_{j}+\alpha_{j}\beta_{i}(i<j)$$D^{l}(g)$$\beta$$\alpha$$g\beta$$g\alpha$(134)$p_{n}^{(1)}(g^{-1} \gamma)=\sum_{m}p_{m}^{(1)}(\gamma)D_{mn}^{1}(g)$for the polynomials defined by$(p_{1}^{(1)},p_{0}^{(1)},p_{-1}^{(1)})=(- \frac{\gamma_{1}+i\gamma_{2}}{\sqrt{2}}$In fact the polynomials$p_{m}^{(1)}$,$\gamma_{3}$,$\frac{\gamma_{1}-i\gamma_{2}}{\sqrt{2}})$

.

(135)

are related to the spherical harmonics by$p_{m}^{(1)}(u)=\sqrt{\frac{4\pi}{3}}r\mathrm{Y}_{1m}(\theta, \phi)$$m=-1,0,1$ .

,

(136)

Next is the case when $l=2$ . First note that$H^{2} \cong H^{2}(\mathbb{R}^{3})=\{\mathrm{t}\mathrm{r}(C^{T}uu^{T})=\sum_{i,j}C_{ij}u_{i}u_{j}|C\in \mathbb{C}^{3\mathrm{x}3}, C=C^{T}, \mathrm{t}\mathrm{r}(C)=0\}$

and we present the following as the bases of$H^{2}(\mathbb{R}^{3})$

,

(137)

;

$q_{-2}$$=$$(u-iv)^{2}/2$ ,(138)$q_{-2}$$=$$w(u-iv)$ ,(139)$q_{0}$$=$$(2w^{2}-(u^{2}+v^{2}))$$q_{1}$$=$$-w(u+iv)$$q_{2}$$=$$(u+iv)^{2}/2$

(140)

,

(141)
(142)

.

Similarly these polynomials are related to the spherical harmonics by$q_{m}(u)=\sqrt{\frac{8\pi}{15}}r^{2}\mathrm{Y}_{2m}(\theta, \phi)$,$m=2,1,0,$ $-1,$ -2(143)that transform subject to$q_{m}(g^{-1})= \sum_{m}q_{n}(u)D_{nm}^{2}(g)$

25

.

(144)

102
For the bases)’,’$\sigma_{0}=\frac{1}{\sqrt{6}}$$(\begin{array}{lll}-1 0 00 -\mathrm{l} 00 0 2\end{array})$,$\sigma_{-2}=\frac{1}{2}(\begin{array}{ll}1-i 0-i-1 000 0\end{array})0$$\sigma_{1}=\frac{1}{2}$($-i00$$-1-i0$$\sigma_{-1}=\frac{1}{2}\sigma_{2}=\frac{1}{2}\{i01100-1-i000i$$-i)00)0^{0}1$

,
(145)

for

$\mathbb{C}_{2}^{3\cross 3}$, the sohd harmonics $qm(u)$ is put in the form
$q_{m}(u)=\mathrm{t}\mathrm{r}(\sigma_{m}uu^{T})$,$m=-2,$ -1, 0, 1, 2,(146)

and the $q_{m}(u)$ have the generating function(147)$Q_{2}(u, t)= \sum_{-2}^{2}c_{m}q_{m}(u)t^{2-m}$with $(c_{-2}, c_{-1}, c_{0}, c_{1}, c_{2})=(2,4,2\sqrt{6},4,2)$ . If we observe that

$Q_{2}$

is expressed as$Q_{2}(u, t)=(\sqrt{2}p_{-1}^{(1)}(u)t^{2}+2p_{0}^{(1)}(u)t+\sqrt{2}p_{1}^{(1)}(u))^{2}$

,

(148)

we find that
,

(149)

$q_{-2}(u)$$=p_{-1}^{(1)}(u)^{2}$$q_{-1}(u)$$=$$\sqrt{2}p_{0}^{(1)}(u)p_{-1}^{(1)}(u)$$q_{0}(u)$$=$$\frac{2}{\sqrt{6}}(p_{0}^{(1)}(u^{2}+p_{1}^{(1)}(u)p_{-1}^{(1)}(u)),$$q_{1}(u)$$=$$\sqrt{2}p_{0}^{(1)}(u)p_{1}^{(1)}(u)$$q_{2}(u)$$=p_{1}^{(1)}(u)^{2}$

,

(150)

,

$(151)$
(152)

.

(153)

Further, define the function,(154)$P_{2}(\alpha, \beta, t)=((\alpha_{1}-i\alpha_{2})t^{2}+2\alpha_{3}t-(\alpha_{1}+\alpha_{2}))((\beta_{1}-i\beta_{2})t^{2}+2\beta_{3}t-(\beta_{1}+i\beta_{2}))$

which expands into
$P_{2}( \alpha,\beta, t)=\sum_{-2}^{2}c_{m}p_{m}(\alpha,\beta)t^{2-m}$

26

,

(155)

103

(156)

,

$p_{-2}(\alpha, \beta)$$=$$\frac{[perp]}{2}(\alpha_{1}-i\alpha_{2})(\beta_{1}-i\beta_{2})$$p_{-1}(\alpha, \beta)$$=$$\frac{1}{2}(\alpha_{3}(\beta_{1}-i\beta_{2})+(\alpha_{1}-i\alpha_{2})\beta_{3})$$p_{0}(\alpha, \beta)$$=$$\frac{1}{2\sqrt{6}}(4\alpha_{3}\beta_{3}-(\alpha_{1}-i\alpha_{2})(\beta_{1}+i\beta_{2})-(\alpha_{1}+i\alpha_{2})(\beta_{1}-i\beta_{2}))$$p_{1}(\alpha, \beta)$$=$$- \frac{1}{2}((\alpha_{1}+i\alpha_{2})\beta_{3}+\alpha_{3}(\beta_{1}+i\beta_{2}))$$p_{2}(\alpha, \beta)$$=$$\frac{1}{2}(\alpha_{1}+i\alpha_{2})(\beta_{1}+i\beta_{2})$The relations with$p_{-2}(\alpha, \beta)$$p_{m}^{(1)}$, (158)
(159)

,

(160)

.

are given by$=p_{-1}^{(1)}(\alpha)p_{-1}^{(1)}(\beta)$$=$$p_{0}(\alpha, \beta)$$=$$p_{1}(\alpha, \beta)$$=$$p_{2}(\alpha, \beta)$$=p_{1}^{(1)}(\alpha)p_{1}^{(1)}(\beta)$$p_{m}(\alpha, \beta)$

(161)

,

$(162)$$\frac{\sqrt{2}}{2}(p_{0}^{(1)}(\alpha)p_{-1}^{(1)}(\beta)+p_{-1}^{(1)}(\alpha)p_{0}^{(1)}(\beta),$$p_{-1}(\alpha, \beta)$

The functions

(157)

,

$\frac{1}{\sqrt{6}}(2p_{0}^{(1)}(\alpha)p_{0}^{(1)}(\beta)+p_{-1}^{(1)}(\alpha)p_{1}^{(1)}(\beta)+p_{1}^{(1)}(\alpha)p_{-1}^{(1)}(\beta)$,$\frac{\sqrt{2}}{2}(p_{1}^{(1)}(\alpha)p_{0}^{(1)}(\beta)+p_{0}^{(1)}(\alpha)p_{1}^{(1)}(\beta))$

(163)
(164)

(165)

form abasis of the space of polynomials associated with$\{\mathrm{t}\mathrm{r}(C^{T}\alpha\beta)=\sum_{i,j}C_{\dot{\iota}j}\alpha_{i}\beta_{j}|C=C^{T}, \mathrm{t}\mathrm{r}(C)=0\}$$\mathbb{C}_{2}^{3\mathrm{x}3}$

;

(156)

and are expressed as$p_{m}(\alpha, \beta)=\mathrm{t}\mathrm{r}(\sigma_{m}\alpha\beta)$

(167)

.

Here we note that $q_{m}(u)$ transforms according to $q_{m}(g^{-1}u)= \sum_{n}q_{n}(u)D_{nm}^{2}(g)$ . Since, we, and since
are subject to the transformation
see that$q_{m}(g^{-1}u=\mathrm{t}\mathrm{r}(g\sigma_{m}g^{-1}uu^{T})$$\sum_{n}q_{n}(u)D_{nm}^{2}(g)=\mathrm{t}\mathrm{r}(\sum_{n}\sigma_{n}D_{nm}^{2}(g)uu^{T})$$\sigma_{m}$$A_{g} \sigma_{m}=\sum_{n}\sigma_{n}D_{nm}^{2}(g)$

27

.

(168)

104
If we apply this transformation rule to$p_{m}(\alpha,\beta)$, we find that$p_{m}(g^{-1} \beta,g^{-1}\beta)=\sum_{n}p_{n}(\alpha,\beta)D_{nm}^{2}(g)$Therefore we found arealization of the component space.

$H^{2}$(169)

in the decomposition(126).

References[1] R. G. Littlejohn and M. reinsch, Rev. Mod. Phys. 69, 213 (1997).
[2] R. G. Littlejohn and M. reinsch, Boundary conditions on internal three-body wave
functions, Physical Rev. AVol. 61, 042502, (2000).
[3] T. Iwai, A geometric setting for internal motions of the quantum three-body system,
J. Math. Phys. 28, 1315 (1987).
[4] S. Tanimura, T. Iwai, Reduction of quantum systems on Riemannian
manifolds
with symmetry and application to molecular mechanics, J. Math. Phys., Vol. 41,
No. 4, (2000).
[5] R. G. Littlejohn, K. A. Mitchell, V. Aquilanti, S. Cavall, Body rames andframe
singularities for three-atom systems, Physical Rev. AVol. 58, 3705, (1998).
[6] R. G. Littlejohn, K. A. Mitchel, V. Aquilanti, S. Cavall, Internal spaces, kinematic rotations, and body frames for four-atom systems, Physical Rev. A, Vol. 58,
3718, (1998).

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