Commun. math. Phys. 31,243--257 (1973) © by Springer-Verlag 1973

Spectral and Scattering Theory for the Klein-Gordon Equation

Lars-Erik Lundberg Nordita, Copenhagen

Received February 5, 1973

Abstract. Eigenfunction expansions associated with the Klein-Gordon equation, are derived in the static external field ease, By employing these, we develop spectral and scattering theory. The results are almost as strong as those obtained in the Schr6dinger case.

1. Introduction

We shall in this paper consider spectral and scattering theory for the Klein-Gordon (K-G) equation

( D + 2iqo(x) Ot -- qo(x) 2 + q~(x) + m 2) u(x, t) = 0, (1.1)

where x e R 3, t ~ R and [] = 0 ~ - A. The functions qo(x) and %(x) are static external potentials coupled like a zero'th component of a four- vector and a scalar respectively.

Equations similar to (1.1) have been studied previously. Thoe [1] developed spectral and scattering theory in the case m = 0, qo(x) = 0 and q~(x) > 0 by employing a method due to Lax and Phillips [2] (only suited for the m = 0 case). Strauss [3] showed the existence and bounded- ness of the scattering operator and its inverse when m > 0 and qo(x) = O. Veselic [4] considered some spectral properties of Eq. (1.1) in the case m > O, q,(x) -- 0 under very restrictive conditions on qo(x) (excluding for example the square well case).

We shall consider Eq. (1.1) when m > 0 and derive eigenfunction expansions. These will enable us to develop the spectral theory of Eq. (1.1) indeta i l and develop scattering theory to the same level as is possible in the Schr6dinger case [5-8].

The main motivation for this investigation (from a physical point of view) is the fact that once we have developed spectral and scattering theory for Eq. (1.1) when considered as a classical field equation, the associated quantum field theoretic problem can be completely solved (and will be considered elsewhere).

In Section 2 we start by specifying the class of potentials we shall con- sider. Furthermore, we write the K-G equation in the form iat ~ = A ku

244 L.-E. Lundberg:

and construct a Hilbert space J¢' associated with it (by employing the field-energy as norm). The operator A is then shown to be self-adjoint and its essential spectrum a~(A) is determined (Section 3). Eigenfunction expansions associated with A are derived and the absolutely continuous spectrum %~.(A) is determined (Section 4).

Finally, the existence and completeness of the wave-operators We are established and the explicit form of the S-matrix is given (Section 5).

2. The K-G Equation Written as i~t ~P = A ~' and the Hilbert Space Associated with it

We shall consider Eq. (1.1) under the following conditions on the potentials qo(X) and qs(x);

i) qo(x) and qs(x) are real-valued and locally H61der-continuous except at a finite number of singularities;

ii) qo(x) z and qs(x) are square integrable; iii) qo(x) and qs(x) behave as (9(Ix[- 3 -~), ~ > 0 for Ix[-~ ~ ; iv) . f dx ( - qo(x) z + qs(x)) If(x)] 2 > - c~.[dx(I V f (x)l 2 + m 2 If(x)12), with

0 < ~ < 1 and f (x ) e C;°(R3).

Provided gq. (1.1)together with {u(x,O),ic?tu(x,O)}={fl(x),f2(x) } defines a well-posed initial-value problem, it is well known that the field energy (energy integral)

E(u, i~tu)-= [dx(IVuI2 + ( m Z - q ~ + q~) IuI2 + l~tul2), (2.1)

is independent of t (if it is finite). Note that condition iv) on qo(x) and qs(x) ensures the positivity of E(u, i3,u).

Let us introduce the pair

7J(x, t) = {u(x, t), i~tu(x, t)}, (2.2)

and the time evolution operator U(t); 71(., t) = U(t) 7%, 0). The operator U(t) is easily seen to be unitary in the Hilbert space 2¢f

consisting of initial data f = { f l , f 2 } such that I[ f l l2=E(f l , f2)<oe, provided the initial-value problem is well-posed.

Definition2.1. Let ~ denote the Hilbert space obtained by com- pleting N = Cg(R a) x Cg(R a) in the norm 9iven by

ilf[]2 = .[dx(lVf~[2 + (m 2 _ qo2 + q~)[/~[2 + [f2]2), (2.3)

and with the scalar product defined in the obvious way. Let furthermore ~ o denote ~ f when qo = q~ = 0.

Klein-Gordon Equation 245

Remark 2.2. The norms in 0~o and ~ are equivalent (this follows from condition ii) and iv) on the potentials), i.e. there exist constants cl and c2 such that

0 < c 1 II f tl o < [] f It < c2 [j f I1 o, (2.4)

where II II 0 denotes the norm in 2/f o. The K-G equation (1.1) can be written

i0, ~ = A ~ , (2.5) where

A = ( O 2qo1)' L = - A + m 2 + q , q = - q ~ + q s , (2.6)

and with ~u given by Eq. (2.2). One can easily check that A is symmetric on ~ in )F.

In the next section we shall prove that A is essentially self-adjoint on ~ in Yf and furthermore determine the essential spectrum of its closure (also denoted by A) in )F.

3. Self-adjointness of A in ~ and the Essential Spectrum ¢r e (,4)

In this section we show that A is essentially self-adjoint on ~ in Yf by using a well-known theorem of Kato and Rellich (generalized to the case of two Hilbert spaces by Thoe [1]). We furthermore determine the essential spectrum of A(a~(A)) in ~ , by a compactness argument.

The operator A can be split into two parts

A = A o + V, (3.t) where

A0 =(0L0 ~), V = ( ~ 20o0)' L o = _ A + m 2 . (3.2)

Let /4" denote the Sobolev space consisting of all functions, which together with their derivatives up to order n, are square integrable.

We note that ~ = ~ o = H1 x H ° = H 1 x L2(R3). Let us start with the following well-known results (when m > 0).

Remark 3.1. A o is essentially self-adjoint on ~ in YF o and its closure (also denoted by A0) has the domain D(Ao) = H z x H 1.

Remark3.2. The spectrum of A o in x4'~o satisfies a(Ao)=a~(Ao) = aa.c.(Ao) = ( - o% - m] u Ira, oo). Here "e" stands for essential and "a.c.'" for absolutely continuous.

Remark 3.3. The unitary one-parameter group e-iAot maps N into 9 . Remarks 3.1 and 3.2 are easily verified by Fourier transformation and

3.3 follows from the finite propagation velocity of solutions to Eq. (1.1) in the free case.

246 L.-E. Lundberg:

Theorem 3.4. The operator A is essentially self-adjoint on N in and its closure (also denoted by A) has the domain D(A) = D(Ao) = H 2 x H 1. The essential spectrum of A and A o coincide, ae(A ) = ae(Ao).

Proof. We have seen that A = Ao + V is symmetric on N in ~ and that Ao is essentially self-adjoint on N in ~o- It then follows from a theorem due to Thoe (see Appendix 1) that the condition

IIVfl lo<31lAofl lo+Yllf[Io, 3 < 1 , f e N , (3.3)

ensures the essential self-adjointness of A on N in )F. We will in fact show that ]? can be chosen arbitrarily small. It is sufficient for (3.3) to hold that

IIVfll2o <fl ' l lAofll ~ + 7'11 fll z , (3.4)

where fl' can be chosen arbitrarily small. The estimate (3.4) follows from the following estimates* (q(x) e L2(R3), qo(x) ~ L4(R3))

Ilqf1112 z <~1 Ilto f~l122 + 61(e~)Ilf~l[z 2 , (3.5) and

ltqof2112<=e2(tofz;f2)2+62(e2) ,2 Ilf211e, (3.6)

where e t , e 2 > 0 are arbitrary (see [9], p. 302 and p. 321). This finishes the proof of the essential self-adjointness of A on N in o~/tL

In order to ensure that o-e(A)= ao(Ao) it is sufficient to show that V is a compact operator from D(Ao) to s4e o (see [10], p. 18) or, equivalently that q(qo) is a compact operator from H2(H t) to H °. This, however follows from conditions ii) and iii) on qo(X) and qs(x) (see [10], p. 104).

4. Eigenfunction Expansions Associated with A and the Absolutely Continuous Spectrum tr .... (A)

In this section we construct eigenfunctions and generalized eigen- functions of A. These constitute a particular set of weak solutions to the "eigenvalue equation"

A ~ =coq~, e)~ o'(A). (4.1)

Equation (4.1) can be written

(0) 2 - 209% - L) u = 0, (4.2)

with 4~ = {u, cou}, and is of SchrOdinger type. This means that we can make use of the well-known properties of solutions to the time- independent Schr/Sdinger equation (see [5-8]).

Let us define L(co) = L + 2coq0. (4.3)

1 The lower index "T" denotes L2(R3).

Klein-Gordon Equation 247

Remark 4.1. The Schr/Sdinger operator L(~o), co e R is self-adjoint on D(L(co)) = H 2 in L2(R 3) with ac(L(co)) = aa.~,(L(co)) = [ m z, oo), L(oJ) has a finite number of eigenvalues (see [ 10], p. 218) and L(0) is positive (follows from condition iv) on the potentials).

Let u, denote a square integrable solution of Eq. (4.2) with o~ = co. e R. Remark 4.1 implies that u, e H 2 and that fe),l < m.

Remark 4.2. The eigenfunctions ~b, of A have the form ~b, = {u,,~o,u~}, i.e. A~ , = c0n~ . with ~ e D(A). We assume q~, to be normalized such that (~,, ~ ) = ~,~.

Remark 4.2 and the fact that at(A)= ( - 0 % - m ] u [ m , oo), implies that the projection operator Pd given by

m

I'd = E(m) - E ( - m) = .i dE(2) (4.4) - i n

with A = 2 dE(2) has the following representation

Pal f = 2 q~,(f , cb,)-- 2 q~,f,, f e W , (4.5) n n

Le. every element in PdY¢ ~ can be expanded in terms of {~,}. Let us introduce

oO - - i n

P+ = .[ dE(2), P - = .I dE()~), P = P+ + P - , (4.6)

and Y'f+ = P± Y f , A ± = P+- A , (4.7)

Our goal is to obtain a formula similar to (4.5) for P±f . It is well-known that Eq. (4.2), under our conditions on the potentials,

has solutions which satisfy the following integral equation (Lippmann- Schwinger equation) (see [5])

U+ (X,k)=eik x 1 e ±ilk[tx-yl - - 4 ~ "(dy I x - y [ (+-2e~(k)q°(Y)+q(y))u+-(Y'k) ' (4.8)

where ~o(k) = l f ~ + m 2.

Remark4.3 . A solution u+(x, k) of Eq. (4.8)has the following pro- perties; a) it is unique, b) it is bounded on R 3 × K, c) u~(x, k ) - e ik~ is uniformly continuous on R 3 x K. Here K is a compact set in R 3 - {0}. (For the proof see [5].)

Remark 4.4. The operator A has as a generalized eigenfunction eb+(x, k )= Ck{U±(X, k), _+ ~(k)u±(x, k)}. This is easily seen to be a weak

1 solution of Eq. (4.1) with ~o = 4-_ ~o(k). We choose Ck = ~r~(2~)3/2o~(k ) •

248 L.-E. L u n d b e r g :

Let ( , )N denote the scalar product in ~ when the integration is only carried out over a sphere of radius N in R 3.

Theorem 4.5. The mappin9 F ± ; ~ - * L2(R 3) defined by

F ± f ( k ) = limo~ (f , q~+-(., k))N = f ± (k), (4.9)

is isometric with initial domain 2/f ± and the whole of L2(R 3) as range. The adjoint F ± * is given by

F ±*9(x) = !irn .[ dk ~b+-(x, k) 9(k), (4.10) Kn

where K, C R 3 is an increasin 9 set of compacts such that UK, = R 3. The operator F ± diagonatizes A ± ;

~± * a/t •-+ (4.11) A ± = _ . . . . ~(.)- ,

where M~o(k) stands for the multiplication operator co(k). The singular spectrum of A fulfills a,(A) < ( - m, m), i.e. aa.¢.(A) = ae(A ).

Remark 4.6. The analog to formula (4.5) reads P± f = F ± * F± f .

Proof of Theorem 4.5. We shall start by showing that F -+ is isometric with initial domain 2/f-+. Our proof is based on the following formula

1(2, 2 ' , f ) = ((E(?.)- E0./)) f , f )

1 z (4.12) = lira - - ! d#((R(p + i e ) - R(# - ie)) f , f )

~+o 27ri ,

where R (z) = (A - z)- 1, as (A) c~ (2', 2) = 4~, and f ~ N. The right hand side of Eq. (4.12) will now be evaluated in a series of

lemmas and takes the form

I0o, ) / , f ) = .I dklF+-f(k)l 2 , (4.13) 2' < w(k) <.~

m 0(3 where the _+ sign is chosen when < 2 ' < 2 < . The integral

- - 0 0 - - m

1(2, 2', f ) turns out to be continuous as a function of 2 and 2' in the interval given above, which implies that as(A) C ( - m, m) and furthermore that F ± maps ~4 ~± isometrically into LZ(R3).

Lemma 4.6. Let f ~ 9 , then

R(z) f = l { 1 , z } G ( z ) ( L f l + z f 2 ) - - l { f l , 0 } , I m z ~ 0 , (4.14)

where G(z) = (L + 2zqo - z 2) - 1, (4.15)

is of the Carteman type in L2(R 3) Jbr tm z + 0.

Klein-Gordon Equation 249

Proof. The operator G(z) is easily seen to exist and to be bounded in L2(R 3) for z sufficiently small (due to the positivity of L and the L-bound- edness of qo). Formula (4.14) can than be verified for small z by straight- forward algebra and the analyticity of R ( z ) f in 24 ° can be employed to extend the validity of Eq. (4.14) to all z such that Im z + 0.

The first resolvant equation for G(z) has the form

G(z) = Go(z ) - G(z) (2zqo + q) Go(z ) , Imz + 0, (4.16)

where Go(z) = (Lo - z2) - 1. The operator Go(z) is of the Carleman type and (2 z qo + q) Go (z) is compact in L 2 (R 3) for Im z ~ 0; thus the boundedness of G(z) in L2(R 3) (which follows from Eq. (4.14)) and Eq. (4.16) implies that G(z) is of the Carleman type for Imz =~ 0 (for more details see [11]). This finishes the proof of the lemma.

Remark4.7 . Let 96 H 2 × LZ(R 3) and h ~ H , then

(9, h) = (Lg 1, hi ) 2 -k- (g2, h2)2- (4.17)

Lemma 4.8. The kernel G(x, y; z) of G(z) saiisfies

+C~ , p 6 a e ( A ) - { + _ m } . (4.18)

For the proof see Ikebe [5].

The second resolvant equation for G(z) has the form

G(z) - G(z') = (z - z') G(z) (z + z' - 2qo ) G(z'), (4.19)

and by combining Eqs. (4.15), (4.17), (4.18) and (4.19) one gets

2e x I()~, 2 ' , f ) = lira .[ d# ]IG~ - ie) fu][~, (4.20)

~0 7/; X'

where fu = L f l +/Lf2- The estimate (4.18) is used to show that the last term in Eq. (4.19) does not survive in the limit e,~0 (for details see [ t 1]).

Lemma 4.9. A solution Eq. (4.8) is related to the kernel of G(z) by the followin 9 formula

lim (lkl 2 + ( _+ co(k) + ie) 2 + m 2) S dy G(x, y; +_ co(k) + ie) e ~k" (4.21)

= u ± ( x , k ) on R3 x K .

Lemma 4.9 is proved in complete analogy with Lemma 9.2 in Ikebe [5]. Eq. (4.13) can now be verified by inserting Eq. (4.21) into Eq. (4.20),

using the Parseval equality and finally employing the following formal identity

1 e 1 t i m - - - b(co(k)2 - #2) , (4.22) ego ~ (co(k)2-- / .12+g2)2-b4k/2e 2 2p

250 L.-E. Lundberg:

(Remark 4.3 and the fact that f , has compact support allows us to freely interchange the order of intergrations, and also to interchange the limit with the integrations.)

It then remains to prove that F -+ maps onto L2(R 3) and that Eqs. (4.10) and (4.11) hold. Eq. (4.10) is easily verified by using the isometric character of F +- and the definition of the adjoint (F±f, 0)2 = ( f , F±*g), f ~2/t°, 9 e L2(R3). Eq. (4.11) is proved as follows: Let f ± = P+- f , with f e D(A); the spectral representation for A together with Eqs. (4.12) and (4.13) then gives

(A f ±, f ±) = .I2 d(E(2) f +-, f ±) = .[ dk + co(k)lF± f (k)l 2 (4.23)

= +_ (Mcol.) f+-f, F+-f)2 = + (F ± * Mcoo)F±f, f ) .

The ontoness of F ± is verified by showing that F ± * has a trivial null- space. We follow a similar proof, given in the Schr6dinger case, by Atsholm and Schmidt [8].

Lemma 4.10. Let 91, g2 ~ L2(R3) and let Zi be characteristic functions corresponding to sets {k; co(k)~ Ii} , where 11 and I 2 are disjoint intervals on R + ; then

(F ± * )~1 g 1, F +- * ;(2 g2) = 0. (4.24)

Proof. See Appendix 2. Let us assume that F + *g = 0. Lemma 4.10 and Eq. (4.10) then give

S dku+(x ,k)g(k)=O, m < 2 ' < ) + , (4.25) ,~' < co(k) <

and .f df2u+(x, lk]f~)g(Iklf2) = 0 fora.e. ]k l>0. (4.25)

l~l = 1

The Lippmann-Schwinger equation (4.8) finally gives

.I df2eqktaXg(]kl~2) =0 , fora.e. Ik l>0 . (4.26) lal = 1

But this shows that g -- 0, due to the Fourier inversion theorem. The same conclusion holds of course for F - * . The proof of Theorem 4.5 is thus completed.

5. Exis tence and Completeness o f the Wave-Operators W± and an Explicit Representation o f the S-Matr ix

In scattering theory one is concerned with the asymptotic behaviour of solutions to some evolution equation for large positive and negative times.

Klein-Gordon Equation 251

We shall in this section study the asymptotic behaviour of solutions to the Klein-Gordon equation (1.1) or, equivalently Eq. (2.4) i~ t ~ = A 7 j, with ~g = {u, i~,u}. We shall in fact show the existence of two wave- operators W e which have the property that

e-~atW+ f --,e-iA°tf, t ~ + o0, f ~ H o . (5.i)

Our results are collected in the following theorem:

Theorem 5.1. a) The wave-operators W e defined by

W+ = s-lim eiAte -iA°~ , (5.2) t--* _+ oo

exist as isometric mappings of Yt~o into P Y f and they intertwine A and Ao, A W+_ ~ W~ Ao.

b) One has P+- W_ = F+-* F # , W+ f = W_ f , (5.3)

i.e. W+ maps onto P Yf .

c) The scattering operator S defined by

S = W+* W_ Po ~ + W* W+ P o , (5.4)

is unitary in 2/f o and commutes with A o. The following representation for S+ = F f S F f * holds

S+-g(Jk lQ)=g( lk iQ)-~z ik .[dg2'T+-(lkl, O,~2')g(iklQ'), (5.5) where

Z--- (tkt, (2, f2') = ( 2 - ~ "f dx e - ~ j< a':,( +_ Roe(k) qo(x) + q(x)) U +(X, Ikl g2), (5.6)

with u + (x, k) = u + (x, k) and u_ (x, k) = u- (x , - k).

Remark5.2. T-+(IkI, O, f2 ') is the phase factor appearing in the asymptotic expansion of u± (x, k) for large IxI,

u ± ( x , k ) = e i k X + 2 r c 2 r +- lkl, ixl, fkt ....... ~ + (9 . (5.7)

The proof of Theorem 5.1 will be divided into three parts. First we show the existence of W+ as defined in Eq. (5.2), then Eq. (5.3) is established, and finally Eq. (5.6) is derived.

Proof of Theorem 5.i. a) By differentiating and integrating W(t) = eiAt e-ia°t we get

t

W(t) f = f + i.[ dt e,AtVe iaot f , f ~ ~ C D(V), (5.8) o

and thus t

t] W(t) f II < llftl + .[ dt tt Ve-iA°t fl l , (5.9) 0

252 L.-E, Lundberg:

which shows that it is sufficient for the existence of W e f that IIVe- iAotf I1 is integrable on (0, -I- oo).

We have ttVe-ia°~fl[2 = tlqfl(t)lt 2 + H2qof2(t)ll~ , (5.10)

where f ( t ) = {fx (t), f2(0} = e-iAo,f. Let Os denote the characteristic function for a sphere of radius N

around the origin in R 3.

Lemma 5.2. The followin9 estimate holds

HONfi(t)l[oo <= const ttt -3/2 , it[ > 4 N , i = 1, 2, (5.11)

where the constant is independent of N.

For the proof see [11], Lemma 8.t, or [12] where a similar result is proved in the Dirac case.

We are now in the position to estimate the right-hand side of Eq. (5.10). The first term is estimated as follows (t = 4N);

Ilqf l (t)tl 2 ~ I1( 1 - ON) q f ~ (t)lI 2 + ll Orvqf l (t)l12

< ][(1 - ON) q[l~ flfl(t)lt2 + [Iqt12 HONfl(t)[l~o

= (9(Itl- 3/2),

(5.12)

where we have used the fact that q(x) = C(]x]- 3 -~), lx]~oo ' and Lemma 5.2. We get a similar result for ][qofl(t)l[v

This shows that TIVeiA°tfll is integrable and thus W_+ exists on N. One can then easily verify that liw+_fll=tlfllo by observing that tlflL 2= ttflL g + (q f l , f l )2 and using an estimate similar to (5.12).

The domain of W e can now be extended by continuity to the whole o f ~ o and we furthermore have PW+ = W+ and AW+ ~ W+_A o (see Kato [13], p. 346). This completes the proof of a) in Theorem 5.1.

b) In this part of the proof we use a different but equivalent definition of W_+.

Lemma 5.3. One has

W+_=W+_(~=s- lim eiatJe -iA°t, (5.13) t~_+oo

where d is defined by (J f , g) = (f , g)o, f , g ~ d/t'.

Proof. See Appendix 3. We now show that P+-W_=F+-*F~. Put Wd(t)=eiatde -iA°t,

B ( t ) = F ~ * F + W j ( t ) , and choose f and e such that f ~ and 0~- e C~( R3 - {0}) [one can easily verify that this means that 9 e D(A)].

Klein-Gordon Equation 253

Let us consider (B(t) f , O)o. By differentiating and integrating it we get

t

(B(O f , 9)o =(Fo+* F+ J f , g)o +i.[ dt(F~* F+ eiAfV' e-iA°t f , g)o , (5.14) 0

where V ' = A J - J A o. The integrand can be rearranged as follows (we put F = F + and Fo = Fo+),

(F~ Fe iAt V' e-iAo~ f , g)o = (V' e-iAotf, e-iAtf*Fog ) = (e-iAo,f, Ve- iAtF* F o 9)o = (e-iao,f, VF* e-i'°(')tF o g)o

(5.15) = (e-iA°*£ V.[ dk ~+ (., k) e-i°'(k)'O~ (It)) o

= .f dk ( f , e i¢A°-°~(k))* V~+ (., k))o O~d (k),

where we have used the definition of J and interchanged the order of integrations, which is allowed due to the absolute convergence of the integrals (n+( ., k) = Vq~+( ., k) e -~o)-

Equations (5.2) and (5.13) ensures the existence of the limit as t ~ - co of Eq. (5.14); thus we are allowed to take the Abelian limit and get

(F~- * F + W_ f , g)o = .[ dk ( f , q~ + (., k)) o O~ (k)

+ lim .[ dk( f , Ro(co(k)+ ie)H + (., k)) o O~ (k) (5"16) e , t o

The Lippmann-Schwinger Equation (4.8) can be written

eb+-(. ,k)=q)f( . ,k)-(Ro(+_co(k)+iO) V)@+-(.,k), (5.17)

where @f( . , k )6L~(R3)×L~(R3)=_(L~) 2 and (Ro(+_e~(k)+iO)V) is a compact integral operator on (L~) 2 (compare [5], p. 14).

By inserting Eq. (5.17) into Eq. (5.16) we finally get

( F ~ * F + W _ f , a ) o = ~ d k ( f , q ~ ( . , k ) ) o O ~ ( k ) P,+ , = ( o f , 9)0 (5.18)

which implies that F~-* F + W_ = Pg , (5.19)

and thus P+ W_ = F+* F~ . (5.20)

In a similar way one shows that P - W_ = F - */~o- and it is furthermore simple to verify that W+ f = W_ f . This completes the proof of part b) of Theorem 5.1.

c) Let f ~ D(Ao) C D(V); we then get

~3

W+ f - W_ f = i .f dt eiA*Ve-~a°* f , (5.21) - o o

and by applying W* we get co

W* W_ f = f - i .[ dt e~A°tW+ Ve-~a°t f . (5.22) - o o

254 L.-E. Lundberg:

Let f , 9 E Wo + be such that f ~ , Og ~ C~( R3 - {0}), ( f , 9 ~ D(Ao)). By inserting Eq. (5.22) into Eq. (5.4) we obtain

cO

( s f , a)o = ( f , 9)0 - i .[ - ~o (5.23)

= ( f , g ) 0 - i - - 0 0

In complete analogy with the SchrOdinger case (see Ikebe [7]) we get

dt(eiA°tW+ Ve-iAot f , g)o

dt(Ve-~aot f , W_ e-iAO,g).

(S f , g)o = ( f , g)o - 2~i .[ dkdk'b(co(k') - on(k))

• ( V q ~ ( . , k), qS+( ., - k ' ) ) f ~ ( k ) O - ~ ( k ' ) , (5.24)

when f , g ~ ~fo +. In the case when J; 9 ~ ~¢~o we obtain similarily

(S f , g)o = ( f , g)o - 2rci .I dkdk'6(o.)(k') - co(k)) (5.25)

• ( W o (., k), (., k')) f o (k) 0o (k'). Equation (5.5) follows directly from Eqs. (5.24) and (5.25), which

completes the proof of Theorem 5.1.

6. Summary and Conclusions

We have in this paper developed the spectral and scattering theory for the Klein-Gordon equation with two static external potentials (coupled like a zeroth component of a four-vector and a scalar, re- spectively)

([--] + 2iqo(x) 0 t - qo(x) z + G(x) + m z) u(x, t) = O .

Our main results, which are contained in Theorems4.5 and 5.1 are similar to those obtained in the Schr/Sdinger case (see [8]). The main difference is that we have to impose a limit of the strength of q0 and on the negative part of qs (see condition iv), p. 244).

Theorem 4.5 can be employed to explicitly diagonalize the quantum- field-theoretic Hamiltonian for a charged scalar field interacting with two kinds of external static fields qo(x) and G(x). Furthermore, Theorem 5.1 can be used to prove that the S-matrix, defined in the LSZ- sense, coincides with the classical S-matrix [Eq. (5.5)] and that asymptotic completeness holds (under our conditions on the potentials). All this will be considered elsewhere.

Acknowledgements. I want to thank Professors L. G~rding, J. Hamilton, L. H6rmander , I. Segal and R. F. Streater for st imulating discussions. Also, I am grateful for the hospitality shown to me at Nordita.

Klein-Gordon Equation 255

Appendix 1

Theorem (Thoe [1], p. 373): Let "V be a complex linear vector space which forms a Hilbert space with respect to each of two scalar products ( ,) and (,)o. Suppose T o and V are closed linear operators on the Hilbert space Y/~o = {~V, (,)o}, possessing the following properties:

i) T o is self-adjoint ; ii) D(V) D D(To);

iii) I[ V f IJ o < a It To f H o + b ][ f ][ o, for f ~ D(To) and 0 < a < 1; iv) The operator T = T o + V with domain D(T) = D(To) is symmetric

in the Hilbert space ~F = {~V, (,)}. Then T is self-adjoint in ~ .

Appendix 2

Proof of Lemma 4.10. Let us start with the following lemma;

Lemma A2. Let h 1 and h 2 be two functions in C~(Ra), such that the compact sets S 1 and $2, defined by

S i = { t ~ R ; 3 k 6 s u p p h ~ with co(k)=t},

are disjoint and contained in R + - {0}. Then

( F -+ * h i , F -+ *h2) = 0. (A 2.1)

We observe that Lemma 4.10 follows from Lemma A2, since )fifl and )~zf2 can be approximated in L2(R 3) by sequences of Cy functions having their support in {k; co(k) ~ 11} and {k; co(k) s 12}, respectively.

Proof of Lemma A2. We start by showing that if 9 ~ C~(R 3) has support in R 3 - {0}, then F+-*g ~ D(A") for each integer n, and

p(A) F+ * 9 = F'+'*(p( +_co( • )) 9), (A 2.2)

for each real polynomial p. To this end we consider

F+-* g = .f dk cI) +-(., k) g(k) .

It is easy [using Fubini's Theorem and our knowledge of ~-+(x, k)J to see that

AF+-* 9 =.f dk (b+-( ., k) (+co(k)) g(k) = r±*( +_co( • ) g) , (A2.3)

weakly. Explicitly this means that

(A f, F +*g) = (f, r -+*(_ co(-) g)), (A 2.4)

for any f e N and thus for any f ~ D(A). Since A is self-adjoint, it follows that F -+*g lies in D(A) and (A 2.3) holds. By iterating this fact we get (A 2.2).

256 L.-E. Lundberg:

Equa t ion (A 2.1) is now easily established. In fact, the self-adjointness of p(A) gives

(p(A) F +- *h 1, F +- *h2) = (F±*hl, p(A) F±*h2), ( a 2.5)

and thus, due to (A 3.4) we get

(F+-*(p( + o•( . )) hl ), F+-* hz) = (V+-* h~, V+-*(p( + oo( • )) h2)). (A 2.6)

We can now choose a funct ion ~p(t) in C~(R) such that tp( t )= 1 for t ~ St and ~v(t) = 0 for t ~ $2. We can approx ima te ~v uniformly on S~ wS2 by a sequence {p,} of po lynomia l s and obta in (A 2.1) in the limit.

Appendix 3

Proof of Lemma 5.3. Let us start by observing tha t J f = { L- 1Lo f l, f 2} when f s ~ . Let f e v~; we then get

HeiAt(J -- t) e- iAot f H = I[(J - 1) e-ia°t f ll (A 3.1)

= I I C - ~ ( C o - c ) e-~A°¢ftl = I IC-~Qe-~a°~f l [ ,

where C = ( 0 01)' Co= (Lo ~), and Q= ( - 0 00)" L e m m a 4 . 7 and

(A 3.1) finally gives

II C- 1 Qe-iAot f II 2 = (Qe-iAot f , C- 1Qe-iAotf)L2@L 2

[t~e-iAot fll 2 < IIC-111L2eL2 ~ J , L 2 ~ L ~

= c o n s t H q f t ( t ) l l ~ O , I t l --- ,oe,

where, in the last step, we have used Eq. (5.12).

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Klein-Gordon Equation 257

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L. E. Lundberg NORDITA Blegdamsvej 17 DK-2100 Kopenhagen/Denmark