Quantum mechanics

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QUANTUM MECHANICS* Hitoshi Kitada t December 28, 2003

*@1998-2003 by Hitoshi Kitada, All Rights Reserved tGraduate School of Mathematical Sciences, University of Tokyo, Komaba, Meguro, Tokyo 153-8914, Japan, e-mail: [email protected], web page: http://kims.ms.utokyo.ac.jp/

Preface I consider in this book a formulation of Quantum Mechanics, which is often abbreviated as QM. Usually QM is formulated based on the notion of time and space, both of which are thought a priori given quantities or notions. However,when we try to define the notion of velocity or momentum, we encounter a difficulty as we will see in chapter 1. The problem is that if the notion of time is given a priori, the velocity is definitely determined when given a position, which contradicts the uncertainty principle of Heisenberg. We then set the basis of QM on the notion of position and momentum operators as in chapter 2. Time of a local system then is defined approximately as a ratio lxl/lvlbetween the space coordinate x and the velocity v, where !xi,etc. denotes the absolute value or length of a vector x. In this formulation of QM, we can keep the uncertainty principle, and time is a quantity that does not have precise values unlike the usually supposed notion of time has. The feature of local time is that it is a time proper to each local system, which is defined as a finite set of quantum mechanical particles. We now have an infinite number of local times that are unique and proper to each local system. Based on the notion of local time, the motion inside a local system is described by the usual Schrodinger equation. We investigate such motion in a given local system in part II. This is a usual quantum mechanics. After some excursion of the investigation of local motion, we consider in part III the relative relation or motion between plural local systems. We regard each local system's center of mass as a classical particle. Then as the relative coordinate inside a local system is independent of its center of mass, we can set an arbitrary rule on the relation among those centers of mass of local systems. We adopt the principles of general relativity as the rules that govern the relations of plural local systems. By the reason that the center of mass and the inner coordinate are independent, we can combine quantum mechanics and general relativity consistently. We give an approximate Hamiltonian that explains partially the usual relativistic quantum mechanical phenomena in chapter 9. In the final part IV, we consider some contradictory aspect of mathematics in chapter 10. Although this does not give directly that mathematics is inconsistent, this will give an introduction to the next chapter 11, where starting with the contradictory nature of the semantics of set theory in the sense that if we consider all sentences of set theory, they are contradictory, we regard that the Universe that is described by ourselves is of contradictory nature, and can be described as a superposition of all possible, infinite number of waves. As this is the state of the Universe, the Universe is described as a stationary state describing a superposition of all waves. We then give a formulation of iii

IV

the Universe and local systems inside it, in the form of a theory described by Axiom 1 to Axiom 5 in cha}ter 11. In the final chapter 12, we will prove that there is at least one Universe wave Junction¢ in which all local systems have local motions and thus local times. This concludes our formulation of Quantum Mechanics.

Hitoshi Kitada Dec. 15, 2003, Tokyo

Notation We here explain some notations which will be used in the text. Ck(mn) (k = 0, 1, 2, · · · , oo) is a space of k times continuously differentiable functions / (x) of x E Rm. R;'1)is a subspace of Ck(mn) whose element f E Ck(R;'I) has compact support in R;'I. In particular, C8°(Rm)is a space of infinitely differentiable functions on Rm with compact support. We also use the notation Cgo(G)for a region Gin R;'I to denote the space of functions with continuous derivatives up to order k with the support contained in G. S = S(mn) denotes a space of rapidly decreasing functions / on R;'I. Namely / E S means that / is an infinitely differentiable function satisfying

CS(

(1) for all integers k = 0, 1, 2, · · · and multi-indices a non-negative integer and

=

(a 1 , · · · , am), where each a; is a

(2) The most important notion is the Hilbert space L 2 (R;'1)with m of functions f(x) on Rm satisfying

= 1, 2, · · ·.

It is a space

11/11 = (/, 1)112 < oo,

(3)

where the inner product is given by

(f,g)

= JR!" f(x)g(x)dx.

(4)

Concretely it is obtained by a completion of Sor of C8°(Rm) with respect to the norm (3). Along with this Hilbert space we use weighted £ 2 space: L~ = L~(R;'I) (s E R1), which is a completion of S with respect to the norm

llflls = where (x)

(lmlf(x)l2(x)

11/IIL? =

= (1 + jxj2 ) 112 • L~(Rm) is also a Hilbert (f,g)s

= (f,g)L? =

f f

28

dx)

112 ,

(5)

space with the inner product:

f(x)g(x)(x)

28

dX.

(6)

:F denotes Fourier transformation from S onto itself: :Ff(f.) = (21r)-m/2

V

e-i(:i:

f(x)dx.

(7)

vi :Fis extended to a unitary operator from L2 (R!") onto itself:

11:Ffll = 11111.

(8)

We define Sobolev space H 8 = H 8 (R!") of orders E R 1 as the Fourier image of L~(R!"). Thus it is a comple,tionof S with respect to the norm

llfllH•=

(

lm

l{Da:}f(x)l 8

2 dx )

1/2 ·

(9)

Here

(10) where ({)8 denoteE:a multiplication operator by ({)8 in £ 2 (~). V( ({} 8 ) is a set of functions f satisfying

Namely its domain

(11) and its value applied to / E 'D(({} 8 ) is ({}8 /({) with the inner product:

E L2 (R!"). H 8 (Rm) is also a Hilbert space

(12) We further use weii~htedSobolev space H;(R!") (6 E R 1 ). This is a completion of S with respect to the norm

(13) and is a Hilbert space with the inner product

(14)

sm-1 denotes the unit sphere of Ii!" with surface element 0 be the mass of the j-th particle. The Hamiltonian of the system is defined by (2.2) 7

CHAPTER 2. POSITION AND MOMENTUM

8

where PJ= Z::!=1 PJk and ½3{x) (x E R3 ) is a real-valued pair potential which describes the interaction between the particles i and j. Since this interaction depends only on the relative position xi - x 3 E R3 of the particles, we can remove the center of mass from the Hamiltonian. Namdy, denoting the old variables xi by Xi with some abuse of notation, we introduce new variables xi as follows. We first define the center of mass of the N-particle system by m1X1 + · · · + mNXN Xe= --------, m1+···+mN

(2.3)

and then define Xi as Jacobi coordinates: m1X1 + · · · + miXi -------, m1 +···+mi

, Xi =)·i+l-

{2.4)

i= 1,2,··· ,n=N-1.

Accordingly, wedelinethemomentumoperatorsPa

= {Pa1,Pa2,Pa

3)

andpi = (pil,Pi2,Pi3 ):

1i 8 1i 8 Pc= -:-8X , Pi= -:--8 . i C i Xi It is clear that these satisfy the canonical commutation relation. Using these new Xe, Pc, Xi,Pi, we can rewrite H 11s

H=H+Hc,

(2.5)

where

H-

1 2 =~ LJ ~Pi+

i=l µ,

1

He=

"

LJ Vij(Xij),

i Ois the reduced mass defined by the relation: 1

1

1

µi

mi+1

m1 + · · · + mi

-=--+-----

The new coordinates give a decomposition L2(R 3N) = L2(R 3 ) ® L 2 (R 3n) and in this decomposition, H is written as

H

= He®

I+ I® H.

He is the well-knc,wnLaplacian, and what we are concerned with is the relative motion of the N-particl~. Thus we have only to consider Hin the Hilbert space 1l = L2(R3n). We rewrite this H as H: n

H =Ho+ V

I

= L ~Pl+ i=l µ,

n

.

~

L½3(Xi3) = - L ~~"'; + L¼3(xi3), i=l µ, i 0 that goes to 0 when Mi 's in the multi-indices Mb 's tend to oo such that as T -+ oo (3.11)

for any pair a= {i,j} with a f/.Ce for all Ce Eb. Here rvfM means that the norm of the difference of the both sides is smaller than €M as T -+ oo, and F(lxol < R) denotes a smooth positive cut off function which is 1 on the set S = {xllx0 I < R} C R 3n and is 0 outside some neigMorhood of S. By this lemma, ·;he third term on the RHS of (3.10) vanishes as t-+ oo within the small error €M > 0 determined by the values of Mi in the multi-indices Mb = (M 1 , • • · , Me). The last term on the RHS of (3.10) also vanishes by X 2 f E 1£. Thus as t -+ oo we have asymptotically

itH ( ~ -

L

D)Ae-itH f '°' t sisH71Re-isHJ-ds+ 2eitHR e-itHfL..t lo

2~J(b)$N

_!t2

rv fM

2

(3.12)

0

b b

2:5lbl:5N O

L

- t~

lt seisHi[Hb,Ab]Ae-isHfds.

2:5lbl:5N O

Taking the inner p1oduct of the last term with that as t -+ oo

L

L ; lt

2:5lbl:5N2:5ldl:5N,d;,!b

we have by (3.6) ru:,t

f = (I -

pMi

)f and noting by Lemma 3.3

s(f, eisHPdi[Hb,Ab]Ae-isHf)ds

rvfM

0,

(3.13)

O

-+ oo

; L

lt s(f,eisHi[Hb,Ab]Ae-isHJ)ds

2:5lbl:5N O rveM

;

L

lt s(f, isH Ai[Hb, Ab]Ae-isHf)ds

2:5lbl:5N O rv

{:

L

lt(f,eisHAi[Hb,Ab]Ae-isHJ)ds,

2:5lbl:5N O

provided that the limit as t -+ oo of the RHS exists, which we will prove below. Here rv means rvo. Letting t(s) = s - mS for mS::; s < (m + l)S for any fixed S > 0, we have

3.2. JUSTIFICATION OF LOCAL TIME AS A NOTION OF TIME

17

by Lemma 3.3 and some commutator arguments as t -too

i

L

1t

2:5lbl:5N

,....,,M

O

L

(T,isH Ai[Hb, Ab]Ae-isHl)ds

i

1t

2:5lbl:5N

(1,ei(s-t(s))HAeit(s)Hbi[Hb,Ab]Ae-isHl)ds.

O

.

This can further be reduced and is asymptotically equal toast-too

L

i

2:5lbl:5N

with an error

fM

>0

1t (1, ei(s-t(s))H Aeit(s)Hbi[Hb, Ab]e-it(s)Hb Ae-i(s-t(s))H1)ds. 0

Noting s - t(s) = mS for mS $ s < (m + 1)8, we rewrite this fort=

nS (3.14)

:Ef

Writing A = =l A,E)3ibl-l with A,Ej being the one dimensional eigenprojection of Hb with eigenvalue Ei, we see that the RHS is bounded by

'°' eimSHRbeiS(Hb-Ei) AbRb,EjR LJ .!_'°' LJ .!_[(/S , L

n-1

e-imSHf-)

lbl-1

j=l n m=O

- . SH- b -. . SH-(!, e•m AA A,Ejf'ibl-le-•m /)]. This is arbitrarily small when S

> 0 is fixed sufficiently large, by our assumption 11 IXblPb,Ej II
1 large enough so that the second term is less than f.M and then we let Tk -+ oo (see the proof of Lemma 3.3 below). We can thus take a sequence {tm} tending to oo and sequences Mf that also tend to oo so that (3.21)

0,

CHAPTER 3. TIME

20 and for all R > 0, and a= {i,j} that is not in any Ce Eb

(3.22) when m-+ oo. (3.7) and (3.8) follow from these by density argument. The case tm-+ -oo is treated similarly. There remains to prove Lemma 3.3.

Proof of Lemma 3.:7:We prove a more general version of Lemma 3.3: Under the assumption of the lemma, we have as T -+ oo

II~1T

B(s)F(lxal < R)F(jxbl < R)Pifi~1e-isHEH(B)dsll rv,M O

(3.23)

for any a= {i,j} such that a tJ.Ce for all Ce Eb. We prove (3.23) by induction on k

= jbj.

Lemma 3.4 {3.23) for lbl= 2 holds.

Proof: Since IIF(lxl > S)F(lxal < R)F(lxbl < R)II-+ 0 as S-+ oo when lbl = 2, R < oo and a¢ Ce for any Ce Eb(£= 1,2), we have only to show

II~

/!1;., 1T B(s)F(lxl

< R)EH(B)e-isHdsll = 0.

(3.24)

The operator F(lxl < R)EH(B) is a compact operator. Thus it suffices to prove the lemma with F(lxl < R)EH(B) replaced by a one dimensional operator Kf = (J,¢)'1/J, where ¢ E 1ic(H). Then

II~

1T B(s)Ke-isHdsf

=

II~

1T eisHK*B(s)*dsf

(3.25)

II.!_

=

sup {T eisHK*B(s)*fdsll2 11/Jl=lT lo

=

sup T\ {T fT(B(s)*f,'ljJ)('ljJ,B(t)*f)(e-i(t-s)H 0, we have for.,\> 0

.r(.-\)f

=

= =

(m)(m- 2)/2(.rf)(m ·) (2.-\tl/4 (m)(m-1)/2g(m) {2.-\)-lf4(21r)-l/21:

ei.,/V.r.rp((p)p))such that supp 1/2

(l:

(2?r)-lf2(2A)-1.'4

(r)-2s dr) 1/2 ([:

(r)2s11Fp(·Thus we have an expression of~~(µ):

(~~ (µ)f,g) = 1$1 2~/(R.o(z)

- R.o(z))Eo(B)/,g)

forµ, EB= (a,b) (0 1/2). We consider termwise boundary values of R.o(z) = R.o(µ± ie) as write for /, g E S using Fourier transform

(R.o(µ+ ie)/, g)

JR"' e2/2 _ ~µ + ie) i({)g({)de

=

= {00 (2,\)(m-2)/2

lo

where

l

(,\- µ) -

J denotes the Fourier transform of /.

. h(,\) = (2,\)tm- 2)/ 2

. &€

f

E

+0. To do so, we

--

(5.32)

f(V2>.w)g(V2>.w)WJJd..\,

lsm-1

Setting

f(V2>.w)g(V2..\w)WJJ = (:F(,\)/,:F(..\)g)L2(sm-1), (5.33)

{

lsm-1

we rewrite the RHS of (5.32) as

100 ,\ ,\2

. -µ

+€

.1 100

2h(,\ + µ)d,\

+ 11"'&-1r

€

(,\

O

-

{5.34)

)2 2h(,\)d,\. µ, + €

The second integral on the RHS is a Poisson integral, thus the limit as e satisfies the estimHte by {5.24):

+0 exists and {5.35)

for s > 1/2, where Caµ> 0 is bounded locally uniformly with respect toµ,> 0. The first term on the RHS of (5.34) is written for 6 > 0 as

!

co

-,,

,\

,\2

+e

2h{,\ + µ)d,\

= 11rj-6 + \ -µ

{5.36)

loo) ,\2 ,\2h{,\ + +e 6

µ)d,\

+

f 6 ,\ -6

,\2

+€

2h{,\ + µ)d,\.

+

The limit as e 0 of the first term is equal to

([:6 + 100)½h.=

rli >,1+8 h(µ + >.)-

lo >,2+ E2

h(µ - >.)d),

>,9

,

where O < 0 < s - 1/2. Thus by applying (5.25) to the definition (5.33) of h(>,), we see that the limit as E .J,.0 exists and is bounded by

for s > 1/2 ands -1/2 > 0 > 0, where Cs8µ > 0 is bounded whenµ moves in a compact subset of (0, oo ). Thus the second term on the RHS of (5.36) is bounded by

for s > 1/2. Combining this with (5.37) and (5.35), we obtain the existence of the boundary value /lo(µ+ i0)f = lime.J.O Ro(µ+ iE)f in L: 8 (Rm) and the estimate

llllo(µ + i0)IIL2-+£2 -• ::; C

.

locally uniformly in µ > 0, where s > 1/2. Similarly, the same estimate holds also for /lo(µ - i0). Reexamining the arguments above, we also see that !lo(µ± i0) is continuous with respect toµ> 0 in operator norm from L~(Rm) into L: 8 (Rm) for s > 1/2. Summarizing, we have proved:

Theorem 5.1 The resolution {E0 (>.)} of the identity for Ho is expressed for f,g E L~(Rm) (s > 1/2) and A > 0 as:

d~(Eo(-A)f,g)

=

=

2~/(Ilo(>.+iO)-/lo(.A-iO))f,g) (F(>,)f,F(>.)g)L2(sm-l)·

(5.38)

Here {Jlo(>,± iO)h>o and {F(-A)h>o are continuous families of bounded operators from L~(~) (s > 1/2) into L: 8 (Rm) and L 2 (sm- 1 ), respectively, defined by for f E L~(Rm)

Jlo(>,± i0)f = lim Jlo(>,± iE)f,

(5.39)

e.l,O

F(>.)f(w)

= (2>.)(m-2)! 4 (.rf)(~w).

(5.40)

In particular, Jlo(>,± i0) and F(>.) are Holder continuous of order 0 (0 < 0 < s -1/2) locally uniformly with respect to>, > 0, in the uniform operator topologyof B(L~(~), L: 8 (~)) and B(L~(Rm),L 2 (sm- 1 )), respectively.

5.2

Spatial asymptotics

Since H 0 is a selfadjoint operator in 1l

of the free resolvent = L 2 (Rm), H 0 generates

Uo(t) = exp(-itHo) = e-itHo (t E R 1 )

a unitary group

(5.41)

CHAPTER 5. FREE HAMILTONIAN

42 such that

Uo(t)Uo(s)= Uo(t+s)

{5.42)

(t,s E R 1).

If we move to the :momentum space ii.=£ 2 (.Rf) by Fourier transformation, we have for g ES= :FS by (5.5)

U0 (t)g({) := (.rUo(t):F-1g)({) = e-itlEl2 l 2g({).

(5.43)

By Lebesgue's dominated convergence theorem, this implies that U0 (t)g is continuous in t E R1 as an £ 2 (.R;/l)-valuedfunction oft E R 1 for general g E 1£. For/ES , e-itHo

f(x)

= =

Uo(t)f(x)

=

(21r)-m

= (:F-1Uo(t):Ff)(x)

(21r)-m/2 {

(5.44)

2 /2(:Ff)({)de ei:z:Ee-itlEl

}Rm

lam ei:z:Ee-itlEl2/2 /am e-iEy f(y)dyde

Let x({) E S such that O $ x({) $ 1 and x{O)= 1, and set for € > 0

Then Xe ES for each€> 0 and xe({)--+ 1 as€ ,I..0 for each { E Rf· Thus we can rewrite (5.44) using Fubini's theorem as

Obviously this limit does not depend on the choice of limits of integrals oscillatory integrals, and write

x

E

S. We call this

type of the

(5.46) For the simplicity of notation, we introduce a variable (5.47) Then {5.46) can b,~written as

,~-itHof(x)= Os-/

{

2/2-MJ(y)dy,if i(:z:E-tlEl

}R2m

(5.48)

We often drop the integration region R 2m, when it is obvious from the context, and write (5.49)

5.2. SPATIAL ASYMPTOTICS OF THE FREE RESOLVENT

43

We introduce some notations. We call a = (a 1 , • '. · , om) with the components a; being nonnegative integers a multi-index. Then we define

Daz =Dai.:i:1 ,,Dam Zm'

xa = xa1 .. ,xam 1 m,

lal = 0!1 + · · · + O!m, (Dz) = (1 + D~)1l 2 = (1- a:i:) 112.

(5.50)

Noting the relation

(5.51) and integrating by parts inside the integral (5.45) with respect toy, we obtain for =

(2'11")-mlimf 40

= =

II II

1:.ffl

f

f ES

i(:i:(-tl(l2 /2-MxE({)f(y)dy~

JR"' 1nm

2 ei(:i:(-tl(l /2-Mxe({)({)-2m((Dy)2mf)(y)dydf

i(z(-tl(l 2 /2-M(e)-2m((Dy}2mf)(y)dydf

If f does not belong to S but just satisfies, e.g. the conditions

for all multi-indices a, we then define citHo f by

and integrate by parts with respect toy and { using the relations (5.51) and Df(e-i(y) = (-l)lalya(e-i(Y). Then we obtain e-itHof(x) =

II

W?

(5.52)

e-i(y(De)2m(ei(z(-t1(12/2)(e)-4mxE({))(y)-2m(Dy}4m(xe(y)/(y))dydf

Noting that xi{) satisfies IDf (xE({))I = le1a 1(Dfx)(e{)I ~ Caflal for any multi-index a, we see that the equality

holds. With some smoothness assumptions on the integrands, we always have in this way an expression of oscillatory integrals that does not contain the damping factors like XE({)

CHAPTER 5. FREE HAMILTONIAN

44

or xe(y). We will 11sethese techniques in the next section in considering the behavior of e-itHo. By (5.14), we ltave the relation forµ E R 1 and e =I0 (5.54)

at least for

f, g

E S. Noting the relation for e > 0

(>.- µ =i=ie)- 1 = i

1±00 O

eit(µ±i•->->dt,

(5.55)

and using Fubini's theorem, we can write

=

(Ro(µ±ie)f,g)

i

1±oo 100it(µ±i,->.)d(Eo(>.)J,g)dt

= i 1±00 (eit(µ±i•-Ho)f, g)dt. Since lleit(µ±i,-Ho) fll ~ e-•ltl llfll in respective signs, we have from this for f E 1£

Ro(µ± ie)f = i 1±00 eit(µ±i•-Ho) fdt,

(e>0,µER

1).

(5.56)

Since eitHof (f E 'H) is continuous in 1l with respect tot E R 1 by the remark after (5.43), the integral can be understood as a Riemann integral. Using the secor.d line of (5.44), we can rewrite (5.56) for f ES:

where J = :Ff. In what follows, assumingµ> 0, we apply stationary phase method to this integral and derive~ asymptotic expansion as r = Jxl--+oo. To do so, we assume f E C8°(R'['- {0}) with supp f C {{I (0 (m + 1)/2, we have an expansion formula for J1 (rw). In particular, as the first approximation we get for large r > 1

l1i(rw) _ (21r)(m+l)/2r(l-m)/2e-(m-1)1ri/4eirtf>(sc,(c)Idet J(sc, ec)l-1/2 j (ec)I = o(r-(m-1)/2) .. Noting that

(sc,ec) = Jiiµ,

I det J(sc,ec)I = (2µ)-(m- 3)12,

we obtain returning to (5.57)

R.o(µ+ i0)J(rw)

= ../2ire-(m-3)1ri/4( 2µ)(m-3)/4ei,/2p.rr-(m-1)/2(F/)(

+ o(r-(m-1)/2)

~)

as r ~ oo for J E Ccf(Jl"&- {O}). Similarly we can show

R.o(µ- i0)J(rw)

= ../2ire(m-3)1ri/4(2µ )(m-3)/4e-i,/2p.r r-(m-1)/2(F

J)(-,v2µw)

+ o(r-(m-1)/2)

as r ~ oo. Reversing the order of expression we obtain a relation between the Fourier transform F(µ) defined by (5.16) and the spatial asymptotics of the free resolvent llo(µ ± i0): Theorem 5.2 For

F(µ)J(±w)

Ff E Ccf (Jl"&- {0}) andµ > 0, w E sm-1 ,

= (21r)-1f2e±(m-3)1ri/4(2µ)1/4

one has

Jim r(m-1)/2e,:.i,/2p.r(Jlo(µ r-+oo

± i0)J)(rw).

(5.75)

CHAPTER 5. FREE HAMILTONIAN

50

5.3

Propai?;ation estimates for the free evolution

In this section we consider some estimates of e-itHowhich are stronger version of Theorem 3.2 in the free Hamiltonian case. For this purpose we introduce the notion of pseudodifferential operator (we call this a 'ljJdoin the following). P is called a '1/Jdowith symbol p(x, e),if it is written for f ES

Pf(x)

= Os- {

{ ei(x-v)ep(x,e)J(y)dya,f,

}RmJR!"

(5.76)

f

where is the variable defined by (5.47). We call p(x, e) the symbol of the 'ljJdoP and write p(x, e) = u(P)(x, e). For (5.76) to be well-defined as an oscillatory integral, we need to assume some smoothness conditions on the symbol p(x, e) of P. E.g. let us assume that p(x, e) is C00 with respect to (x, e) and satisfies the estimates

1a~O:p(x,e)I< oo

sup

(x,{)ER 2m

(5.77)

for all multi-indices a: and /3. Let as before x E S(R"') such that x(O) = 1, and set Xe(e) = x(1:e),and define (5.76) by

Pf(x)

II

= 1~1:f

ei(x-y)ep(x,e)xe(e)J(y)dyd[

(5.78)

Then using the relation 2 )e-ive, (1 + D;)e-iy{ = (1 + 1{1

we integrate by parts inside the integral (5.78):

1:m-! I i(x-y){p(x,{)xe({)(l + 1e1tm(1 + n;rf(y)dya,f

Pf(x)

2

=

JJ

i(x-y){p(x,

e)(l+ ,e,)-m(1 + v:r 2

f(y)dyd[

Thus Pf is well-defined for f E S as an oscillatory integral independently of the choice of the damping factor Xe· We write Pf as Pf= p(X, Dx)f to indicate the symbol p(x, {) used in the definition (5.76). As other forms of the definition of 'ljJdo,we can adopt

Qf(x)

= q(Dx, X')f(x) =Os-ff

or

Pf(x)

= p(X, Dx,X')J(x) = Os-/

where the symbols q({, y) and p(x,

ei(x-v)eq({,y)f(y)dydf,

I

i(x-y)ep(x,e,y)f(y)dydf,

e,y) satisfy

sup

1ara:q(e,y)I < oo,

({,y)ER2m

sup (x,{,y)ER3m

l~of 8Jp(x,{,y)I < oo

for all multi-indices a:,/3,'Y· The relation between these expressions is given by

5.3. PROPAGATION ESTIMATES FOR THE FREE EVOLUTION

51

Proposition 5.3 Let p(x, e, y) be as above. Then Pf= p(X, Dx, X')f is written as

Pf(x)

pL(X, Dx)f(x) =Os-I

I ei(x-y)ePL(x,e)J(y)dyd[

PR(Dx,X')f(x) =Os-I

I ei(x-y)ePR(e,y)f(y)dyd[,

(5.79) (5.80)

wherePL(X,e) and PR(e,y) are defined by PL(x,e) = Os- I I e-iyr,p(x,e + 'f/,X + y)dydrf, PR(e,y)

= Os- I

I iZT/p(y+ z, e + 'f/,z)dzdrf.

(5.81) (5.82)

Proof is easily done by using Fourier transformation, and is left to the reader. It is easy to see that for P

= p(X, Dx, X'),

P*f(x) =Os-I

the adjoint operator P* is given by

I ei(x-y)ep(y,e, x)f(y)dyd[

Thus if we consider the operator P*P, we have the integral operator

P*Pf(x)

= Os-/ I i(x-y)er(x,e,y)f(y)dyd[,

where

r(x, e, y) =Os-I

I e-iZT/p(x+ z, { + 'r/,x)p(x + z, e, y)dzdrf.

If we define the semi-norms for the symbol p(x,

IPlt= for f, = 0, 1, 2, ···,then

sup

sup

(5.83)

e,y) by

1a~af8Jp(x,ty)I

l0 1+1.01+h•l9 x,{,yER!"

we have for r above (5.84)

for some constant C,. > 0, where

i' = f, + 2mo, mo= 2[m/2 + l]. Here for a real number s, [s] denotes the greatest integer that does not exceed s. This is seen by integrating by parts in (5.83) by using the relations (1 + D;)e-izr, = (1 + lrJl2)e-izr,, (1 + D~)e-izr,= (1 + lzl2)e-izr,,

(5.85)

and noting that

(5.86)

CHAPTER 5. FREE HAMILTONIAN

52 For the product of v + 1 (v

P;f(x) =Os-//

~

1) ¢do's

ei(:i:-y){P;(x,~. y)J(y)dyd[,

(j = 1,2, · · · , v + 1),

we rewrite Pjf:

f

= Os-/

P;f(x)

e-iy{P;(x,~,x

+ y)f(x + y)dyd[

and calculate the product Qv+1=Pi··· Pv+l· Then we have

Qv-cif(x)

fI

= Os-

1df ei(:i:-:i:'){qv+1(x,~,x')J(x')dx

Here

qv+1(X,~. x')

(5.87)

2v ,....-,.._)

=Os-/···

V

j e-iE'i= yiqil]P;(x 1

+ yi- 1 ,t + rf ,x + ti)Pv+1(X+ ft,~, x')dyvd:,t,

3=1

where

fi°= 0,

yi = y 1 + · · · + yi (j = 1,2, · · · , v),

dy v = dly • • • dvy '

d~v J,:;:V 'T/ = d--J. TJ • • • UTJ •

Using (5.85), we integrate by parts in (5.87):

qv+i(x,~,x')

(5.88)

2v ___.....___

V

=Os-I ...I e-iE'J=1 II

(1 + IYtlmo)-1(1+ D~o)

yiqi

£=1 V

X

lIP;(x

+ "f- 1,t + rf,x + "f)Pv+1(X+ f/,t,x')dyvdrt.

i=l

We then make a change of variables

zi =y1+···+yi

(J. = 1 2 · · · v)

' '

' '

which is equivalent to

yi

= zi - zi-1,

zo = 0.

Noting V

LYirf j=l

V

= Lzk(r/k=l

r,k+l), T/k+l= 0,

53

5.3. PROPAGATION ESTIMATES FOR THE FREE EVOLUTION we again integrate by parts in (5.88):

q11+1(x,e,x')

(5.89)

211

=Os-/ ..·I ,.._......___

II

e-iEk=I zk(71k-71k+I)

II(1 + l1l-17k+11motl(l

+ n;;,o)

k=l II

X

II(l + lze- ze-11motl(l + n;eo) i=l II

X

IIP;(x

+ zi- 1,e + ryi,x + zi)P11+1(X + z",e,x')dz"dij".

j=l

By (5.86), we now obtain the estimate: 11+1

lq11+1(X, e, x')I ~

co+lII IP;l3mo j=l

for some constant C0 > 0 independent of v. Differentiating the both sides of (5.87) and estimating similarly, we have with f; being 3-dimensional mulit-indices 11+1

L II IP;l3mo+li;I·

lq11+1(x,e,x')le ~ Co+l

(5.90)

1e1 + +iv+1l~t j=l 00

Note that the constant Ce in (5.84) depends on£, while in the present estimate (5.90) it is replaced by the sum Eie 1 + ..·Hv+il 9 , which enables us to take the constant Co independent off. Using this estimate we prove Theorem 5.4 Let p(x,e,y) satisfy !Pie< oo for all£~ 3mo, Then P defines a bounded operator from 1l = L 2 (Rm) into itself satisfying

= p(X,D.,,X') (5.91)

for the constant C0

> 0 in (5.90}.

Proo/4: Let XE C8°(B), x(0) = 1 and 0 ~ x(x) ~ 1, where B = {(x, 1}, and set

p,(x,e,x')

= X(Ex,fe,fx')p(x,e,x').

Set K(x, x') 4 Proof

= f

here follows that of [34] p.224

i(x-x')ep,(x,

}Rm

e,x')df

e,x')I max(Jxl,1e1,lx'I) ~

CHAPTER 5. FREE HAMILTONIAN

54 Then

PJ(x) = piX,Da:,X')f(x)

f

=

K(x,x')f(x')dx'

}Rm

satisfies the estima1;e for f E S (5.92} where½ > 0 is the volume of the ball Now set for v = 2e (l = 0, 1, 2, · · ·)

1(1 0): If/ is localized in a region G of R"",physically the region G should move to the region G +tv = {x +tvl x E G} after time t. Thus the localization factor Pa on the LHS should effect so that IIPacitHoPaf II decays as t -+ oo. Theorem 5.5 tells that the rate of this decay is t-s. In the case of ,:5.102)of Theorem 5.6, P+ on the RHS of PacitHoP+ restricts the initial function to the phase space region where cos(x, v) ~ 0 + p by (5.97). Then the state e-itHoP+f propagates toward the direction v ':/:0 which is almost parallel to x. Thus the location of the state should be separated from 0 when t -+ oo. Theorem 5.6 gives the rate of this separation. In the case of Theorem 5.7, similarly to (5.102) of Theorem 5.6, the state citHop+f propagates toward the region in the phase space where x and v are almost parallel to each other. Since P_ on the LHS restricts the state to the region where x and v are anti-parallel, the state P_e-itHop+f should decay. Theorem 5.7 gives the rate of this decay. We now prove these theorems. Proof of Theorem ~·.5: It suffices to prove (5.101) for even integers s ~ 0, because we have only to interpolate them to get the desired estimates. The case s = 0 is obvious. Let s > 0 be an even iILteger.We recall (5.49):

e-itHof(x)

=Os-//

2/2-Et1) iCzE-tlEl f(y)dy 0 + p/3, IYI? a, 1e1? a} supp p __ (x,{) C {(x,{)I

and (5.110) for any k, l

= 0, 1, 2, · · ·.

Then we have to estimate the four terms:

(x}op __e-itHoP+_(x}o,

(x}op __e-itH0 P++(x}o,

(x}op_+e-itHoP+-(x}o,

(x}op_+e-itHop++(x}o.

By Theorem 5.6 a:ad (5.110), we have

ll(x}oP_+e-itHop+_(x}oll ~ Cso(t}-s, ll(x}oP_+e-itHo P++(x}oll ~ Cso(t}-s, ll(x}op__e-:-itHop+_(x}oll ~ Cso(t}-s. Thus we have only to consider

(x}op __e-itHoP++(x}of(x)

=Os-ff

(5.111)

i 0 and Ca > 0 independent of x E ~. H is a generalization of a two-body Hamiltonian that describes the two-body system in R3 to a general dimension m = 1, 2, · · ·. Vs and VL are called short- and long-range potentials respectively. The assumption (6.2) can be weakened to allow some local singularities with respect to x E Rm. E.g. (6.4} is known sufficient ([8]) for some results we prove below. (6.4} includes the Coulomb singularities of order 1/lxl at x = 0, thus together with the long-range part VL, V covers the Coulomb type long-range potentials. As the inclusion of singularities is of rather technical nature, we restrict the description below to the potentials satisfying (6.2} and (6.3). As is well-known (see, e.g. [43], Chapter 4.), the perturbed Hamiltonian (6.1) has eigenvalues in general, unlike the unperturbed Hamiltonian H0 • Thus we have first to specify eigenvalues and eigenspace 1lp(H) of H. Then restricting the total Hilbert space 1l to the continuous spectral subspace 1lc(H) = 1lp(H)1.,we will consider the properties of scattering states f E 1lc(H). Before going to the investigation of eigenvaluesof H, we see that H defines a selfadjoint operator in 1l. In the case of the unperturbed Hamiltonian H0 , the selfadjointness was 61

CHAPTER 6. TWO-BODY HAMILTONIAN

62

trivial by virtue of the expression {5.5). But in the case of the perturbed Hamiltonian, getting such an expression is a problem that will be solved later. We recall the definition of the adjoint open.tor H* of H. Let V(H*) be defined as the set of all g E 1i such that there exists an f E 1i satisfying (g, Hu)

= (f, u)

for all u E V(H).

(6.5)

Since V(V) = 1i, tile domain V(H) of the Hamiltonian H = H 0 +Vis equal to V(H 0 ) = H 2 (Rm), which is dense in 1i. Thus the relation (6.5) determines f uniquely. Then we define H*g = f for g E V(H*). In general, His called symmetric if H* is an extension of H (in notation JI CH*) and selfadjoint when H* = H. A symmetric operator His called essentially selfadjoint if the closure H** of H is selfadjoint. In our caseof Ho and V, it is clear that Ho and V are selfadjoint with V(H 0) = H 2 (Rm) = V(H 0 ) by (5.5), and V(V*) = 1i = V(V) by our assumptions (6.2) and (6.3). Thus V(H*) = V(H 0)nV(V*) = H 2 (Rm) = V(H); hence H is selfadjoint. Turning to the eigenvalues of H, we define i[H, A] as a form sum (i[H, A]f, g)

= i(Af,

Hg) - i(H f, Ag)

{6.6)

for f, g ES, where as before A= (x · D., + D., · x)/2 = x · D., + m/(2i) = D., · x - m/(2i) is a selfadjoint operator with V(A) = Ht(~), the weighted Sobolev space, as defined in the section of Notation at the beginning. We first prove the non-existence of positive eigenvalues of H. To do so we set P(>.) = EH(>.) - EH(>. - 0) for ).. > 0. (See the section of Notation.) Then letting B = (>. Ejµ, >. + Ejµ) with f. > 0 sufficiently small and µ > 1 sufficiently large, and using the equalities as form mms i[Ho, A] = 2Ho and i[VL, A] = -x · v'., VL(x), we have for any u E 1i (EH(B)i[H, A]EH(B)u, u)

(6.7)

= i(ABH(B)u,

H EH(B)u) - i(H EH(B)u, AEH(B)u)

= i(ABH(B)u,

VsEH(B)u) - i(VsEH(B)u, AEH(B)u)

+(i[VL, A]EH(B)u, EH(B)u)

= i(x

+ (2HoEH(B)u,

· D.,EH(B)u, VsEH(B)u) - i(VsEH(B)u,x +m(VsEH(B)u,EH(B)u)-

-(2V EH(B)u, EH(B)u)

EH(B)u)

· D.,EH(B)u)

(x · v'.,VLEH(B)u,EH(B)u)

+ (2H EH(B)u,

EH(B)u).

By (6.2) and (6.3), EH(B)Vs (x · D.,)EH(B), EH(B)VsEH(B), EH(B)(x · v'., VL)EH(B) and EH(B)VEH(B) are compact operators defined on 1i. In (6.7), we replace u by un = I, where 0 is the constant in (6.2). By Theorem 5.5 and (6.2), we have (6.18) Thus the integral h (6.17) converges for such a g and the proof is complete. D By the definition (6.16) of W±, we easily see that W± are isometric operators from 1-l into 1-land the relation holds: (6.19)

6.2. WAVE OPERATORS

65

By (5.56) and its variant for H, we have for

E

> 0, >..E R 1 and f

E

1l

1±00 eis(>.±iE-Ho) fds R(>..± ie)f = i 1±00 eis(>.±ie-H) f ds. R.o(>.. ± ie)f

=i

(6.20) (6.21)

Taking the Laplace transforms of both sides of (6.19), we thus have (6.22) It follows from this that -21 .(R(>..+ iE) - R(>..- ie))W±f 1ri

= W±-21 .(R.o(>.. + ie)- R.o(>.. - ie)). 1ri

(6.23)

Let EH(a) = ½(EH(a- 0) + EH(a)), etc. Then it is not difficult to see (see [50] p. 325 or [21] p.359) by using (5.10)-(5.12) that for -oo < a < b < oo s-lim

~ la fb (R(>..+ iE)-

R(>..- iE))d>.. = EH(b) - EH(a),

(6.24)

~ t (R.o(>..+ iE) 21ri la

R.o(>.. - ie))d>..= E 0 (b) - E 0 (a).

(6.25)

e.j.0 21ri

s-lim 40

From these and (6.23), we now have (6.26) for any Borel set BC (-00,00), where EH(B) and E 0 (B) are spectral measures for H and H0 respectively. From (6.26), we have for any continuous function F(>..), (6.27) This relation is called the intertwining property of the wave operators W±. We define the absolutely continuous spectral subspace 1lac(H) of a selfadjoint operator H as the space of functions f E 1l such that

EH(B)f

=0

if

IBI= 0,

IBIis the

Lebesgue measure of a Borel set B ofR 1lv(H)l. is equivalent to the condition where

(6.28) 1.

We remark that f E 1lc(H)

(EH(>..)J,J) is continuous with respect to >..E R 1 .

=

(6.29)

In fact, the eigenspace of H for an eigenvalue >..E R 1 is equal to P(>..)1l:= (EH(>..)EH(>..- 0) )1l, where EH(>..- 0) = s- limµt-XEH(µ). Thus 1lp(H) is spanned by P(>..)1l with >..E R 1 , and the orthogonal complement 1lp(H)1. consists of those elements f that satisfy (6.29). In the case of the free Hamiltonian H0 , it follows from Theorem 5.1 and 1lv(Ho) = {O}that

1lac(Ho)= 1lc(Ho) = 1l.

(6.30)

CHAPTER 6. TWO-BODY HAMILTONIAN

66

Combining this wi1;hthe intertwining property (6.26), we see that for range R(W±) = Wi:1i,

f = W±g in the (6.31)

for a Borel set B with the Lebesgue measure IBI= 0. Namely W±J (f E 1i) belongs to the absolutely continuous spectral subspace 1iac(H)(c 1ic(H)) of H. Thus we have shown (6.32) For the perturbed Hamiltonian H, 1ip(H) =I {O} in general as seen above, thus (6.30) does not necessarily hold. Nevertheless by Theorem 3.2, it is expected that (6.33) holds. If this equality holds, 1ic(H) is unitarily equivalent to 1ic(H0 ) = 1i = L2 (Irn) by the unitary operators

W:J,: 1i = 1ic(Ho) = 1iac(Ho)-+ 1ic(H) = 1iac(H),

(6.34)

which intertwine Hand H 0 • The relation (6.33) is called the asymptotic completeness of the wave operators W±. Assuming the asymptotic completeness holds, we define unitary operators (6.35) where :Fis Fourier transformation as before. Then by the intertwining property (6.27), we have for f E 1ic(H) n V(H)

:F±HJ({) = :FW±Hf({)

= :FHoW±f({) = l~2 :F±J({).

(6.36)

Thus :F±are consid,~redas generalizations of Fourier transformation that diagonalize Hon 2 /2):Ff({) 1ic(H) as the usual Fourier transformation :F diagonalizes H0 : :FH0 f({) = (1{1 (f E 1ic(Ho) n V(ffo) = V(Ho)). Thus the asymptotic completeness gives a spectral representation of tbe perturbed Hamiltonian H = Ho + V.

6.3

Asymptotic

completeness

In this section, we will be devoted in the proof of the asymptotic completeness (6.33). To do so, since by definition

1iac(H) C 1ic(H),

(6.37)

we have only to show that (6.38)

67

6.3. ASYMPTOTIC COMPLETENESS

The definition of wave operators W± in {6.16) is known working only for short-range potentials satisfying {6.2) or {6.4). To extend {6.38) to include the long-range potentials satisfying (6.3), it is necessary to modify the definition {6.16). We here choose a modification of the form W f ±

=

lim itH Je-itHof t--+±oo

'

{6.39)

where J is an identification operator or time-independent modifier introduced in [19], and will be defined as a Fourier integral operator of the form

J f(x)

=

=

Os-ff f(y)dy~ c.nf !(~)d{. i(,p(z,{)-y{)

{6.40)

i,p(z,{)

J is the Fourier transform of f E S (Rm), c;,. = (271" )-m/ 2 , and the phase function cp(x,~) will be constructed as a solution of an eikonal equation Here

in the forward and backward regions, in which x E Rm and ~ E ~ are almost parallel and anti-parallel to each other, respectively. With cp(x,~) being well-defined, we argue as follows: We appeal to the Cauchy criterion of convergence in proving the existence of W± and its asymptotic completeness: 'R(W±) = 1ic(H), with using Theorem 3.2. For instance, the argument for the asymptotic completeness is as follows: We consider the case t -+ +oo only. The other case is treated similarly. Noting that the vectors of the form EH(B)g with g E 1ic(H) and B being a Borel subset of the interval {0,oo) are dense in 1ic(H), we evaluate the difference for g E 1ic(H) with g = EH(B)g (B C {d2/2, b), 0 < d, 0 0 so small that C0 p°0 < 1/2 we obtain the following

Proposition 6.4 Take p > 0 so that C0p°0 < 1/2. Then for ±t 2: ±s 2: 0 one can

construct diffeomorphismsof~ X

H

eH

(6.64) (6.65)

y(s,t,x,e) 11(t,s, x, e)

such that { q(s, t, y(s, t, X, e), ~) = X p(t,s,x,17(t,s,x,e)) = e

.

(6.66)

CHAPTER 6. TWO-BODY HAMILTONIAN

70

y(s, t, x, {) and rJ(t,s, x, {) are C00 in (x, {) E R 2m and their derivatives8~8f y and 8~8f rJ are C 1 in (t, s, x, (1. They satisfy the relation { y(s, t, x, {): q(t, s, x, rJ(t,s, x, {)) rJ(t,s, x, {) - p(s, t, y(s, t, x, {), {)

(6.67)

and the estimates for any a, /3 1a~af[V.,y(s,t,x,{)-J]I:::;

0 (s}- 61 , Ca(3p°

0 (s}-l-oi. 1a~af[v.,,,,(t,s,x,{)]I:::; Ca(3p°

l8f[rJ(t,s,x,{)-{]I ~ Cap°0 (s)- 61 l8f[y(s, t, x, {) - x - (t - s){]I :::;Cap°0 min( (t}l-oi, jt - sl (s}- 61).

(6.68) (6.69) (6.70) (6.71)

Furtherfor any la + /31 2: 2 0 (s}- 61 , ,a~af,,,(t,s,x,{)I:::; Ca(3p°

(6.72)

0 (t - s}(s}- 01 . 1a~afy(s, t, x, {)I :::;Ca(3p°

(6.73)

Here the constants Ca, Caf3> O are independent oft, s, x, { The following illustration would be helpful to understand the meaning of the diffeomorphisms y(s, t, x, {) and rJ(t,s, x, fr Let U(t, s) be the map that assigns the point (q,p)(t, s, x, rJ)to the initial data (x, rJ). Then times

U(t, s)

c(t,:x,eJ f--t

(6.74)

We now define i~(t,x, {) by

cp(t,x,{)

= u(t,x,rJ(t,0,x,{)),

(6.75)

where

u(t, x, rJ)= x · rJ+ lot (Hp - x · v' .,Hp)(r,q(r, 0, x, rJ),p(r,0, x, rJ))dr.

(6.76)

Then it is shown by a direct calculation that cp(t,x, {) satisfies the Hamilton-Jacobi equation

= il{l 2 + Vp(t,v'e¢(t,x,{)), cp(0,x,{) = X • {,

(6.77)

v'.,cp(t,x,{) = rJ(t,0,x,{), v'e¢(t,x,{) =y(0,t,x,{).

(6.78) (6.79)

8t¢(t,x,{)

and the relation

6.3. ASYMPTOTIC COMPLETENESS

71

We define for (x, ~) E R2m

±(x,e) = t--.±oo lim ((t,x,e)-(t,O,e)).

(6.80)

We will show the existence of the limits below. We set for R, d > 0 and a 0 E (-1, 1)

r ± = r ±(R, d, a0 ) = {(x, e)E R2mllxl2: R, lei2: d, ± cos(x, ~) 2: ±ao}. Proposition

(6.81)

6.5 The limits (6.80} exist for all (x, e) E R 2m and define C 00 functions of

(x,e). The limit ±(x,e) satisfies the eikonal equation: For any d > 0 and ao E (-1, 1), there is a constant R = Rd = Rduo> 1 such that for any (x, e) E r ± = r ± (R, d, O'o),the following relation holds: (6.82)

Furtherfor any a, /3 we have the estimate: (6.83)

where C013> 0 is independent of (x, 0 Er±· Proof: We consider ¢ = ¢+ only. can be treated similarly. We first prove the existence of the limit (6.80) fort -t +oo. To do so, setting (6.84) we show the existence of the limits

t

= t--.oo lim 8~8f8tR(r,x,e)dr+8~8f(x·e). }

lim8~8fR(t,x,e)

t--.oo

(6.85)

0

By Hamilton-Jacobi equation (6.77),

BtR(t, x, e)

= Bt(t, x, e) - Bt(t, 0, ~) = Vµ(t,Ve(t,x,e))- Vµ(t,Ve(t,o,e))

=

(6.86)

(Ve(t,x,e)- "ve(t,o,e)) · a(t,x,~)

= (y(O,t, x, e) - y(O,t, 0, ~)) · a((t, x, ~) VeR(t,x,0

· a(t,x,e),

where

a(t,x,e)

= fo\vxVp)(t, "ve(t,o,e) +0"veR(t,x,~))d0,

VeR(t,x,~)

= x · fo\Vxy)(O,t,0x,~)d0.

(6.87) (6.88)

CHAPTER 6. TWO-BODY HAMILTONIAN

72 By (6.68), we have for any a, f3

l~BtVeR(t,x,{)I By (6.71) and (6.W), for

(6.89)

$ CQp(x}.

l/31 ~ 1 18tVe¢(t,o,{)I ::; Cpltl.

(6.90)

From this, (6.87), and (6.89), we have

l~Bta(t,x,{)I

$ CQp(t}-1-.s/2(x}IQl+I/JI_

Thus by (6.86), (6.H9)and (6.91), there exists the limit for any

= lo ('° a~a: (VeR(t,x,{)

lim a~atR(t,x,{) t--+oo

(6.91)

a,/3

· a(t,x,{))dt+~at(x

· {).

(6.92)

In particular, ¢, = ¢>+(x, {) = limt--tooR(t, x, {) and 77(00,0, x, {) = limt--+oo V :i:¢(t,x, {) exist and are C 00 in (x, {). Next we show (ti.82). By the arguments above, the following limit exist:

= =

lim V:,;¢,(t,x,{) = lim 17(t,O,x,{)

t--+oo

t--+oo

(6.93)

lim p(O,t, y(O,t, x, {), {).

t--+oo

Thus for a sufficiently large lxl (i.e. for lpxl ~ 2) we have

. )12+ VL(x) -- -i lim . I( ( 2 -21 IV:i:¢+(a,,{ 2 t--+oo p O,t,y 0,t,x,{),{)I + Vp(O,x).

(6.94)

Set for O $ s $ t < oo

ft(s,y,{)

= ~lp(s, t,y,{)1 2 + Vp(s,q(s, t, y,{)).

(6.95)

Then by (6.53) we have

8ft as (s, Y, {) ·- p(s, t, y, {) · 8sp(s, t, y, {)

(6.96)

a';(s, q(s, t, y, {))

+(V :i:Vp)(s,q(s, t, y, {)) · 8sq(s, t, y, {) + 8

a';(s, q(s, t,

-- 8

y, {)).

On the other hand we have from {6.66) and {6.67)

q(s, t, y(O,t, x, {), {) = q(s, t, q(t, 0, x, 17(t,0, x, {)), {) = q(s,O,x,17(t,O,x,{)), p(s, t, y(O,t, x, {), {) = p(s, t, q(t, 0, x, 17(t,0, x, {)), {) = p(s,O,x,17(t,O,x,{)).

(6.97) (6.98)

6.3. ASYMPTOTIC COMPLETENESS

73

Now using Proposition 6.3, we have for cos(x,e) ~ a0

jq(s,t, y(0, t, x, e), e)I = jq(s,0, x, 17(t,0, x, e))I ~ Ix+ sp(s,0, x, 17(t,0, x, e))I - Cop°0 (s) 1- 81 = lx+sp(s,t,y(0,t,x,e),e)I-Cop° 0 (s}1- 81 ~ c(lxl + slel) - Cop°0 (s}1- 81 - Cop°0 (s}1- 81 , where c > 0 is a constant independent of s, t, x, lei~ d, and from the definition (6.50) of Vp(t,x)

e. By (x, e)E r +(R, d, ao), we have

av:

supp a((s,x) c {xll::; (log(s))lxl/(s}::; Thus there is a constant S

(6.99)

= Sd,uo> 1 independent

2}.

(6.100)

oft such that for any s E [S,t]

aft as (s, y(0, t, x, e), e) = 0.

(6.101)

For s E (0, S], taking R = Rs > 1 large enough, we have for lxl ~ Rand cos(x,e) ~ a0

aft

88 (s, y(0, t, x, e),e)= 0.

(6.102)

Therefore we have shown that for (x, e) Er +(R, d, ao)

ft(s, y(0, t, x, e), e) = constant for 0 ::; s ::; t < oo.

(6.103)

In particular we have

ft(o, y(o, t, x, e), e) = ft(t, y(o, t, x, e), e),

{6.104)

which means 1 2lp(0, t, y(0, t, x, e), e)I2 + v;,(o,x)

Since Vp(t,y) {6.94)

~

= 211e12+ Vp(t,y(0, t, x, en.

0 uniformly in y E Rm when t

~

(6.105)

oo by (6.52), we have from this and

{6.106)

if R > l is sufficiently large. We finally prove the estimates (6.83). We first consider the derivatives with respect toe: af(¢+(x,e) - x · e) =

foaf atR(t,x,e)dt, 00

(6.107)

where as above R(t, x, e) = ¢(t, x, e) - ¢(t, 0, e). Set

'Y(t,x, e) = y(0, t, x, e) - (x + te)

(6.108)

CHAPTER 6. TWO-BODY HAMILTONIAN

74

r

for (x, e) E +(R, d, cro)- Then by (6.71) we have for 0 E [O,1] 1Ve¢(t, o,e) --0VeR(t, x,e)I

= = =

ly(O,t, o,e) + 0(y(O,t, x,e) - y(O,t,o,e))I (6.109) lte + -y(t, 0, e) + 0(x + -y(t,x, e) - -y(t,0, en1 10x+ te + (1 - e)'Y(t,o, e) + 0-y(t,x, e)I ~ Co(0lxl + tie!) - c1l 0 min( (t}l-oi, ltl)

e,

for some constants Co,c1 > 0 independent of x, 0 and t ~ 0. Thus there are constants p E (0, d) and T = Td,uo> 0 such that for all t ~ T and (x, e) Er +(R, d, cro)

('ve¢(t,o,e) +0VeR(t,x,e)}-

1 ::;

C(0lxl +tlel}- 1.

(6.110)

Therefore a(t, x, e) defined by (6.87) satisfies by (6.89) and (6.90) lafa(t,x,e)I::;

Cp

fo1 (0lxl +t1e1}- 1- 6d8.

(6.111)

Using (6.109), we f:ee that (6.111) holds also fort E [O,T] if we take p Therefore for all (a:,e) E +(R, d, cro) we have from (6.86) and (6.89)

r


c,sufficiently small and R (x, e) Er +(R, d, ere,)

= Rd,uo,P>

1 sufficiently large, we have for

lq(T, 0, X, 17( 00, 0, X, e))I ~ Co(lxl + rl{I)

(6.115)

6.3. ASYMPTOTIC COMPLETENESS

75

for some constant Co > 0. Therefore we obtain (6.116) For higher derivatives, the proof is similar. For example let us consider

1 1 00

ae{C'vxVp)(r,q(r,0,x,17(00,0,x,e)))}dr

00

(V X "xVp) (r, q(r, 0, x, 77(00,0, x, e))) V{q. V(17dr,

(6.117)

where we abbreviated q = q(T, 0, x, 11( oo, 0, x, en and 77 = 77(oo, 0, x, e). The RHS is bounded by a constant times (6.118) for (x, e) Er +(R, d, ao) by (6.57) and (6.70) of Propositions 6.3 and 6.4. Other estimates are proved similarly by using (6.57), (6.62), (6.70) and (6.72). D Now let -1

< a_ < a+ < 1 and take two functions t/J±(a) E C ([-1, 1]) such that 00

(6.119) (6.120) (6.121) and set

X±(x,e)

= t/J±(cos(x,e)),

(cos(x,e)

=

:i"

1

; ). 1 1

(6.122)

We then define the phase function cp(x,e) by

cp(x,e) = {(+(x, e) -

X • e)x+(x,

e) + (-(x, e) -

X

(6.123) • e)x_(x, e)} ¢(2e/d)¢(2x/ R) + X • e,

where ¢(x) is the function defined by (6.51). cp(x,e) is a C 00 function of (x,e) E R 2m. Noting that x+(x,e) + X-(x,e) 1 for x =/:-o,e =/:-0, we have proved the following theorem.

=

Theorem 6.6 Let the notations be as above. Then for any d > 0 and -1 < a_ < a+ < 1, there is R = Rd = Rdn± > 1 such that Rd > 1 increases as d > 0 decreases and the followings hold:

i) For lei ~ d, lxl ~ R and cos(x, e) ~ a+ or cos(x, e) ::; a_ (6.124)

CHAPTER 6. TWO-BODY HAMILTONIAN

76

ii) For any multi-indices a, /3 there is a constant Cap > 0 such that

,a~af(cp(x,e) -

Cap(x) 1- 0 be the constant in (5.91). Then we

1 sup sup 18~0:BJ(l- J(x,17,y))I $ Cm0 R-6o < 2C, la+/3+~·19mo :i:,71,y O

(6.142)

by taking R > 1 large enough. Then we have from Theorem 5.4 that (6.143)

III-JJ*ll:5~Thus J J* is invertible with the inverse 00

(JJ*)-1

= (I -

(I - JJ*))-1

= L(I - JJ*)i,

(6.144)

j=O

whose RHS conveqes in operator norm by (6.143). This implies that the range 'R,(J) equals 1£ and J* is one-to-one. Furthermore, {6.142) implies that the symbol r(x, 17,y) = 1 - J( x, 17,y) of the ¢do I - J J* is small so that the series of symbols 00

q1(x,17,y)= I:r;(x,17,y)

(6.145)

j=O

converges in the Fr,~chespace 8 0,0 of symbols p(x, 17,y) whose semi-norms are

IPle =

sup sup IB:0:8Jp(x, 11,y)I< oo l•:t+/3+-rl:5l :i:,71,y

(l

= 0, 1, 2, · · · ).

(6.146)

Here in (6.145), r;(x, 17,y) is the symbol of the ¢do (I - JJ*)i (j = 0, 1, 2, · · · ). To see that (6.145) converges in 80,0 , we note that the inequality (5.90) stated before Theorem 5.4 implies lr;lt $ (Co);

~=IT

lrl3mo+ltkl$ (Co);(lrlamo);-t(lrlamo+d

ll1+···H;l:5!k=l

(

L

1) {p.147)

ll1+··+l;l:5l

where r(x,17,y) = r1(x,17,y) = (1- J)(x,17,y). Since

L

t(

1= ll1+··+l;l:5l k=O

3i

+:-

1 ) $ Gel

(6.148)

6.3. ASYMPTOTIC COMPLETENESS

79

for some constant Ce > 0 independent of j multi-indices in (5.90)), we have

= 0, 1, 2, · · ·

(recall that £i's are 3-dimensional (6.149)

Thus (6.142):

1

Colrlam < 2, 0

(6.150)

implies the convergence of the series (6.145) in the symbol space S0,0 , and we have from (6.144) with Q 1 = q1(X, D.,, X')

Qi(JJ*)

= (JJ*)Q1 = I.

(6.151)

We next consider for g E S

:FJ* J:F-1g({)

=Os-ff e-i(cp(y,f.)-cp(y,11))g('fJ)dydrj.

(6.152)

Similarly to (6.134), we write

cp(y,{) - cp(y,'TJ) =

({- 'TJ)· Veep({,Y,'TJ),

(6.153)

where

Vecp(e,Y,'T/)=

foVecp(y, + 0({ - 'TJ))d0. 1

'T/

(6.154)

Then noting that the inequality similar to (6.135) holds also for Vecp({,Y,'TJ), we make a change of variable:

z

= Vf.cp({,Y,'TJ),

(6.155)

and obtain

:FJ* J:F- 1g({)

=Os-ff e-i(f.-11J({, z, 'fJ)g('fJ)dzdrj, )z

(6.156)

Here J({,z,'TJ)= Jdet(VyVecp- 1 (e,z,'f/))I is a Jacobian, which belongs to S0,0 . Arguing similarly to the case of J J*, we can now construct a 1/JdoQ2 = q2(X, D.,, X') that satisfies q2 E So,oand

Q2(J*J)

= (J* J)Q2 = I.

(6.157)

(6.157) and (6.151) show that J has an inverse J-1: 1{-+

1{

(6.158)

that is expressed as (6.159)

80

CHAPTER 6. TWO-BODY HAMILTONIAN

Since J* is bounded as we have seen, and Q2 = q2(X, Dz, X') is bounded on 1-lby q2 E So,o and Theorem 5.4, 1- 1 = Q2 J* is also a bounded operator on 1-l. We are now prEpared to prove the existence and asymptotic completeness of the wave operators {6.39). Before going to the proof, we remark that the definition {6.39)should be understood as follows with taking into account the dependence of the identification operator J = Jd on d > 0. Namely for f E :,:- 1cr(R"'), the support of whose Fourier transform is contained in a set :E(d) := {{I 1{1 ~ d}, we define W±f by {6.39)with taking J = Jd. Noting that the phase function cp(x,{) of Jd is equal to cp(x,{) = +(x,{)x+(x,{) + -(x,{)x-(x,{)

(6.160)

in :E(d)n {xi lxl ~ R}, we have a definition of W± independent of d > 0 and R =Rd> l by extending this W± to the whole space 1-lby preserving the boundedness. Since the proof of the existence of W± is quite similar to and simpler than that of the asymptotic completeness, we only prove the latter. We have already stated the outline of the proof of the asymptotic completeness at the beginning of this section. There what should be noted is that we prove the existence of the limit (6.47) for g = EH(B)g, where the Borel set B is a subset of (d2/2, b) for some d > 0 with 0 < d2/2 < b < oo. Then by (6.14), the energy restriction EH(B) is translated into the restriction E 0 (B) on the state e-itmHg = EH(B)e-itmHg as m -t oo. Thus asymptotically as tm -t oo, we can assume that the {-support ,)f the symbol P+({, y) of P+ satisfies (5.100) with u = d/2 on the state e-itmHg. Then we can let J = Jd in (6.41), and take P+ thereafter so that its symbol P+({, y) satisfies (5.97), (6.42), and (5.100) with u = d/2. What remains to be proved is then the estimation of the following factor in (6.44):

{6.161) By (6.159) we have

(6.162) By (6.15), the constant 0 +pin (5.97) can be taken arbitrarily as far as -1 < 0 + p < l. We here take 0 + p = u+ + p < l with p > 0, where u+ E (-1, 1) is the number specified in Theorem 6.6. w~write

(6.163) The last factor 1-:. is bounded and can be omitted in the estimation. So we have to estimate

(6.164) By a calculation

(6.165)

6.3. ASYMPTOTIC COMPLETENESS

81

Here the symbol r+(Y,'TJ)is given by

r+(Y,'TJ)=

ff

os-

i(y-z)c;:P+((

+ Vx'P(Y,'TJ,Z),z)(z)012 dzd(

(6.166)

and satisfies by (6.42) lr+l~-l,6/2) := sup supl(y)-

012 (y)101 a;a:r+(Y,'TJ)I

are v-dimensional Laplacians and Me and µft) are the reduced masses. We introduce the inner product in the space R!' = Rv(N-l) as in (7.9): (X,

y)

=

((xa, xa), (Ya,Ya)) = (xa,Ya) + (x°, ya) k

ICtl-1

LMexe·Ye+

k-1

LL

e=l

e=l

i=l

µfelxft)

·yftl,

(7.13)

and velocity operator

V = (Va,Va) = M-1 P = ( ma-1 Pa,( µ a)-1 Pa),

(7.14)

where

M

=(

ma 0

O)

µa

(7.15)

is the n = v(N - 1) dimensional diagonal mass matrix whose diagonals are given by (Ck) Th en H,o 1s • wn·t ten as M 1, · · · , M k-1, µ (01: , · · · , µ 10k 1_ 1 . 1

Ho= ~(v, v) =Ta+ H8 = ~(va, Va}+ ~(va, va).

(7.16)

We need a notion of order in the set of cluster decompositions. A cluster decomposition b is called a refinement of a cluster decomposition a, iff any Ce E b is a subset of some Dk E a. When b is !'I.refin~ment of a we denote this as b ::; a. b 1:.a is its negation: some cluster Ce Eb is not a subset of any Dk E a. Thus for a pair a= {i,j}, a ::; a means that a = {i, j} C Dk for some Dk E a, and a 1:.a means that a = {i, j} 1.

7.2. SCATTERING SPACES

89

Definition 7.4 Let real numbers r, a, 8 and a cluster decomposition b satisfy O::; r::; 1, a, 8 > 0 and 2 ::; Jbl::;N. 6 (~) for O < r::; 1 by i) Let~ cc R 1 - 'T be a closed set. We define Sb"'

II

Si;"'6(~) = {f E EH(~)1-l J e-itHf,..., For r

F(lxal ?: at)F(lxbl ::; 8ne-itH f as t-+ oo~7.30) ai,b

= 0 we define sg"'(~)by S'g"'(~)= {f E EH(~)1-l I

II

F(Jxal ?: at)F(Jxbl ::; R)e-itH 111 = 0}. ai,b

lim limsuplle-itH f -

R.->oo

t.->oo

(7.31)

We then define the localized scattering space S-;(~) of order r E (0, 1] for H as the closure of

unst

0 (~)

e-itHf,...,

sg(~) is

= {f

E EH(~)1-l J 3a

> 0,W > 0:

II

F(lxal?: at)F(lxbl::; 8ne-itH f as t-+ oo}. ai,b

(7.32)

defined as the closure of

O

lim limsuplle-itH R->oo

t.->oo

f-

II

F(lxal ?: at)F(Jxbl ::; R)e-itH111 = O}. ai,b

ii} We define the scattering space Sb of order r E

(7.33)

[O,1] for H as the closure of (7.34)

We note that

6(~), s;;"'

Sg"'(~), Sb(~) ands;; define closed subspaces of EH(~)1-l and

1-lc(H),respectively. Proposition 7.5 Let~ cc R 1 - 'T and f E S;;"'6 (~) for O < r ::; 1 or f E Sg"'(~) for r = 0 with a, 8 > 0 and 2 ::; IbI ::;N. Then the following limit relations hold:

i} Let a 1:.b. Then for O < r ::; 1 we have when t -+ oo (7.35) For r

= 0 we have lim limsup IIF(Jxal< at)F(Jxbl ::; R)e-itH

R.->oo

t.->oo

!II= 0.

(7.36)

CHAPTER 7. MANY-BODY HAMILTONIAN

90

ii) For O < r ~ 1 we have when t

-+ oo (7.37)

Farr=

0

lim limsup IIF(lxbl >

R---+oo t---+oo

R)e-itH

ill = 0.

(7.38)

iii) There exists a sequence tm-+ oo as m-+ oo depending on f E Sr'76 (Li) or f E Sg17(Li) such that (7.39)

for any function 0, a ~ a' > 0 and o' ~ o > 0 and Li CC R1 - T. then sgu(Li) C sga'(Li), Sg17 (Li) C sr 16 (Li) C S(a'o'(Li), Sg(Li) C Sb(Li) CS( (Li), Sg(Li) C Sb(Li) C Sb, and sg C Sb CS(. Proposition 7. 7 Let b and b' be different cluster decompositions: b "f'b'. Then for any 0 ~ r ~ 1, Sb and Bb,are orthogonal mutually: Sb..l Sy.

7 .3

A partition of unity

To state a proposition that will play a fundamental role in our decomposition of continuous spectral subspace by Sl, we prepare some notations. Let b be a cluster decomposition with 2 ~ lbl ~ N. For any two clusters C1 and C2 in b, we define a vector Zbi that connects the two centers of mass of the clusters Ci and C2. The number of such vectors when we move over all pairs of clusters in bis

kb=

(

l~I ) in total. We denote these vectors by

Zbl, Zb2, • • • , Zbk&·

Let Zbk (1 ~ k ~ kb) connect two clusters Ce and Cm in b (£ "f' m). Then for any pair a = { i, j} with i E Ct and j E Cm, the vector Xe, = Xij is expressed like (zbk, x(Ct), x(Cm)) E Rv+v(ICtl-l)+v(ICml-1), where x(Ct)( E Rv(ICtl-1)) and x(Cm)( E Rv(ICml-1)) are the positions of ·;he particles i and j in C,.and Cm, respectively. The vector expression

7.3. A PARTITION OF UNITY

= (zbk,x(Ce),x(C,,.))is in the

91

If we express it in the larger space R~bkx R;iN-lbl), it would be Xa = (zbk,xb), and lx0 ,l2= lzbkl2 + lxbl2 • Thus if lzbkl2 is 2 , e.g. if lzbkl2 > p > 0 and lxbl2 < 0 sufficiently large compared to lxbl2 ~ lx(Ce)l2 + lx(C,,.)1 2 > p/2 for all with p » 0 > 0 (which means that p/0 is sufficiently large), then lx0 1 a i b. Next if c < b and lei= lbl + 1, then just one cluster, say Ct E b, is decomposed into two clusters q and Cf in c, and other clusters in b remain the same in the finer cluster decomposition c. In this case, we can choose just one vector Zck (1 ::; k ::;;kc) that connects clusters C~ and Cf in c, and we can express xb = (zck,xc). The norm of this x0

space

Rv+v(ICel-l)+v(IC,,.l-l).

vector is written as (7.41) Similarly the norm of x

= (xb, xb) is written

as (7.42)

We recall that norm is defined, as usual, from the inner product defined by (7.13) which changes in accordance with the cluster decomposition used in each context. E.g., in (7.41), the left-hand side (LHS) is defined by using (7.13) for the cluster decomposition b and the RHS is by using (7.13) for c. With these preparations, we state the following lemma, which is partially a repetition of [24], Lemma 2.1. We define subsets n(p, 0) and Tb(P,0) of R;l = Rv(N-l) for cluster decompositions b with 2 ::; lbl ::; N and real numbers p, 0 with 1 > p, 0 > 0:

T,(p, 0)=

2 > plxl'}) n {x I lx,I'> (I - 0)lxl'}, (Q{x I lz.,1

T,(p,0)=

(Cl { I lz.,,1> p)) n { I lx,1> 2

x

x

2

I - 0).

(7.43) (7.44)

Subsets Sand 89 (0 > 0) of Rn= Rv(N-l) are defined by

{x I lxl2 ~ 1}, 89 = {x I 1+0~ lxl2 ~ l}. S

=

Lemma 7.8 Suppose that constants 1 ~ 01 > Pi > 0i > PN > 0 satisfy 0i-1 = 2, 3, · · · , N - 1. Then the followings hold:

~

0i + Pi for

j

i)

SC

U

Tb(Plbl,0lbl)-

(7.45)

25:1bl$N

ii) Let 'Yi > 1 (j

= 1, 2) satisfy "Yl"Y2 < ro :=

~n {pi/0i}25:35:N-1

(7.46)

CHAPTER 7. MANY-BODY HAMILTONIAN

92 If b 1:e with lbl ~ lcl, then

n(,1 1Plbl,'Y201b1) n Tc('Y11Picl,'Y201c1) = 0.

(7.47)

iii) For 'Y> 1 and 2 $ lbl $ N 71,(P 1b1,01b1) n s(JN-1

c Tb(P1b1,0 1b1) n s(JN-1 1 cc n(-rPlbl,-y0lbl)n s(JN-1 c n(,~ -l Plbl,-r~0,b,)n s(JN-1'

(7.48)

where (7.49)

iv) If ~~~r < ro, then for 2 $ lbl $ N Tb(,~- 1Plbl,'Y~0lbl) C {x

I lxal2 > Pibdxl2/2

for all a

1:b}.

(7.50)

v) If-y(l + -y) < ro cmd b 1:e with lbl ~ lei, then Tb(,~- 1PJbl, -y~0lbl) n Tc('Y~-l Piel,-y~01c1) = 0.

(7.51)

Proof: To prove (7.45), suppose that lxl 2 ~ 1 and x does not belong to the set

A=

u

[(n{x I lzbkl2

> Pibdxl2}) n {x I lxbl2 > (1-01b1)lxl2}] .

2:5lbl:5N-1 k=l

Under this assumption, we prove lxal2 > PNlxl2 for all pairs a = {i,j}. (Note that Zbkfor lbl = N equals some Xa,) Let lbl = 2 and write x = (zb1,xb). Then by (7.41), 1 $ lxl 2 = lzbll2 + lxbl2, Since x belongs to the complement Ac of the set A, we have 2. If lzbll2 $ Pibdxl2, then lxbl2 = lxl 2 - lzbll2 ~ (1 lzbll2 $ Pibdxl2 or lxil 2 $ (1- 0lbJ)lxl 2 2 2 2 Pibl)lxl ~ (01-PlbJ)l:cl ~ 01bdxl by 0j-1 ~ 0j+Pj· Thus lxbl2 = lxl 2- lxbl2 $ (1-01b1)1xl for all b with lbl = 2. 2. Then by x E Ac, we can choose Next let lel = 3 and assume lxcl2 > (1 - 01c1)1xl 2 2 Zck with 1 $ k $ kc such that lzckl $ Picdxl. Let C,. and Cm be two clusters in e connected by Zck,and let b be the cluster decomposition obtained by combining C,. and Cm into one cluster with retaining other clusters of e in b. Then lbl = 2, xb = (zck,xc), and lxbl2 = lzckl2 + l:ccl2- Thus lxbl2 = lxl 2 - lxbl2 = lxl 2 - lzckl2 - lxcl2 = lxcl2 - lzckl2 > 2 ~ (1-0lbl)lxl 2, which contradicts the result of the previous step. Thus (1-0Jcl - PJc1)1xl 2 ::orall e with lei = 3. 2 lxcl $ (1 - 01c1)1xl 2, thus lxdl2 = Repeating this procedure, we finally arrive at lxdl2 $ (1 - 01d1)1xl 2 2 2 2 2 lxl - lxdl ~ 01ddxl > PNlxl for all d with ldl = N - l. Namely lxal > PNlxl2 for all pairs a= {i,j}. The proof of (7.45) is complete.

7.3. A PARTITION OF UNITY

93

We next prove {7.47). By b ~ c, we can take a pair a= {i,j} and clusters C1.,CmE c such that a $ b, i E Ct, j E Cm, and f. =/:-m. Then we can write Xa = (zcA:, xc) for some 1 1 1 $ k $ kc, Thus if there is x E Tb('Y Plbl, 'Y281bl) n Te('Yi'" Plel,'Y281e1), then 1 2 > 'Y11P1eilxl 2, 2 > lxbl2 ~ lxal 2 = lzbkl2 + lxel2 ~ lzcA:1 'Y281bilxl

(7.52)

But since lbl ~ lcl, we have Piel> 'Yl'Y281bl when lbl = lcl by {7.46), and Piel> 'Y1'Y281el ~ 'Y1'Y2(81bl + Plbl)> 'Y1'Y281bl when lcl < lbl by 8;-1 ~ 8; + P;, which both contradict the inequality (7.52). This completes the proof of {7.47). · (7.48) follows by a simple calculation from the inequality lxl 2{1 + 8N_1)- 1 $ 1 that holds on SsN-i. {7.50) follows from the relation lx 0 12 = lzbkl2 + lxbl2 stated before the lemma, and (7.51) from 'Yt72= -y{l +-y) and ii). D In the followings we fix constants 'Y> 1 and 1 ~ 81 > P; > 8; > PN > 0 such that 8;-1 ~ 8; + P; (j

= 2, 3, .. · , N

- 1),

max {-r{l +-y), 22Yi'Y~} < ro = i;mn {p;/8;}, --y 1 2~!:,N-1

(7.53) (7.54)

where 1; (j = 1, 2) are defined by {7.49). Let p(.~)e C 00 (R 1) be such that 0 $ p(A) $ 1, p(A) = 1 (A$ -1), p(A) = 0 (A~ 0), and p'(A) $ 0. Then we define functions 0 in (7.60)-(7.61}. Proof: We have only to see (7.67) and (7.68). But (7.67) is clear by (7.48), (7.50), (7.54), (7.62) and (7.63), and (7.68) follows from (7.56), (7.60) and (7.64). D

7.4. A DECOMPOSITION OF CONTINUOUS SPECTRAL SUBSPACE

7 .4

A decomposition

95

of continuous spectral subspace

The following theorem gives a decomposition of 1lc(H) by scattering spaces St(2 ::; lbl::;

N). Theorem 7.10 Let Assumptions 7.1 and 7.2 be satisfied. Then we have

1lc(H) =

EBst.

(7.69)

2::,lbl::,N

Proof: Since the set

is dense in 1lc(H), and St (2 ::; lbl::;N) are closed and mutually orthogonal, it suffices to prove that any 4'(H)f with 4' E C0 (R 1 -T) and f E 1l can be decomposed as a sum of the elements fl in St: 'P(H)f = I:2:s,lbl:SNfl. We divide the proof into two steps. In the first step I), we prove existence of certain time limits. In the second step II), we prove existence of some "boundary values" of those limits, and conclude the proof of decomposition (7.69). I) Existence of some time limits: We decompose 'P(H)f as a finite sum: 'P(H)f = I:Jonite'lj;i0 (H)f, where 'l/JioE C0 (R 1 - T). In the step I), we will prove the existence of the limit L

t~~

L eitHab,>.t(t)*Jb(vb/re)Gb,>.t(t)e-itH'l/Jjo(H)f,

(7.70)

£=1

under the assumption that supp'I/JioC .6.CC!::,,.for some intervals .6.CC!::,,.CC R 1 - 7 with EE!::,,. and diam!::,,.< d(E), where d(E) > 0 is some small constant depending on E E R 1 - 7 and diam S denotes the diameter of a set S C R 1. The relevant factors in (7.70) will be defined in the course of the proof. We will write f for 'l/Ji 0 (H)f in the followings. We take 'lj; E C0 (R 1 ) such that 'lj;(>.)= 1 for >. E .6.and supp 'lj; C !::,,. for the intervals .6.CC A above. Then f = 'lj;(H)f = EH(ll.)f E EH(!::,,.)1£ C 1lc(H) and citH f = 'lj;(H)citHf. Thus we can use the decomposition (7.22) for the sequences tm and M{:' in Theorem 1.3:

e-itmHf = 'lj;(H)e-itmHf = 'lj;(H)

L

f5:/:I' e-itmHf.

(7.71)

2::,ldl::,N

By Theorem 1.3-(7.25)

'lj;(H)P:/3'e-itmHf ,...., 'lj;(Hd)P:/3'e-itmHf

(7.72)

as m -+ oo. Since (7.73)

CHAPTER 7. MANY-BODY HAMILTONIAN

96

where Pd,E;is onE, dimensional eigenprojection for Hd with eigenvalue Ej, the RHS of (4.4) equals

(7.74) By supp '1/Jc !),,, cc R 1 - T, Ej E T, and Td 2'.'.0, we can take constants Ad > Ad > 0 independent of j = 1, 2, .. · such that Ad 2'.:Td 2'.'.Ad if '1/J(Td + Ej) =I-0. Set Ao = maxdAd >Ao= mind Ad> 0 and

:E(E) = {E- A I A Er, E 2'.'.A}.

(7.75)

Note that we can take Ao> Ao> 0 so that

:E(E) CC (Ao,Ao) c (0,oo).

(7.76)

Let IJ!E C[f (R 1) s,:1.tisfyIJ!(A)= 1 for A E [Ao,Ao] and supp IJ!C [Ao- K, Ao+ K] for some small constant K > 0 such that the set [A~,Ao]U [Ao,A~] is bounded away from :E(E), where A~ = Ao - '.2K> 0 and A~ = Ao+ 2K. Then the RHS of (7.74) equals for any

m= 1,2,···

IJ!2(Td)'I/J(Hd)P!;1:I' e-itmH f.

(7.77)

On the other hand, by Theorem 1.3-(7.24) and (7.26), we have

1::1 'I/J(Hd)J5!;1:I' 2

f

e-itmH

0

rv

(7.78)

m

and

IJ!2(Td)'lj;(Hd)P!;1:I' e-itmH f"' IJ!2 (lxdl2 /(2t!i))'I/J(Hd)J5!;1:I' e-itmH f

(7.79)

as m -t oo, whern to see (7.78) we used (7.24) and i[Hd, lxdl2 /t 2] = i[Hg, lxdl2 /t 2] = 2Adjt 2 where Ad==(xd ·pd+ pd· xd)/2, and to see (7.79) the fact that lxdl2 /t~ and Hd commute asymptotically as m -t oo by (7.26). Thus by lxl2 = lxdl2 + lxdl2 we have

1J12(Td)'IJ•(Hd)J5!;1:I' e-itmH f"' IJ!2 (lxl 2 /(2t!i))'I/J(Hd)P!;13' e-itmH f as m -too.

(7.80)

From (7.71)-(7.72}, (7.74}, (7.77) and (7.80), we obtain

(7.81) as m-t oo. Let constants,>

1 and 1 2'.:01 > pj > 0j > PN > 0 be fixed such that

0·J- 1>0-+p· J max

{1 (1 +

3

(1·=2 ' ...

'

N-1)

2,',,' } < ro = Tl

,), 2 1 -

A~

,

~ {pj/0;}

2:5J:5N-l

(7.82) (7.83)

7.4. A DECOMPOSITION OF CONTINUOUS SPECTRAL SUBSPACE for

'YJ(j = 1, 2) defined by

97

{7.49). Set

(7.84) with A&= Ao - 2K defined above. Let To> 0 satisfy 0

< 16ro < A~(< A~ < Ao)-

(7.85)

We take a finite subset {.\Jf= 1 of 7 such that L

r C LJ(.Xe - To,.Xe+ To).

{7.86)

i=l

Then we can choose real numbers At E R 1 , Te> 0 (f

re < ro, uo < To, IAt -

= 1, 2, · · · , L)

and u0 > 0 such that

.Xel < To,

L

r C LJ(Ae -

Tt, At+ Tt),

(At - Tt, At+ Tt) C (>.t- To,>.t+ To),

i=l

dist{{At - re, At+ re), (Ak - Tk,Ak + rk)} > 4uo(> 0) for any£ =I-k.

(7.87)

We note that for £ = 1, · · · , L

{A I re~ IA- (E - Ae)I~ Tt + 4uo} n I;(E) = 0. Now let the intervals A and

(7.88)

Li be so small that

diam Li < diam A

< fo := min {u0 , rt}.

(7.89)

1::,;e::,;L

Returning to (7.74), we have

Td + Ej E suppt/J,

(7.90)

if t/J(Td+ Ej) =I-0 in (7.74). By suppt/J C A, diam A < f 0 , and E E A, we have from (7.90) -fo ~ Td - (E - Ei) ~ fo.

(7.91)

Thus we have asymptotically on each state in (7.74)

2 -2f. 0 < - lxl t2 - 2{E - E-) 3 m

< 2f.o. -

By (7.87), Ej E 7 is included in just one set (At - Tt, At+ re) for some £ 1 ~ f(j) ~ L. Since IEi - At(j)I < Te(j),we have using (7.89)

-2Te(j) - 2uo ~

lxl2t2 m

2{E-

Ae(j))~ 2Te(j)+ 2uo

(7.92)

= f(j)

with

(7.93)

CHAPTER 7. MANY-BODY HAMILTONIAN

98 on each state in ('r.74). Thus L

L ;(1lxl ft;. - 2(E - At)I < 2re+ 2uo) = 1 0

2

i=l

asymptotically as m--+ oo on (7.74). Now by the same reasoning that led us to (7.81), we see that (7.71) asymptotically equals as m--+ oo L

L L

2 ; Ae)I< 2rt + 2uo)W2 (lxl2 f(2t2.n))P;t:te-itmH J. (7.94) 0 (jlxl ft;. - 2(E-

t=l 2:$ldl:$N 2 ft!, - 2(E - At)I < 2rt + 4uo) Since q>.,. 0 (11xl 2re+ 2uo), (7.94) equals

= 1 on supp

2 ft!, - 2(E - At)I < q>.,. 0 (11xl

L

L :~ ;(jlxl ft2.n- 2(E - At)I < 2rt + 4uo) 0

2

l=l 2:$ldl:$N

I

I

2 2 2 x; e-itmH f. 0 ( lxl ft2.n- 2(E - Ae) < 2re+ 2uo)\Jl (lxl f (2t2.n))P;t:r

(7.95)

Set

B

= (x;,-1/ 2A(x)-112,

A= ~(x · p + p · x)

= ~((x,v)+ (v,x)). -Mm

We note by Theorem 1.3-(7.24) and (7.26) that on the state Pd d

. e-•tmH

(7.96)

f

(7.97) 2 asymptotically as tm --+ oo. Using this, we replace q>.,. 0 {Ilxl ft!, - 2(E - At)I < 2re + 2 2uo) by q>.,. (IB -· 2{E Ae)I < 2rt + 2uo) in {7.95). Let cp(A) E C8°((y'2(Ao - 2K), 0 y'2{A0 + 2K))), 0 ::; cp(A) ::; 1, and cp(A) = 1 on [y'2{Ao- K), y'2(A 0 + K)j(::) supp 2 {B) into {7.95) and then remove the factor 2 f2) n {O,oo)). We insert a factor cp W{A \Jl2 (lxl2 f (2t'!,)) using (7.81):

L

L L

2 ft;. ;/llxl

- 2(E - At)I < 2rt + 4uo)

i=l 2:$ldl:$N

x; (IB 2 0

2 (B)P;t:t e-itmH 2{E- Ae)I< 2re+ 2uo)cp

f.

(7.98)

2 /t!, - 2(E- At)I < 2re+ 4uo) we have On supp .,. 0 (11xl

lxJ2

0 < 2{E - At) - 7ro ::; T

::;2(E -

Ae)+ 7ro.

(7.99)

m

Since (7.76), !At- >-el < ro and (7.85) imply

2(E - At) + 7ro 1 14ro 14Agf16 0 2{E - Ae)- 7ro - = 2{E - Ae)- 7ro < 30A0 f16 - Ag < N-1,

(7.100)

7.4. A DECOMPOSITION OF CONTINUOUS SPECTRAL SUBSPACE

99

we can apply the partition of unity in Proposition 7.9 to the ring defined by (7.99). Then we obtain L

e-itmHJ

"'L L L

2 Jb(xb/(rttm)); 0 (11xl/t;.-2(E--\l)l:/:I' e-itmHf, 0 (jB - 2(E-

X

(7.101)

where

rt= J2(E - -\e)- 7r0 > 0 (£ = 1, · · · , L). By the property (7.67), only the terms with d

~

(7.102)

b remain in (7.101):

L

L L L Jb(xb/(retm)); (llxl /t;. - 2(E-

e-itmHf "'

2

0

Ae)j< 2Tt+ 4ao)

i=l 2::,jbl::,N d::,b

4>;/jB2 -

X

2(E- At)I < 2Tt+ 2ao):/:I' e-itmHf.

(7.103)

Using Theorem 1.3-(7.26), we replace xb/tm by vb, and at the same time we introduce a pseudodifferential operator into (7.103): (7.104) with u > 0 sufficiently small. Then (7.103) becomes L

e-itmHf "'

L L L P;(tm)Jb(vb/rt);(11xl/t;. - 2(E 0

2

At)I < 2Tt+ 4ao)

i=l 2::,jbj::,N d::,b

X

2 ; e-itmHf. 0 (jB - 2(E - At)I < 2Tt + 2ao),p2(B)f>:/:I'

(7.105)

We rearrange the order of the factors on the RHS of (7.105) using that the factors mutually commute asymptotically as m -+ oo by Theorem 1.3. Setting 2 2 - 2(E - -\e)j < 2Tt+ 4ao) Gb,>.e(t)= Pb(t)u 0 (1ixl /t 2 x u 0 (jB - 2(E - Ae)I < 2Tt + 2ao)'P(B),

(7.106)

we obtain L

e-itmHf"'

L L L Gb,>.e(tm)* Jb(vb/rt)Gb,>.e(tm)P:/:I' e-itmHf. i=l 2$lbl$N

(7.107)

d::,b

Now by some calculus of pseudodifferential operators and Theorem 1.3 we note that A(t)Jb(vb/rt) yields a partition of unity ]b(xb/(ret)) asymptotically as m -+ oo whose support is close to that of Jb(xb/(rtt)). Then we can recover the terms with d "i b, and -Mm using 7.22, we remove the sum of Pd d over 2 ~ ldl~ N: L

e-itmHf"'

L L i=l 2$lbl$N

Gb,>.e(tm)* Jb(vb/re)Gb,>.e(tm)e-itmH f.

(7.108)

CHAPTER 7. MANY-BODY HAMILTONIAN

100

We note that on. the RHS, the support with respect to B 2/2 of the derivative 2 ¢>~ 2(E - ..Xe)I< 2re + 2uo) is disjoint with '£(E) by (7.57) and (7.88), and the 0 (jB support of cp'(B) iBsimilar by (7.76) and the definition of cpabove. We prove the e,cistence of the limit (7.109) for£= 1, ···,Land b with 2 :s;lbl:s;N. For this purpose we differentiate the function

(7.110) with respect to t, where f, g E EH(b..)1l. Then writing

Dtg(t)

= i[Hb,g(t)] +

!!

(t)

(7.111)

for an operator-valued function g(t), we have

!

(eitHab,>.1 (t)*Jb(vb/re)Gb,>.1(t)e-itH f, g)

2 - 2(E - -Xe)I = (eitHDZ(cp(B))uo(IB < 2re+ 2uo)

1 2 - 2(E - ..Xe)I Xu < 2re+ 4uo)Pb(t)Jb(vb/re)Gb,>.t(t)e-itH f, g) 0 (Ilxl /t 2 - 2(E+(eitHcp(B)DZ(uo(IB ..Xe)I < 2re+ 2uo)) 2 - 2(E- ..Xe)I x¢>uo(llxr!/t < 2re + 4uo)A(t)Jb(vb/re)Gb,>.tCt)e-itH f,g) 2 - 2(E - ..Xe)I +(eitHcp(B}Puo(IB < 2re + 2uo) 2 2 - 2(E- ..Xe)I xDZ (4>u < 2re + 4uo)) A(t)Jb(vb/re)Gb,>.e(t)e-itH f,g) 0 i'.llxl/t 2 +(eitHcp(B)•Puo(IB- 2(E - ..Xe)I < 2re + 2uo) 2 x¢uo(I1xr!;t - 2(E - ..Xe)I < 2Te+ 4uo)DZ(Pb(t)) Jb(vb/re)Gb,>.e(t)e-itH f, g)

+((h.c.)f,g:, +( eitHi[Ib,ob,>.,(t)Jb( Vb/re)Gb,>.t (t)]e-itHf, g),

(7.112)

where (h.c.) denotes the adjoint of the operator in the terms preceding it. We need the foli.owinglemmas (see [24],Lemmas 4.1 and 4.2): Lemma 7.11 Let Assumption 7.1 be satisfied. Let E E R1 - T. Let F(s) E C0 (R1) satisfy O :s;F :s;1 aiid the condition that the support with respectto s2 /2 of F( s) is disjoint

with '£(E). Then there is a constant d(E) > 0 such that for any interval b.. around E with diam b..< d(E), one has 2 00 {

-

1 -F(B)e-itH EH(b..)f

1-ooM

for some constant C > 0 independent off E 1l.

dt :s;Cllfll 2

(7.113)

7.4. A DECOMPOSITION OF CONTINUOUS SPECTRAL SUBSPACE

101

Lemma 7.12 For the pseudodifferentialoperatorPb(t) defined by (7.104} with u > 0, there exist norm continuous boundedoperatorsS(t) and R(t) such that

D!A(t)

= ¼s(t) + R(t)

(7.114)

and S(t) 2: 0,

IIR(t)II::;C(t)- 2

(7.115)

for some constant C > 0 independent oft E R 1 . We switch to a smaller interval D..if necessary in the followings when we apply Lemma 7.11. For the first term on the RHS of (7.112) we have

D!(ip(B)) = ip'(B)i[Hb,B] + R1

(7.116)

with

JJ(H+i)- 1(x)112 i[Hb,B](x)112(H +i)- 111< oo, ll(H + i)- 1(x)R1(x)(H + i)- 111< oo.

(7.117) (7.118)

(See section 4 of [24] for a detailed argument yielding the estimates for the remainder terms R 1 here and S1 (t), etc. below.) By the remark after (7.108), the support with respect to B 2 /2 of ip'(B) is disjoint with 'E(E). Hence the condition of Lemma 4.2 is satisfied. Thus using (7.117)-(7.118) and rearranging the order of the factors in the first term on the RHS of (7.112) with some integrable errors, we have by Lemma 7.11: the 1st term where

= (eitHB~1\t)*Bi 1\t)e-itH f,g) + (eitHS1(t)e-itHf,g),

Bj1>(t) (j = 1, 2) and

(7.119)

S1 (t) satisfy

1:

IIBJ1>(t)e-itH fll 2dt::; Cllfll 2 ,

ll(H + i)- 1S1(t)(H + i)- 111::; cr

(7.120) 2

(7.121)

for some constant C > 0 independent off E EH(D..)1land t E R 1 . Similarly by another remark after (7.108) and Lemma 7.11, we have a similar bound for the second term on the RHS of (7.112): the 2nd term where Bj2\t) (j

= (eitHB~2)(t)*B?>(t)e-itHf,g) + (itHS 2 (t)e-itHf,g),

= 1, 2) and

(7.122)

S2 (t) satisfy

1:

IIBJ2>(t)e-itH fll2dt::; Cllfll2,

ll(H + i)- 1 S2(t)(H + i)- 111::; Cr 2

(7.123) (7.124)

CHAPTER 7. MANY-BODY HAMILTONIAN

102

for some constant C > 0 independent off E EH(L::!,.)1£ and t E R 1 . For the third tum on the RHS of (7.112), we have

2(E- .\e)I < 2re+ 2ao) 2 /t 2 - 2(ExD! (uo(llxl .\e)I < 2re+ 4ao)) 2 2 - 2(E - .\e)I < 2re + 2ao) = tcp(B)u 0 (IB

2 p(B)u 0 (IB

x (~ -

-

;!2)(11xl/t

1

¢~0

2 -

2

2(E - .\e)I < 2re+ 4ao)

+Sa(t),

(7.125)

where S 3 (t) satisfies

(7.126) 2 2 On the support of ,p~ 0 (Ilxl /t - 2(E - .\e)I < 2re + 4ao), we have

lxl/t 2::J2(E - .\e) - 2re - 5ao > 0 by (7.85) and (7.87). Thus there is a large T (x) = lxl, and hence

2

(7.127)

> 1 such that fort 2:'.T we have lxl > 1 and

= (x) (~ . D- El) + (D.~ - El) (x) (A -E1:) t (x) "' t "' (x) t t t t 2

=

2 (;) ( B -

1; 1)

+ tS4(t)

(7.128)

with IIS4(t)11~ Ct- 2 fort 2:'.T. By (3.16), we have

(7.129) with

Ii= [--2re - 5ao, -2re - 4ao], 12= [2re+ 4ao, 2re+ 5ao],

(7.130)

and ¢~0 (1s1 < 2re+ 4ao) 2::0 for s E 11, ¢~0 (1s1 < 2re+ 4ao) ~ 0 for s Eh Consider the case

l:d2 /t 2 -

lxl2

t2 By the factor

(7.131) (7.132)

2(E - .\e) E 12. Then

E (2(E - .\e) +

2re + 4ao, 2(E - .\e) + 2re+ 5ao].

(7.133)

2 - 2(E - .\e)I < 2re + 2ao), we have cp(B),Puo(IB

B 2 E (2(E - .\e) - 2re- 3ao, 2(E - .\e) + 2re+ 3ao]

(7.134)

7.4. A DECOMPOSITION OF CONTINUOUS SPECTRAL SUBSPACE and B ~

103

J2.¾>0. Thus .,t(t)*Jb(vb/re)Gb,>,At) restricts the coordinates in the region: lxal2 > Pibdxl2 /2. Summarizing we have proved that (7.112) is written as

!

(eitHGb,>.t(t)* Jb(vb/re)Gb,>.tCt)e-itH f, g) 2

= (eitHA(t)*A(t)e-itH f,g) + L(eitH Bf'\t)* Bfk>(t)e-itHf,g) k=l

+(85(t)f,g),

(7.139)

where with some constant C > 0 independent of t > T and

£ 00

f

E 1l

2 $ Cll/11 2, IIBt>(t)e-itHEH(~)/11 (j, k = 1, 2)

II(H + i)-185(t)(H + i)-111$

(7.140) (7.141)

crt-min{e,e1}.

Integrating (7.139) with respect to t on an interval [T1 , T2] c [T, oo), we obtain

(eitHGb,>.t (t)*Jb(vb/re)Gb,>., (t)e-itHf, 9)

l~:T 1

= rT2(A(t)e-itHf, A(t)e-itHg)dt lT1

+

t k=l

{T2(Bfk>(t)e-itHf, Bt\t)e-itHg)dt lT1

+ fT\S5(t)f,g)dt. lT1

(7.142)

CHAPTER 7. MANY-BODY HAMILTONIAN

104

Hence using (7.140), (7.141) and the uniform boundedness of Gb,>.,(t)int>

1, we have (7.143)

for some constant C' > 0 independent of T2 > T1 ~ T and g E EH{A)1{.. (7.143) and {7.U2) with (7.140) and (7.141) then yield that

I(eitHG,,,>.,(t)*Jb(vb/re)Gb,>.,(t)e-itHf,g)l~~TiI$.,(t)* Jb(vb/re)Gb,>.,(t)e-itH f

(7.145)

l=l

exists for any f E EH(A)1{, and bwith 2 $ lbl$ N if A is an interval sufficiently small around E E R 1 - 7: diam A < d(E). Then the asymptotic decomposition (7.108) implies

(7.146) for f = ,,P(H)f = Eu(A)f. we see that satislies

R

Further by the existence of the limit (7.145) and J

= EH(A)f, (7.147)

in a way similar to 1;heproof of the intertwining property of wave operators. Now returning to the first 'P(H)f, and noting that supp'P is compact in R1 - 7, we take a finite number of open intervals D.;0 CC R 1 - 7 such that E;0 E A; 0 , diam A;0 < d(E; 0 ), and supp'P cc b.;0 cc R1 - 7. Then we can take '1/J;oE C8°{A;0 ) such that 'P(H)/ = Thus from (7.145)-(7.147), we obtain the existence 0 (H)f. of the limit for 2 $ lbl$ N:

u;;te

E;:ite'I/J; finite

L

LL

fl= t~::!,

jo

itHGb,>.,(t)* Jb(vb/re)Gb,>.e(t)e-itH,,p 30(H)J,

(7.148)

l=l

and the relations

'P(H)J =

L

fl'

EH(b.)fl

= fl

(7.149)

2~lbl~N

for any set A CC R 1 - 7 with supp'P CA. Set

a3 = .;:;--1p3>.. 0/2,

83 = ,/y0 3A'o (j = 2, 3, · · · , N,

0N = 0).

(7.150)

105

7.4. A DECOMPOSITION OF CONTINUOUS SPECTRAL SUBSPACE

Then by (7.148), some calculus of pseudoclifferentialoperators, and 2 supp (Jb(xb/re).e(t)*'l/J1 (Hb)Jb(vb/rt)Gb,>.i(t)e-itH,,P; (H)f, 0

(7.160)

The proof of the existence of these limits is similar to that of jl in (7.148), since the change in the present case is the appearance of the commutator [H,,,P 1 (Hb)]= [lb,,,P(Hb)] whose treatment is quite the same as that of the commutators including lb in (7.138). We introduce the decomposition (7.22) into hb and 9b on the left of e-itH,,p; 0 (H)f as in (7.71). Then by the factor Jb(vb/re)Gb,>.e(t), we see that only the terms with d::; bin the sum in (7.71) remain asymptotically as t = tm -+ oo by the arguments similar to step

-Mm

I). On each summand Pd,E;ljd1: 1 in these terms (see (7.73)), Hb asymptotically equals H~ = Tj + Hd = Tj + E; ,,., lx~l2/(2t'!i) + E; ,,., lxbl2/(2t'!i) + E;, where for d ::; b, H~ = Tj + Hd = Hd - n, Tj = Td - n and xb = (x~,xd) is a clustered Jacobi coordinate inside the coordinate xb. Thus we have -'FJm

b

·t H

'l/J1(H)Jb(vb/re)Gb,>.e(t)Pd,E;ljdl: 0 (H)f 1e-' m 'l/J; b2

-'FJm

2

"''I/J1(lxI /(2tm) + E;)Jb(vb/re)Gb,>.i(t)Pd,E;ljd 1: 1e-'

·t H m

'l/J; 0 (H)f

(7.162)

2/(2t'!i) + E;) # 0, then for some£= 1, · · · , Lb as m-+ oo. If ,,P 1 (la:bl - (>..b - E·)I < f.b. Ilxbl2 2t2 t J-0

m

If£ is a (unique) £(.i) such that E; E (>..~(j)-

lxbl2 < b 2t2 -

Te(j)

m

(7.163)

T/(j), At(j) + T/(j)),we have

-b

b

-b

+ To..~ - E; ~ 4a~.

(7.165)

7.4. A DECOMPOSITION OF CONTINUOUS SPECTRAL SUBSPACE

107

Thus from (7.163) (7.166) Setting a'

= yi6f8we then have for £ =/-f (j) lxbl~ a'tm.

(7.167)

hb=ft' +gb;,

(7.168)

Therefore hb can be decomposed as

where

ft'= t~~

finite

L

io

l=l

finite

L

io

l=l

L LeitHGb,,\,(t)*F(lxbl ~ o't)'ifJ1(Hb)Jb(vb/rt)Gb,,\t(t)e-itH1Pio(H)J, (7.169)

gb; = t~~

LL

itHcb,,\t(t)*F(lxbl ~ a't)'lfJ1(Hb)Jb(vb/ri)Gb,,\t(t)e-itH'I/J; 0 (H)f, (7.170)

The existence of the limit (7.169) is proved similarly to that of (7.160) by rewriting the factor F(lxbl ~ o't) as a smooth one and absorbing it into Jb(vb/re)Gb,,\,(t)with changing the constants in it suitably. The existence of (7.170) then follows from this, (7.148) and (7.168). For gb, similarly to gt; we obtain finite

gb = t~~

L

LL io

eitHGb,,\e(t)* F(Jxbl~ a't)(I - 'I/J1)(Hb)

(7.171)

£=1

x Jb(vb/re)Gb,,\t (t)e-itH1Pio (H)f.

Setting u'

gb

= gbl + gb (T,

(7.172) L

finite

=

t~~

LL eitHGb,,\t(t)*F(lxbl ~ a't)Jb(vb/rt)Gb,,\e(t)e-itH'I/J; (H)f, 0

io

l=l

we obtain a decomposition of fl: (7.173) where ft' and g't' satisfy

e-itHft' e-itHg(

II II

F(lxal ~ (J'lblt)F(lxbl ~ o't)e-itHft', at;b "' F(lxal ~ a1b1t)F(Jxbl ~ olblt)F(Jxbl~ a't)e-itHg(. at;b "'

(7.174) (7.175)

CHAPTER 7. MANY-BODY HAMILTONIAN

108 We can prove the e:cistence of the limits

1 l'imgbCT1 Yb= 1 CT.j.0

(7.176)

in the same way as in Enss [11], Lemma 4.8, because we can take 'lj;1 in (7.157)-(7.158) monotonically decrnasing when f8..j.0 and the factors F{lxbl ::; c5't)and F(lxbl 2::u't) can be treated similarly to 'lj;1 by regarding xbft as a single variable. Further we have as in (7.147)

EH(fl)ff

=ff,

EH(!l)gt

= gt,

(7.177)

which, (7.174) and (7.176) imply (7.178) Thus we have a decomposition: (7.179)

g'( can be decompoued further by using the partition of unity of the ring u' ::; lxbl/t::; c5Jbl with regarding xb a.,a total variable x in Proposition 3.2. Arguing similarly to steps I) and II), we can prove that gt can be decomposed as a sum of the elements fJ of SJwith d < b. Combining this with (7.146), (7.178) and (7.179), we obtain (7.69). 0

We remark that Theorem 7.10 implies the asymptotic completeness when the longrange part vanishes for all pairs a, because in this case we see straightforwardly that st= 'R.(Wb±),where wb± are the short-range wave operators defined by

V;

(7.180) For the case when long-range part does not vanish, we have the following

Theorem 7.13 Lei Assumptions 7.1 and 7.2 be satisfied. Then i) For 2(2 + 1:)- 1 < r :=:;1 Sbr

ii) If

1: >

2(2 + 1:)- 1 , i.e. when

1: >

v'3Sbr

iii) If

1: >

-sl b·

(7.181)

1, we have for all r with O::; r :=:;1

-sl b·

(7.182)

1/2 and li~(x°') 2:'.0 for all pairs a, then we have for all r with O::; r::; 1 Sbr

-sl b·

(7.183)

7.4. A DECOMPOSITION OF CONTINUOUS SPECTRAL SUBSPACE

109

Proof: i) and ii) follow from Proposition 5.8 of [6] and Proposition 7.6 above. (7.182) for r = 0 follows from the proof of Proposition 5.8 of [6]. iii) follows from Theorem 1.1 and Proposition 4.3 of (25] and (4.116) below: Note that 'R.(Ot) in Theorem 1.1-(1.31) of (25] constitutes a dense subset of St,when '1/Jvaries in cr(R 1 D From Theorem 7.13-ii), iii) and Theorem 7.10 follows

n.

Theorem 7.14 Let Assumptions 7.1 and 7.2 be satisfied with E > 2(2 + E)- 1 or with f > 1/2 and Vf (x 0 ) 2::: 0 for all pairs a. Then we have for all r with O $ r $ 1

EB~ = 1lc(H).

(7.184)

2:5lbl:5N

In the next section, we will construct modified wave operators: (7.185) with Jb being an extension of J of (19] to the N-body case. We will then prove

'R.(W{)

= Bg,

(7.186)

which and Theorems 7.13 and 7.14 imply

Theorem 7.15 Let Assumptions 7.1 and 7.2 be satisfied with E > 2(2 + E)- 1 or with f > 1/2 and Vf (x 0 ) 2::: 0 for all pairs a. Then we have/or all r with O $ r $ 1 (7.187)

and

EB'R.(W{) = 1lc(H).

(7.188)

2:5lbl:5N

One might expect that (7.184) and (7.188) are always true, but it is denied:

Theorem 7.16 Let Assumptions 7.1 and 7.2 be satisfied and let N 2:::3. Then the followings hold: i) Let 2 $ lbl $ N and let Eb(r) be the orthogonal projection onto s;;(0 $ r $ 1). Then Eb(r1 ) $ Eb(r2 ) for 0 $ r 1 $ r2 $ 1, and the discontinuous points of Eb(r) with respect tor E [O,1] in the strong operator topology are at most countable. ii) Let O < f < 1/2 in Assumption 7.1. Then there are long-range pair potentials V0 (x0 ) such that for some cluster decomposition b with 2 $ lbl $ N, Eb(r) is discontinuous at r = ro, where f < r0 := (E+ 1)/3 < 1/2. In particular, there are real numbers r1 and r2 with O $ r1 < ro < r2 $ 1 such that

5;1 is a proper subset of S;2.

(7.189) .

CHAPTER 7. MANY-BODY HAMILTONIAN

110

Proof: i) By Proposition 7.6, s;;(0 ::; r ::; 1) is a family of closed subspaces of a separable Hilbert space 11.that increases when r E [O,1] increases. Thus the corresponding orthogonal projection Eb(r) (0::; r ::; 1) onto Sbincreases as r increases, and hence has at most a countable number of discontinuous points with respect tor E [O,1] in the strong operator topology. ii) holds by Theorem 4.3 of [49], Theorem 7.10 and Proposition 7.6, for b, lbl= N -1, with a suitable choice of pair potentials that satisfy Assumption 7.1. In fact, the sum of the ranges R(Wi) of Yafaev's wave operators Wn (n = 1, 2, · · ·) in Theorem 4.3 of [49] constitutes a subspace of (Eb(ro+ 0) - Eb(ro- 0))1£ for b with lbl= N - 1 by his construction of Wn, which means that Eb(r) is discontinuous at r = r0 • Here Eb(r0 ±0) = s- limr---+ro±O Eb(r). D

7. 5

A characterization tors

of the ranges of wave opera-

The purpose in this section is to prove relation (7.186) for general long-range pair potentials Va(xa) under Assumptions 7.1 and 7.2. The inclusion

(7.190) is a trivial relation :for any form of definition of the wave operators wb±. Thus our main concern is to prove the reverse inclusion

sgC R(Wj).

(7.191)

The proof of this inclusion is essentially the same for any definition of wave operators and is not difficult in th~ light of Enss method [8]. As announced, we here consider the wave operators of the form

(7.192) where Jb is an extension of the identification operator or stationary modifier introduced in [19]for two-body long-range case. The first task in this section is to construct Jb. Our relation (7.191) then follows from the definition of the scattering spaces sgand properties of Jb by Enss methc,d. To make the des,~riptions simple we hereafter consider the case = 0 for all pairs a. The recovery of the short-range potentials in the following arguments is easy. Let a C 00 function xo(x) of x ER." satisfy

V;

xo(x)

={

1

0

(lxl~ 2) (lxl::;1).

(7.193)

To define Jb we intr,)duce time-dependent potentials Ibp(xb,t) for p E (0, 1): kb

hp(xb, t)

= h(xb, 0) ITXo(pzbk)Xo((log(t) )zbk/(t) ). k=l

(7.194)

7.5. A CHARACTERIZATION OF THE RANGES OF WAVE OPERATORS

111

Then hp(xb, t) satisfies (7.195) for any£ 2::0 and O < Eo 0 is a constant independent of t, Xb and p. Then we can apply almost the same arguments as in section 2 of [19] to get a solution 'Pb(xb,eb) of the eikonal equation: (7.196) in some conic region in phase space. More exactly we have the following theorems. Let

cos(zbk,(bk) :=

Zbk · (bk

IZbke I 1,bkeI ,

where lzbkle= (zbk· Zbk)112 is the Euclidean norm. We then set for Ro, d > 0 and 0 E (0, 1)

where (bk is the variable conjugate to Zbk·

Theorem 7.17 Let Assumption 7.1 be satisfied with Vf = 0 for all pairs a. Then there exists a C 00 function t(xb, eb) that satisfies the following properties: For any O < 0, d < 1, there exists a constant Ro > 1 such that for any (xb,eb) E f ±(Ro, d, 0) (7.197)

and (7.198)

where C0 13> 0 is a constant independent of (xb,eb) E f ±(Ro,d,0). From this we can derive the following theorem in quite the same way as that for Theorem 2.5 of [19]. Let O < 0 < 1 and let 'l/J±(r)E C 00 ([-1, 1]) satisfy Q ~

1P±(T)~ 1,

'l/J+(r)=

{

~

for 0 ~ T ~ 1, for - 1 ~ T ~ 0/2,

'lj;_(r) =

{

~

for -0/2~r~l, for - 1 ~ T ~ -0.

We set

kb

X±(Xb,eb) =

II1P±(cos(zbk,(bk)) k=l

CHAPTER 7. MANY-BODY HAMILTONIAN

112

and define 'Pb(Xb,{t)= 'Pb,8,d,Ro(xb,{b) by

'Pb(Xb,{b) = {(¢,t(xb,{b) - Xb. {b)X+(xb,{b)+ (

0. Note that 'Pb,8,d,Ro(xb,{b) = 'Pb,8,d',~(xb,{b)when lzbkl ~ max(~,R:i), We then have

l(bkl~ max(d,d') for all k.

Theorem 7.18 Lft Assumption 7.1 be satisfied with V; = 0 for all pairs a. Let O < 0 < 1 and d > 0. Then there exists a constant ~ > 1 such that the C 00 function 'Pb(xb,{b) defined above satisjies the following properties. i} For (xb, {b) Er+(~, d, 0) u r -(~, d, 0), 'Pbis a solution of

(7.200) ii} For any (xb,{b)

E

R 2v(Jbl-l)and multi-indices a,/3, 'Pbsatisfies

a=O a=_;ifO.

(7.201)

In particular, if a 7~ 0,

(7.202) for any 1:0, 1:1 ~ 0 with 1:0 + 1:1

= f.

Further

(7.203) iii} Let

(7.204) Then

(7.205) and

lfl"·"'bti!ebab(xb,.,,b)I 0 so that II! - hll< o. Then we have for any sufficiently large R > 0 limsup e-itHht--too

ITF(lxal ~ ut)F(lxbl ~ R)e-itHh


0 is a small n11mberwith a' < a, F(IPbl ~ S) comes from EH(Ll.)in h = EH(6.)h, and .F(r ~ S) is a nmooth characteristic function of the set {r E R 1 I r ~ S} with a slope independent of S, and Qk is a pseudodifferential operator

Qkg(xb) = (~7r)-v(lbJ-l)

r

} Rv(lbl-1)

r

e("'b'eb-Yb-eb)qk(Zbk, (bk)g(yb)dybdeb (7.218)

} Rv(lbl-1)

with symbol qk(Zbk:(bk) satisfying

lctalbkqk(zbk, (bk)I ~ C,0..,(zbk}-1.01((bk}-1-r1, qA(zbk,(bk) = 0 for cos(zbk,(bk) ~ 0 or lzbkl~ R_o.

(7.219)

The order of products in (7.215) of factors in (7.217) and Jb1 may be arbitrary because these factors are rr..utually commutative asymptotically as t -r oo by virtue of (7.216). We note that d > 0 in the definition of Jb = Jb,9,d,ll-0can be taken smaller than a' > 0 beforehand since W,; is independent of d > 0 as mentioned. Thus we can assume the following in addition to (7.219): (7.220) We now insert the decomposition (7.22) to the left of e-itmHh in (7.215) with noting = f b;yf E 1-lc(H). Then by II(/- pMf')h - hll < 28 and by inserting the factor (7.217) to the left of e-itmHh after the insertion of (7.22), we have

(I - pMf')J

limsup

II(/-

Mm

Pb 161)4>e-itmHhll< 38.

(7.221)

m-+oo

-

.

Mm

By the factor F(lxbl ~ R) in (7.217) and EH(Ll.)in h = EH(Ll-)h,Pb 161in (7.221) converges to Pb as m -r oo in operator norm in the expression (7.221). It thus suffices to consider the quantity (7.222) Mmo

Mmo

for some large but Jixed mo with an error 8 > 0. Since Pb 161 = I:;~t1 Pb,E;(0 ~ Mj;j°< oo) with Pb,E;bein.~ one dimensional eigenprojection of Hb corresponding to eigenvalue E;, (7.222) is reduced to considering (7.223)

7.5. A CHARACTERIZATION OF THE RANGES OF WAVE OPERATORS

115

By Assumptions 7.1-7.2, the factor Pb,E;bounds the variable xb and yields a short-range error of order O((min(zbk))-l-e) on the left of e-iuT&when we replace Ib(xb,xb) by Ib(Xb,0), and we have that (7.223) equals

1 8

r -E))

e-iuE;eiuHPb,E;0( (xb))( (n + h(xb, 0) )Jb - JbTb + 0( (min(zbk)

1

(7.224)

e-iuT&J-lif!due-itmHh

X

b

'

(xb)- 1 0(

where 0( (xb)) is an operator such that (xb)) is bounded. Using (7.210) and the estimate (7.206) in Theorem 7.18-iii) and applying the propagation estimates in Lemma 3.3-ii) of [19] (again with a slight adaptation to the present case), we now get the estimate: ll((Tb+ h(xb, 0))Jb - Jbn

+ O((min(Zbk))-l-E))e-iuT&J,;-1 0 independent of u 2: 0. On the other hand (7.216) yields that

is asymptotically less than 28 as m--+ oo. This and (7.225) prove that the norm of (7.222) is asymptotically less than a constant times 8 as m --+oo. Returning to (7.214) we have proved that limsupsup 11(1 - eisHJbe-isH&J,;-l)e-itmH fll m--too S,:'.0

~o limsupsup ll(J - eisHJbe-isH&J,;-l)ptlble-itmH fll ~ C8, m--too s,:'.0

(7.226)

where a ~ 0 b means that la - bl ~ C8 for some constant C > 0. Since wave operator wb+= s- lims--tooeisHJbe-isHbPb exists, (7.226) yields limsup ll(J m--too

Mm

W/J,;-l)Pb l&le-itmHfll

~

C8.

(7.227)

Mm

By the arguments above deriving (7.221) we can remove Pb l&Iand get limsup ll(J m--too

W/J,;-1)e-itmH!II ~ C8.

(7.228)

Since we assumed (7.212), f is orthogonal to R.(wb+). Thus taking the inner product of the vector inside the norm in (7.228) with e-itmHf, we have

11!112 = lim l(e-itmHf, e-itmHf)I = lim l(e-itmHf, (I m--too

As 8 > 0 is arbitrary, this gives f

m--too

= 0, proving

W/J,;-l)e-itmHf)I ~ C8llfll-

(7.213). The proof of (7.186) is complete.

CHAPTER 7. MANY-BODY HAMILTONIAN

116

Exercise 1. With W± being defined by (6.39) we consider

for f belonging to a. suitable subspace 'D of 1-lc(H). Show that the operator JJ*-I

defines a compact operator on 'D, and prove the asymptotic completeness of W± without utilizing the existence of the inverse J- 1 of J. We remark that the existence of the inverse is required in section 7.5 as the inverse Ji:1 of Jb. 2. Let x(x) (x E R,1 ) be a real-valued C 00 function with compact support such that x(O) = 1, and let .7 = g(x) be a complex-valued C 00 function with compact support

defined on R 1 . Set for t > 0

f(t, x) = lim

1

fiC:;_

E,I.Ov2nt

where argi i) Fort>

=

100 e'-~ g(y)x(Ey)dy, 2t

-oo

'i"Show the following.

0 lim

1 fiC:;_

E.j.0 V

21rit

100 . e'

(o,-y)2 2t

X(Ey)dy= 1.

-00

ii) For t > 0 the following holds uniformly in x E R 1

lf(t, x) - g(x)I :5 C../i, where C > 0 is a constant depending only on g. 3. Let f be a bounded, uniformly continuous function from R 1 to C. Let L 1 be the totality of the Lesbegue integrable functions on R 1 . For 0, there is a unitary operator S(E) on £ 2 (8 2 ), 8 2 being two dimensional sphere with radius one, such that for a.e. E > 0 and w E 8 2

(11J.

= (s{E}h{v'E·)) (w), h E £ 2 {~) = L2 {{0,oo},L2 {8;), l{l 2dl{I). be written as S = {S(E)}E>O· It is known [20) that S(E) can be expressed

(Sh}(v'Ew)

Thus Scan 88

(S(E)ip)(0) = ip(0)-21riv'E

L

S(E,0,w)ip(w}diu

for 'PE £ 2 (8 2 ). The integral kernel S(E, 0,w) with w being the direction of initial wave, is the scattering amplitude S(E, 0) stated in the above and IS(E, 0,w)l2 is called differential cross section. These are the most important quantities in physics in the sense that they are the only quantities which can be observed in actual physical observation. The energy level E in the previous example thus corresponds to the energy shell T = E, and the replacement of E by E' in the above means that T is replaced by a classical relativistic quantity E' = cy'-p2+ m 2c2-mc2. We have then seen that the calculation in the above gives a correct relativistic result, which explains the actual observation. Axiom 9.1 is concerned with the observation of the final stage of scattering phenomena. To include the gravity into our consideration, we extend Axiom 9.1 to the intermediate .process of quantum-mechanical evolution. The intermediate process cannot be an object of any actual observation, because the intermediate observation would change the process itself, consequently the result observed at the final stage would be altered. Our next Axiom 9.2 is an extension of Axiom 9.1 from the actual observation to the ideal observation in the sense that Axiom 9.2 is concerned with such invisible intermediate processes and modifies the ideal intermediate classical quantities by relativistic change of coordinates. The spirit of the treatment developed below is to trace the quantum-mechanical paths by ideal observations so that the quantities will be transformed into classical quantities at each step, but the quantum-mechanical paths will not be altered owing to the ideality of the observations. The classical· Hamiltonian obtained at the last step will be "requantized" to recapture the quantum-mechanical nature of the process, therefore the ideality of the intermediate observations will be realized in the final expression of the propagator of the observed system.

CHAPTER 9. OBSERVATION

130

9.3

The second step

With these remark:; in mind, we return to the general k cluster case, and consider a way to include gravity in our framework. In the scatterin:~ process into k 2: 1 clusters, what we observe are the centers of mass of those k clusters C1 , • • · , Ck, and of the combined system L = (L 1 , · · · , Lk), In the example of the two body case of section 9.2, only the combined system L = (L 1 , L 2 ) appears due to H 1 = H 2 = 0, therefore the replacement of T by E' is concerned with the free energy between two clusters C1 and C2 of the combined system L = (L 1 , L 2). Following this treatment of Tin the section 9.2, we replace T = Tb in the exponent of exp(-itLhb) = exp(-itL(Tb + h(xb, 0))) on the right hand side of the asymptotic relation (9.1) by the relativistic kinetic energy T£ among the clusters C 1 , · · · , Ck around the center of mass of L = (L 1 . · · · , Lk), defined by k

T£ =

L (cJp;

+ m;c 2 -

mic 2 ).

(9.3)

j=l

Here mi > 0 is the rest mass of the cluster Ci, which involves all the internal energies like the kinetic energies inside Ci and the rest masses of the particles inside Ci, and Pi is the relativistic momentum of the center of mass of Ci inside L around the center of mass of L. For simplicity, we assume that the center of mass of L is stationary relative to the observer. Then we can set in the exponent of exp(-itL(T£ + h(xb, 0)))

(9.4) where to is the observer's time. For the factors exp(-itLHi) on the right hand side of (9.1), the object of the ideal observation is the centers of mass of the k number of clusters C 1 , · · · , Ck. These are the ones which now require the relativistic treatment. Since we identify the clusters C 1, · · · , Ck as their centers of mass moving in a classical fashion, tL in the exponent of exp(-itLHi) should be replaced by c- 1 times the classical relativistic proper time at the origin of the local system Li, which is equal to the quantum-mechanical local time ti of the sub-local system Li. By the same reason and by the fact that Hi is the internal energy of the cluste,r Ci relative to its center of mass, it would be justified to replace the Hamiltonian Hi in the exponent of exp(-itiHi) by the classical relativistic energy inside the cluster Ci arou:1d its center of mass

(9.5) where mi > 0 is the same as in the above. Summing up, we arrive at the following postulate, which has the same spirit as in Axiom 9.1 and incbdes Axiom 9.1 as a special case concerned with actual observation: Axiom 9.2 In either actual or ideal observation, the space-time coordinates (ctL, xL) and the four momentum p =(pl')= (EL/c,pL) of the observed system L should be replaced by classical relativistic quantities, which are transformed into the classical quantities (eta, x 0 )

9.3. THE SECOND STEP

131

and p = (Eo/c,po) in the observer's system Lo according to the relativistic change of coordinates specified in Axioms 8.1 and 8.2. Here tL is the local time of the system L and XL is the internal space coordinates inside the system L; and EL is the internal energy of the system L and PL is the momentum of the center of mass of the system L.

In the case of the present scattering process into k clusters, the system Lin this axiom is each of the local systems Li (j = 1, 2, · · · , k) and L. We continue to consider the k centers of mass of the clusters C 1, · · · , Ck, At the final stage of the scattering process, the velocities of the centers of mass of the clusters C1, · · · , Ck would be steady, say v1, · · · , vk, relative to the observer's system. Thus, according to Axiom 9.2, the local times ti (j = 1, 2, · · · , k) in the exponent of exp(-itiH 1), which are equal to c- 1 times the relativistic proper times at the origins Xj = 0 of the local systems Li, are expressed in the observer's time coordinate t 0 by ti= t 0 J1 - (vi/c)

where we have assumed lvi/cl transformation:

2

«

~ to

(1- v;/(2c 2 )),

j

= 1, 2, · · · , k,

(9.6)

1 and used Axioms 8.1 and 8.2 to deduce the Lorentz

(For simplicity, we wrote the Lorentz transformation for the case of 2-dimensional spacetime.) Inserting (9.3), (9.4), (9.5) and (9.6) into the right-hand side of (9.1), we obtain a classical approximation of the evolution:

+ h(xb, 0) +Hf+···+ H~) - (m1vU2 + · · · + mkvV2)l) assumption that lvi/cl « 1 for all j = 1, 2, · · · , k.

exp (-ito[(Ti

(9.7)

under the What we want to clarify is the final stage of the scattering process. Thus as we have mentioned, we may assume that all clusters C1, · · · , Ck are far away from any of the other clusters and moving almost in steady velocities v1, · · · , vk relative to the observer. We denote by rij the distance between two centers of mass of the clusters Ci and Ci for 1 ~ i < j ~ k. Then, according to our spirit that we are observing the behavior of the centers of mass of the clusters C1, · · · , Ck in classical fashion following Axioms 8.1 and 8.2, the clusters C1, · · · , Ck can be regarded to have gravitation among them. This gravitation can be calculated if we assume Einstein's field equation, lvi/cl « 1, and certain conditions that the gravitation is weak (see [36], section 17.4), in addition to our Axioms 8.1 and 8.2. As an approximation of the first order, we obtain the gravitational potential of Newtonian type for, e.g., the pair of the clusters C1 and U1 = LJ7= 2 Ci: k

-G

L m1mi/r1i, i=2

where G is Newton's gravitational constant. Considering the k body classical problem for the k clusters C1 , · · · , Ck moving in the sum of these gravitational fields, we see that the sum of the kinetic energies of C1, · · · , Ck

132

CHAPTER 9. OBSERVATION

and the gravitational potentials among them is constant by the classical law of conservation of energy:

L

m1iU2+ · · · + mkvV2 - G

mim;/ri;

= constant.

1$;i../(fim)= 271".

Thus p(v)

= hm/>..= 2hm/[(m0 )2c2].

This gives a period of the local system with the clock on the energy shell H exp(-itH/fi)

In particular, when v

= >../m:

= exp(-it>../(fim)).

= 0, the period p(O) takes the minimum value:

This we call the least period of time (LPT) of the local system. This gives a minimum cycle or period proper to the local system. The general p(v) is related to this by virtue of the Proposition 1 as follows: Proposition 2. p(v) = hm/>..= p(O)/

J1 - (v/c) 2 ('?_p(O)).

This means that the time p(v) that the clock of a local system, moving with velocity v relative to the observer, rounds 1 cycle when it is seen from the observer, is longer than the time p(O) that the observer's clock rounds 1 cycle, and the ratio is given by p(v)/p(O) = 1/ y'I"'=-(v/c) 2 ('?.1). Thus time, measured by our QM clock, of a local system moving with velocity v relative to the observer becomes slow with the rate Jl - (v/c) 2 ,

9.3. THE SECOND STEP

139

which is exactly the same as the rate that the special theory of relativity gives. This yields that the QM clock obeys the same transformation rule as that for classical relativistic clocks like light clock discussed at the beginning of this exercise, and shows that quantum mechanical clock is equivalent to the relativistic classical clock. These mean also that the space-time measured by using QM clock defined as the QM evolution of a local system follows the classical relativistic change of coordinates of space-time. Thus giving a consistent unification of QM and special relativistic CM. As for the validness of the name LPT, we see how it gives the Planck time:

tp = JGh/c

5

= 1.35125 x 10- 43 s,

where G is the gravitational constant. In fact, given Planck mass: mo = mp

= /hc1G = 5.45604 x

10- 5 g,

our LPT yields

p(O) = 2h/( /hc!Gc

2)

= 2.jhG/c5 = 2tp,

which is 2 times Planck time. II. N-particles case with masses m; (j

= 1, 2, · · · , N)

We consider the case after the local system L of N particles are scattered sufficiently. Then the system's solution asymptotically behaves as follows as the system's time t = tL goes to oo (see (9.1)):

exp(-itLHL/li)f

rv

exp(-itLhb/li)go ® exp(-itLHi/1i)g1 ® · · · ® exp(-itLHk/li)gk,

where hb =Tb+ Ib(xb,0) and k 2'.:1. Now for getting the above result also in this case it suffices to note that the Hamiltonians H1.(.e= 1, 2, ..., k) of each scattered cluster are treated just as in the case I) above but with using a general theorem on the spectral representation of self-adjoint operators H1., H1.are not necessarily free Hamiltonians and we cannot use the Fourier transformation, but we can use spectral representation theorem so that each He is expressed, unitarily equivalently, as )..e/ Me in some appropriate Hilbert space, where Me is the mass of the £-th cluster. Then it is done quite analogously to I) to derive the propositions 1 and 2 above in the present case. Of course these relations depend on .e,and show that the dependence of mass and time of each cluster on the relative velocity to the observer is exactly the same as the special theory of relativity gives as in the case I) above. One of the important consequences of these arguments is that the quantum clock is equal to the classical relativistic clock, which has remained unexplained as one of the greatest mysteries in modern physics in spite of the observed fact that they coincide with high precision.

Part IV Conclusions

Chapter 10 Inconsistency

of Mathematics?

To begin with stating our conclusive thought of this book, we consider that almost all of what we think would be able to be translated into mathematical words insofar as we consider about physical universe as we see in ordinary physical works after Galileo, Descartes, Newton, till the present age. Thus to consider our description of the universe, it would be inevitable to think about metamathematics and set theory which construct the basis of the modern mathematics. In the next chapter, we will try to describe the universe as a contradictory aspect of our language, or exactly speaking, of our mathematical language, in that it could include all sentences as at least meaningful ones so is regarded contradictory as a whole of those meaningful sentences. To be prepared for that purpose, in this chapter we will try to see if mathematics or set theory which is thought to be a basis of modern mathematics is consistent or not. As is well-known, in 1903, Russell's paradoxical set produced immense discussions about the foundation of mathematics, so follows Hilbert's formalism of mathematics, but this direction was negatively answered by Godel [15]. We will see in this chapter how this theorem of Godel seems to give a problem that looks telling that mathematics itself is inconsistent. We consider a formal set theory 8, where we can develop a number theory. As no generality is lost, in the following we consider a number theory that can be regarded as a subsystem of 8, and will call it 3(o). Definition 10.1 1} We assume that a Godel numbering of the system 3(o) is given, and denote a formula with the Godel number n by An. 2) A (O)( a, b) is a predicate meaning that "a is the Godel number of a formula A with just one free variable {which we denote by A(a)), and b is the Godel number of a proof of the formula A(a) in 3(o)," and B( 0l(a, c) is a predicate meaning that "a is the Godel number of a formula A(a), and c is the Godel number of a proof of the formula ,A(a) in 3(o)." Here a denotes the formal natural number corresponding to an intuitive natural number a of the meta level. Definition 10.2 Let P(xi, · · · .xn) be an intuitive-theoretic predicate. We say that P(x 1 , · · • , Xn) is numeralwise expressible in the formal system 3(o), if there is a formula P(xi, · · · , Xn) with no free variables other than the distinct variables xi,··· , Xn such that, for each particular n-tuple of natural numbers xi,··· , Xn, the following holds: 143

CHAPTER 10. INCONSISTENCY OF MATHEMATICS?

144

i) ifP(x1,···

,xn) is true, thenl-P(x1,···

,Xn).

and ii) if P(x1, · · · ,xn) is false, then I- ,P(x1, · · · ,Xn). Here "true" means "provable on the meta level," and I- P means that a formula P is provable in the formal system, e.g., 5(o). 10.3 There is a Godel numbering of the formal objects of the system 5(o) such 0)(a, c) defined aboveare primitive recursive and hence that the predicatesA(o)(a, b) and B< numeralwise expres.Yible in 5(o) with the associatedformulas A(q< 0>,k< 0>) Hence, for any integ,~r£, B< £) is false. In particular, B< 0), ... , B< 0>(a,c), from these follows that are false. By virtue of the numeralwise expressibility of B
(q(o.l) or -,Aqc"'>(q(o.l)) is defined recursively. Thus ii(,) is recursively well-defined for,::; a:. This completes the proof of the lemma. Assume now that a: is a countable limit ordinal such that there is an increasing sequence of recursive ordinals O:n< a: with 00

a:= Uo:n.

(12.9)

n=O

An actual example of such an a: is the Church-Kleene ordinal wfK. In the system B(o.), the totality of the added axioms A("Y) ( 1 < a:) is the sum of the added axioms A("Y) (, < o:n)of 3(0.n). By the lemma, ii(,) is recursively defined for,< O:n. Thus in each 3(0.n) we can determine recursively whether or not a given formula Ar is an axiom of 3(0.n) by soeing, for a finite number of ,'s with ii(,) ::; rand,< O:n,if A("Y) = Ar or not. This is extended to 3(0.). To see this, we have only to see the ,'s with ii(,) ::; rand 1 < a:, and determine for those finite number of ,'s if A(-y)= Ar or not. By (12.9),

ii(,) ~; r and , < a:-# 3n such that q(,) ::; r and , < O:n. Then by induction on n with using the above result for 3(0.n) in the preceding paragraph and noting that the bound r on q(1 ) is uniform in n, we can show that the condition whether or not ii(1 ) ::; r and , < a: is recursively determined. Then within those finite number of ,'s with ii(,)::; rand,< a:, we can decide recursively if for some,< a: with ii(,) ::; r, we have Ar = A("Y) or not. Therefore we can determine recursively whether or not a given formula.Ar is an axiom of 3(0.). Therefore Godel predicate A(o.)(a,b) and Rosser predicate B(o.)(a,c) are recursively defined, and hence are numeralwise expressible in 3(0.). Then the Godel number q(o.)of the formula \ib[-,A(o.)(a,b) V 3c(c::; b & B(o.)(a,c))] is well-defined, anc, hence Rosser formula Age"'> (q(o.)) is well-defined and Rosser-Godel

theorem applies to the system 3(0.). Therefore we can extend 3(0.) consistently by adding one of Rosser formula or its negation A(o.) (= Age"'> (q(o.))or -,Age"'> (q(o.l)) to the axioms of 3(0.) and get a consistent system 3(0.+1). In particular if we assume a least nonrecursive ordinal wfK exists and take a: = wfK, we get a consistent system 3{wfK+1). This contradicts the case ii) of theorem 10.6 in chapter 10. We leaYethe following problem to the reader.

12.3. A REFINED ARGUMENT

163

Question. The least nonrecursive ordinal, the so-called Church-Kleene ordinal wfK has been assumed to give a bound on recursive construction of formal systems (see [12], [44], [47]). However the above argument seems to question if wfK really exists in usual set theoretic sense. How should we think?

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the

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of

Time,

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東大数学教室セミナリーノート

1 Pyatetski-Shapiro 杉浦光夫訳

典型領域の幾何学と保型函数の理論 I

2 Pyatetski-Shapiro 杉浦光夫訳

典型領域の幾何学と保型函数の理論 II

3 河田敬義

多元環の整数論

4 河田敬義

一変数保型函数の理論 I Fuchs 群

5 佐武一郎述浅枝陽．記二次形式の理論前編 6 鈴木通夫述松本英也記有限群論

7 Bruhat 述伊原信一郎記

p 進体上の代数群のユニタリー表現論

8 Cartan 述笠原乾吉記連接層と多変数解析函数論への応用 9 河田敬義

一変数保型函数の理論 II~chs 型式

10 竹内外史述八杉満利子記数学基礎論講義 11 岩堀長慶

対称群と一般線型群の表現論上

12 岩堀長慶

Lie 環と Chevalley 群 I

13 岩堀長慶

Lie 環と Chevalley 群 II

14 小林昭七述落合卓四郎記 G-Structures andP s e u d o G r o u p s 15 小谷正雄述渡辺二郎記理論物理学概論 16 佐武一郎述浅枝陽記二次形式の理論後編

1 7D .C .Spencer 述落合卓四郎記同次線型偏微分方程式の解の層の 分解

18 久保田富雄

Eisenstein 級数の初等理論

19 小平邦彦述諏訪立雄記複素多様体と複素構造の変形 I 20 小平邦彦述山島成穂記代数曲面論 21 清水英男

近似定理，ヘッケ環，ゼータ函数

22 小松彦三郎

佐藤の超函数と定数係数線形偏微分方程式

23 佐武一郎述森田康夫記対称領域の正則うめ込みについて

2 4L .G紅ding

述河合隆裕記定数係数双曲型方程式の lacuna につ

いて

25 熊ノ郷準述井上淳記擬微分作用素とその周辺

26 角谷静夫述高橋陽一郎，金英淳記 s y s t e m

Thes t r u c t u r eo fd y n a m i c a l

27 楕円型偏微分作用素のスペクトル 28 松田道彦述外微分形式系の理論 29 最近の多様体論の発展 30 河合隆裕述中村如記線型偏微分方程式論序説

31 小平邦彦述堀川穎二記複素多様体と複素構造の変形 II 32 小平邦彦述赤尾和男記複素解析曲面論 33 増田久弥最適問題序説 34 小平邦彦述酒井文雄記

35 田中洋

Newnlinna 理論

Bolt7tmann 方程式の確率論的取扱い

36 亀高惟釦述伊藤正幸記

コルモゴロフ・ペトロフスキー・ピスク

ノフ型の非絲型拡散方程式 37

飛田武幸述

田中健一記複素ホワイトノイズと無限次元ユニタ

リ群 38 本田平述伊吹山知義記代数的偏微分方程式

39 森本明彦

擬軌道追跡の方法と力学系の安定性

40 西田孝明

粘性圧縮流体の方程式

41 小林昭七

正則ベクトルバンドルの微分幾何

42 柏原正樹述清水勇二記

Hodge 構造と Holonomic 系

4 3A .D .Weinstein 小野薫，杉山健一記 B r a c k e t s 44 P i e r r eBchapira 戸瀬信之記 S h e a v e s

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M i c r o l o c a lS t u d yo fC o n s t r u c t i b l e

4 5A d r i a nOcneanu 河東泰之記 QuantumS y m m e t r y ,D i f f e r e n t i a lGeｭ o m e t r yo fF : i n i t eGraphsandC l a s s i f i c a t i o no fS u b f a c t o r s

NewS e r i e s :Lecturesi nMathematicalS c i e n c e s TheU n i v e r s i t yofTokyo 東京大学数理科学セミナリーノート

M o r d e l l W e i lLattice の理論とその応用

1 塩田徹治

2S .AngenentN o t e sbyN .I s h i m u r a ,L e c t u r e sonMeanC u r v a t u r eFlow 3 佐武一郎編

久賀道郎遺稿集

4S .P i l i p o v i c ,C o l o m b e a u ' sg e n e r a l i z e df u n c t i o n sandpseud«r皿erential o p e r a t o r s 5 P .A .D e i f tandX .Z h o u ,L o n g t i m eb e h a v i o ro ft h en o n f o c u s i n g n o n l i n e a rS c h r o d i n g e re q u a t i o n-ac a s es t u d y 6 中村周

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7A .N .K i r i l l o v ,D i l o g a r i t h mI d e n t i t i e s 8 金子晃•谷島賢二編

偏微分方程式駒場セミナー I

9N .S .T r u c l i n g e rN o t e sbyN .I s h i m u r a ,L e c t u r e sonN o n l i n e a rE l l i p t i c E q u a t i o n so fS e c o n dOrder 10 柏原正樹山本敦子記 Crystal B a s i so fM o d i f i . e dQ u a n t i z e dU n i v e r s a l

E n v e l o p i n gA l g e b r a 11 松本久義

E n v e l o p i n galgebra 入門

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戸田正人記 Lectures

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1 3R .G o r e n f l o ,AbelI n t e g r a lE q u a t i o n sw i t hS p e c i a lEmphasisonApｭ p l i c a t i o n s 14 青木貴史

無限階擬微分作用素の表象理論

15 中原幹夫

経路積分とその応用

16 山崎昌男

Distribution を初期値とする Navier-Stokes 方程式の解

17 大沢健夫斎藤郁生記多変数複素解析入門

ー補間理論を中心に一

1 8J 6 z s e f: E ' r i t z AnI n t r o d u c t i o nt ot h eTheoryo fHydrodynamicL i m i t s 19S e r g e iDuzhin L e c t u r e sonV a s s i l i e vknoti n v a r i a n t s 20 前田吉昭，梶浦宏成著高村亮記変形量子化入門 21 佐武一郎述浅枝陽記改訂版補遣：渡部隆夫二次形式の理論

（前 22 佐武一郎述浅枝陽記改訂版補遣：渡部隆夫二次形式の理論 （後編） 23 北田均

Quantum M echanics

以下続刊

東京大学大学院数理科学研究科 セミナー刊行会

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東哀大学数理科学セミナリーノート

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