KfM(z,,)dp

A N O N - R E L A T I V I S T I C M O D E L O F T W O - P A R T I C L E D E C A Y IV. R E L A T I O N TO T H E S C A T T E R I N G T H E O R Y ,

SPECTRAL C O N C E N T R A T I O N , A N D B O U N D STATES J. Dittrich, P. Exner*)

Nuclear Physics Institute, Czechosl. Acad. Sci., 250 68 l~e~ near Prague, Czechoslovakia

The present paper which represents the f o u r t h part of the series devoted to analysis of a simple Lee-type model of two-particle decay deals with three problems. The first one concerns relation of the model to the scattering theory. We prove asymptotic completeness for the elastic scattering of the two light particles and show that for a sufficiently weak coupling this system has just one resonance whose position is the same as t h a t of the pole which yield the m a i n c o n t r i b u t i o n to the decay law. The second problem concerns spectral concentration; we prove its occurrence for families of intervals a r o u n d E that shrink slower t h a n quadratically in g. Finally, necessary a n d sufficient conditions for the existence of b o u n d states are discussed.

1. I N T R O D U C T I O N

This is the fourth part o f the paper devoted to a detailed analysis o f a Lee-type model of two-particle decay. Let us recollect briefly the contents of the previous parts [ 1 - 3 ] , hereafter referred to as I - I I I . The model itself has been described in I, where we have proven also its Galilean invariance. After separating the centre- of-mass motion, the analysis simplifies to the perturbation problem o f a simple eigenvalue embedded into a non-simple continuous spectrum. Its solution depends crucially on the properties o f the reduced resolvent. In II, we show that for small enough values of the coupling constant 9, the reduced resolvent has a simple second- sheet pole whose position depends analytically on 9.

Finally, in III we study the pole approximation in which the analytically continued reduced resolvent is replaced by the pole term alone. We show that under some mild regularity requirements on the function ~ that characterizes the interaction, the devia- tions of the true decay law from the exponential one resulting from the approxima- tion is of order 9 4. This gives us further a possibility to establish the rigorous validity o f Fermi golden rule for the present model.

Here we shall be concerned with three problems. In the next section, we shall discuss relations of the model under consideration o f the scattering theory. We establish the existence and completeness of the wave operators for the situation when the two light particles that have played the role o f decay products scatter elastically.

Then we express the R-matrix and show that for a sufficiently weak coupling the system has just one resonance. Its position as a position o f the R-matrix pole is the same as that of the pole considered in Theorem II.3.6 and further on.

It is much more difficult to prove that the wave operators are also asymptotically complete, i.e., that o-sl.g(Ho) = 0. This will be done in Section 3 under some additional

*) Present address: JINR, 141 980 Dubna, USSR.

Czech. J. Phys. B 39 (1989) 121

J. Dittrich, P. Exner: A non-relativistic m o d e l . . . I V

assumptions concerning smoothness of the function t3. As a technical tool, we compute here the full resolvent of the Hamiltonian H a. Section 4 deals with the problem of spectral concentration: we show that the spectral projections EH,(Ag ) on a family of intervals A o that shrink around E tend to the projection E,, provided the shrinking is slower than quadratic in g. In this way, the decaying system remembers the embed- ded eigenvalue which "dissolved" once the perturbation was turned on. Finally, the existence of bound states is discussed in Section 5.

The next part will deal with the quantum-kinematical aspect of the decay under consideration, and with the relations of an appropriate kinematical model to the dynamical one discussed in I - I V .

2. C O N N E C T I O N W I T H T H E S C A T T E R I N G T H E O R Y

Consider the scattering process between the two light particles of masses ml and m2 which played the role of decay products in the previous parts of the paper. In view of (II.2.4), the non-trivial part of the scattering problem concerns the centre- of-mass coordinates only. In the following, therefore, we shall use the "relative"

quantities without mentioning this fact explicitly.

The wave operators for the pair (Hg, Ho) are defined by (2.1) (2+_(Ha, Ho) = s-lira e iu"' o - m ~ Pac(Ho),

t--* + oo

where P,c(Ho) is the projection referring to the absolutely continuous subspace of H o. We have the following

Theorem 2.1: The wave operators (2.1) exist and are complete, i.e., (2.2) Ranf2+(Ho, Ho) = Ran f2_(Ha, H 0 ) = RanPa~(Ho) .

Furthermore, assume ( a ) - (c) and (e) (cf. Section III.2). If the function vi has a piece- wise continuous derivative, then f2+(Ha, Ho) are also asymptotically complete, i.e.,

O ' s i n g ( U g ) = 0 .

Since the interaction Hamiltonian gVis of a finite rank, the existence and complete- ness follow from Kato-Rosenblum theorem (cf. Ref. [4], Theorem XI.8). It is much more difficult to establish the asymptotic completeness; we postpone it to Section 3.

The wave operators (2.1) determine the S-matrix of our problem by

(2.3) S = ~ * Q _ .

In order to derive an explicit expression for S, we employ the following Lippmann- Schwinger-type result (Ref. [5], Proposition 6.11): suppose that the wave operators exist and are complete, H o = H o + gVis self-adjoint and D(V) ~ D(Ho) , then (2.4) S = P~c(Ho) + s-lim s-lim S, JR(2 - i t / , H o ) - R(2 + it/, Ho) ]

t / ~ 0 + 6--*0 +

x [g V - g2 VR(2 + i8, Ha) V] P~,(Ho) dEi ~ ,

] 2 2 Czech. J, Phys. B 39 (1989)

J. Dittrich, P. Exner: A non-relativistic model...IV

where, as in Sec. III.5 the decomposition of unity {E(~ ~ refers to the free Hamilto- nian Ho. The assumptions are obviously valid in our case, so (2.4) holds. For brevity, we s'hall work in the following with the R-matrix defined by

(2.5) R = S - Pac(Ho).

Our goal is now to express matrix elements of this operator. It is clear that both S and R commute with P,~(Ho); in fact, S is a partial isometry o f the subspace P,~(Ho) ~ = ~ a into itself (Ref. [5], Proposition 4.6). Hence we shall consider vectors of the form

(;) (0)

d ~0d

CO~[R3~

only, restricting our attention to those with Od, ~bd e o ~ J"

Let us first express the integral on the rhs of (2.4). We can write it as

(2.6a) C - j'~ B(2)Pac(Ho)dE(z ~ ,

where B(2) denotes product of the square brackets, because the support of dE(2 ) is R+. Moreover, in the corresponding expression of (ku, C~) the integral is actually taken over a finite interval A = [0, 4o], since the supports of ~d, ~d are by assumption contained in a ball of radius x/(Zm2o) for some 4o. The integral is defined (cf. Ref. [5], Section 6.1) as a strong limit of the corresponding Riemannian sums, so we have (2.68) (~, CO) = lira 2(B(tt~)* ~u, E o ( A s ) ~ ) ,

6n-+O j

where 4y are the points of some partition of A, #y e Ay = (4~., 4y+ 1] and 6, = max 5y, J where 6y is the length of Ay. The vectors appearing in the last expression can be written down explicitly in the momentum representation,

(2.7a) and (2.7b)

(eo(a7) +),(p) = z + .

f 3~(k ) 2it/

(8(4)* = - o r.(4 + ia, : d(k) d k ,

where the term linear in g is absent in view of the condition (II.3.3). Substituting from here into (2.6b), we get

(2.8a) (7/, C4~) = lim ~C(/~., 2~, 6~.) 6~.,

6~--* 0 j

where

(2.Sb) C(#, 4, e) = e -1 J'M(z,~)(B(tt)* ~g)d(P) ~bd(P) dp

Czech. J, Phys. B 39 ('1989) 1 2 3

J. Dittrich, P. Exner: A non-relativistic model... I V

and M ( 2 , ~ ) = B3(~/[2m(2 + ~)])\B3(~/(2m2)). The sum on the rhs of (2.8a) is just integral of the step function F, that assumes the value

Vo(2) = C ( g , 27", 67) for 2 E Aj. So we have n

(2.8c) (gt, C~b) = lim ~F,,(2) d2,

n--* oo

provided lim 6, = 0.

n--* oo

Suppose nowthat the function ~ fulfils the assumption (d). Since(2.7b) is a contin- uous function of 2 and p, one checks easily that

(2.9a) lim F,(2) = m x/(Zm2) Ss2 (B(2)* ~)a (x/(Zrn2) n) 0a(x/(Zrn2) n) dl2.,

n - ~ o o

where $2 is the unit sphere and 2 e A. Further (2.78) and (2.S8) gives the estimate

IC(p,

2, e)l _-< ~ (l~l, I~el) IM(Z,~)I~(P) ~e(P)I dp.

In view of the stated assumption (recall that ~e e

Cg(R3)),

modulus of integrand in the last integral is bounded by a positive K within B3((~/(2m2o)). Then the integral may be estimated by

KfM(z,,)dp

=< 4~m~/(2rn20) e for 0 =< 2 < 2 + e =< 20 , and we have the following e-independent bound

(2.9b) ]C(/z, 2, e)] <

8rcmK

~/(2m2o) g2([O[, I~d]) . qc5

The relations (2.8c) and (2.9) together with the dominated-convergence theorem then show that

(7 ~, C~) = I~ 02 m ~/(2m2) Is~ (B(~.)* V)d (x/(Zrn2) n) Ca(x/(2rn2) n) dO. =

Combining now the relations (2.4), (2.5) and (2.7b), we get (2.10) (llY ' Rl~) ^{= ~ 2} lim lim .~R3 dp J'n3 dk ~-~) r

jl--* 0 + 6 - - * 0 +

• r. + H . (k2/2'n _ 2/2m) + Assume now that for each 2 > 0, there is a finite limit

lim

r.(z, Hal

= r.(2 + i0, Ha)

I r n z > 0

1 2 4 Czech. J, Phys. B 39 (1989)

J. Dittrich, P. Exner: A non-relativistie model... 1 V

and that the function (2, fi)~-~ r,(2 + ifi, Hg) is bounded in (0, 20) x (0, Co) for some e o. This is true, e.g., under the assumptions ( a ) - ( e ) for a sufficiently small coupling constant 9 =~ 0 (cf. Lemma III.2.4). Since Oa, ~d have compact supports, the dominated-convergence theorem implies

(~, R+) = 9 2 lim fR~ dp fR~dk ~(p) +a(P)

• r. + io, (k /2m _ / / 2 m / +

Next we must interchange the limit q ~ 0 + w!th the integrals. Consider first the second integral: we can write it as

lim (~176 2it/k2 ~,(k) ;s2~-~kn) d~2..

,_.o+d ~ (kZ/Zm ^_ p2/2m)2 + ,2

Since ~b a ~ C~~ the last integral is a bounded continuous function of k; we denote it as I~,(k). Introducing the variable y = (k 2 - p2)/2rmh the last expression can be rewritten as

(2.11)

/

2ira lim ,-*o+ -p2/<2,,,)1 + y2 x/(p2 + dy 2m'ly) v'(x/(P2 + 2mqy))I* (x/(p2 + 2mqy)).

However, ~1 is bounded by the assumption (d), and the limit can be therefore cal- culated by the dominated-convergence theorem to be 2~imp~l(p)I~(p). Using once more the fact that the integral in (2.11) can be estimated independently of q in an interval (0, ~/o), we are able to justify application of the dominated-convergence theorem to the interchange of the limit with the first integral. Together we get

(7 j, Rob) = 2rrimg 2 [el(p)] 2 pr, + i0, H o dQ. =

I (p2 ) f S

= 2riling z dp

@.(~ lel(p)] z

pr,, ~ + i0, H o ~pa(Pn') dO,,,,

*J R3 2

where the second equality follows from Fubini theorem. Since C~(R a) is dense in ~Cd, we get

(+ )r

(2.12a) (Reb)a(p) = 2gimg 2 [el(p)l 2 pr, -~m + i0, H o ~a(pn') d e . , .

d $2

The operator R is bounded, therefore by standard density arguments the validity of the last relation may be extended to each 9 e ~ .

Let us conclude the above discussion.

Czech, J. Phys. B 39 (1989) ] 2 5

J. t)ittrieh, P. Exner: A non-relativistic m o d e l . . . I V

Theorem 2.2: Assume ( a ) - ( e ) , then the relation (2.12a) holds. In other words, R is a decomposable operator on Jt~a = L2(~+ ; L2($2); m x/(2m2) d2) and its component R(2) for a given 2 ~ R + is the Hilbert-Schmidt operator with the kernel

(2.12b) R(~.; n,

n')

= 2=img2lO, (4(2m~.))12 ~/(2m2) r,(2 + i0, Hg).

N o w we are able to write down the amplitude of the elastic scattering with the initial momentum p' = pn' and final momentum p = pn. It equals (cf. Ref. [5], Eq.

(7.48)) - 2ni(2m2) -s/2 R(2; n, n'), i.e.,

(2.13) f(2; p' --+ p) = 4rc2mg21~l(x/(2m)O)] 2 ru(2 + i0, Hg).

Furthermore, the assumptions of Theorem 2.2 ensure that both 2 ~-+ [es(~/(2mX))l 2 and 2 ~-~ r,(2 + i0, Hg) can be continued analytically from the upper half plane to the region ~2 that contains (0, oo) (as for the first of them, recall that we choose the cut of the square root conventionally along the negative real axis). For g small enough, the continued function 2 ~-+ ru(2 + i0, Hg) has, according to Theorem II.3.& a simple pole whose position depends analytically on g. The same is then true for the scattering amplitude. In the commonly accepted terminology, this fact can be expressed as

Proposition 2.3: Under the assumptions ( a ) - ( e ) , there are a positive go and a com- plex region f2 s m (0, oo) such that for 0 < Igl < go, the scattering system described above has just one resonance in f2 s whose position is given by the relations (II.3.16).

In order to illustrate typical features of the resonance scattering, let us calculate the cross section and the phase shift. The differential cross section is equal to the squared modulus of (2.13). In order to get the squared modulus of r,(2 + i0, H0), one must use the relations (II.3.11) and (II.3.15). A short calculation then gives

(2.14) ~-~do- (2; p' ~ p) --- 16n4rn2gr

lel(~/(Rm'Z))P*

[ 2 - E - 4~g2I(2, v)] 2 + 32n4m3gr

les(\/(2m2))l ~"

It is clear that the scattering is isotropic as a consequence of the fact that v is rota- tionally invariant. In that case, the total cross section does not depend on the chosen initial direction and equals 4n(do-/dQ). It means that both of them have the same shape as functions of energy. For {)'tot, e.g., we have

o-,o,('~) = '~ r(g, ~.)2

2m2 (2 -- E(g, 2)) 2 + 88

~)2'

(2.15a) where (2.15b) (2.15c)

E(g, 2) = E + 4rig 2I(2, v), F(g, 2 t = 8n2mg21~,(x/(2m2))l 2 ~/(2m2).

If the coupling constant g is sufficiently small and the function ~, is slowly varying

] 2 6 Czech. J. Phys. B 39 (1989)

J, Dittrich, P. Exner: A non-relativistic model l . . I V

around E, then we obtain an approximative expression to (2.15a) replacing

E(g, 2)

by

E(g, E)

and

F(g, 2)

by

F(g, E) = FF(g).

The cross-section formula acquires the familiar Breit-Wigner shape with the peak situated at

E(g, E),

and with the width given again by the Fermi-rule expression (III.5.2). At the same time, it is clear that the cross section may have accidental peaks connected with the shape of the function ~ . Let us turn now to the phase shift. According to (2.5) and Theorem 2.2, S is a decomposable operator on o ~ and its component for a given 2 is S(2) = I + R(2).

Only s-wave part is non-trivial according to (2.12a). The s-wave phase shift 6o(2 ) is determined by the relation

(2.16a) So(k) = e 2i~~ I = (1 + 4r~R(2; n, n')) I ;

the higher phase shifts 5,(2) = 0, l = 1, 2 . . . Using now the abbreviations (2.15b,c) one finds after a short calculation that

- 2 + E(g,

2) +

89 4)

(2.16b) So(A) =

- 4 + E(g,

4) -

89

2 ) ' and therefore

. 89 r ( g , 2)

(2.17) 5o(4 ) = arctg . . . (mod r 0 . e(g, 2) - 2

For a weak coupling and a slowly varying ~1, one has an approximative expression o f this function around

E(g, E),

namely

(2.18)

89 r ~ 0 ) arctg

E(g, E) - 2

50(4) ~ 89

+ arctg 89 FF(g) E(o, ~) - 2

. . . a < e(g, e)

. . . 2 = E(g, E) . . . 2 > ~(g, E)

again modulo re. It shows that the phase shift has a sheer increase which changes its value on about re. This is another characteristic feature of a resonance.

The last approximative formula holds in the resonant region only. The asymptotic behaviour for large and small 2 can be also easily obtained under some additional, not very restrictive assumptions. From Lemmas II.3.4 and III.2.2, we know that I ( . , v) is bounded,

1I(2, ~)I--< c 2 ,

under the assumptions (a), (b) and (d). Therefore Eq. (2.17) gives (2.19a) 60(2) ~

-4rcZmgZJel(~(2m2))l 2

~/(2rn/2) (mod g)

as 2 ~ o% the asymptotic region being specified by the requirement

2 ~> E + 4rcgZC2.

Czech. J. Phys. B 39 (1989) 1 2 7

J. Dittrich, P. Exner." A non-relativistic m o d e l . . . I V

In particular, one has

(2.19b) lim 60(2) = 0 (mod n).

).'-~ o 0

To find the behaviour for 2 --* 0 +, we need to know the limit of 1(2, v) first.

Lemma 2.4: Assume (a) and (d), then

(2.20) I ( 0 + , v) - - l i m I ( 2 , v) = - 2 m ~ [ ~ l ( p ) [ 2 d p .

2--,0 +

Proof: Choose a positive A and consider only 2 > 0 satisfying A 2 > 4m2. Then 1(2, v) exists according to Lemma II.3.3 and

1(2, v) -- ~ f] le'(P)[~ p=

- (p /2m) By the dominated-convergence theorem,

dp + f ~ IO'(P)I~ p~

~l-~

(-~P_m)

d p .

lim (co ]~a(p)lZp2 d p = - 2 m f ~ [ ~ l ( p ) 1 2 d p JA - ( / / 2 . , ) 3 A

(note that ~1 ~ L2(~+) according to the assumptions (a) and (d)). a simple calculation gives

(2.21) ~ I a ]~-l-(OI2PZ dp =

Jo 2 -- (p2/2m)

j

^{*A Vz(p ) _} V2(X/(2m)~)) P dp + rnvz(~/(2m2))In 2

- - _ . ~

o 2 -- (pZ/2m) (A2/Zm) - 2

Since [vz(~/(2rn2))] < C1 x/(Zm2) by the assumption (d), the second terms tends to zero as 2 ~ 0 +. With the help of the mean-value theorem, one checks that the integrand in the first term on the rhs of (2.21) is bounded uniformly with respect to 2.

So the integral exists in the Lebesgue sense and the dominated-convergence theorem yields

lim. N ~ 5 ~ , 2 m ) d p = - 2 m I~I(P)I dp.

L ~ O + 0

Summing the two contributions, we arrive at (2.20). []

Let us now denote

(2.22) gcr = E1/2[ 8~m ~ ]~I(P)I 2 d p ] - 1 / 2 . Proposition 2.5: Assume ( a ) - ( e ) and ]g[ + 9r Then

(2.23a) lim 60(2 ) = 0 (mod rQ.

2 ~ 0 +

128 Czech. J. Phys. B 39 (1989)

J. Dittrich, P. Exner: A non-relativistic model...IV

Furthermore, if

MI 2

~ c([o, oo)) and I~1(0)1 4= 0, then

(2.23b)

60(2)

~ 4n~ma2le~(~ (mod re)

E l 1 - (g/go,y]

as 2 --* 0 +.

Proof: The relations (2.23) follow from (2.17) and Lemma 2.4. []

The relation (2.23b) shows that 60(2 ) approaches the limit from above or from below for Igl < go, o r Igl > go,, respectively. For Igl -- go,, we are able to deduce the analogous conclusion under an additional assumption on v. We need an estimate

on 1(2, v) first.

Lemma 2.6: Assume (a), (d) and d/dPl~l(p)l 2 bounded in (0, o~) Then there is a positive c such that

(2.24) ]I(2, v ) - 1(0+, v)] < e(lln (2/,01 + a) 2 f o r 2 > 0.

Proof: Let us choose A > 0 and denote v4(p) --

]e,(p)l 2.

Since I ( ' , v) is bounded, it is sufficient to consider 2 e (0, A2/4m). By Lemma 2.4,

(2.25) We estimate

= ~ ( ~ v,,(p) dp 1(2, v) - I(0+, v) 2m2

Jo

j

^{.oo v4.(p)} dp < 4m ~v,,(p) dp < o9.

A 2 - - ~ - 2 r n ) = A-T J A

2mf] ~;(r p+,/(2m2)

+ 42) m Furthermore,

aaf A vg(p) dp=vg(x/(2m2)) / ( 2 2 ) l n A + \ / ( 2 m 2 )

o 2 - (pZ/am) A - x/(2m2)

where 4(2, p) is a number between ~/(2m2) and p. Since

0 < \/(2_~)in A + ,~/(2m2 ) ~ / ( m ) ( 2 x/(2rn2) '/ < 2(2

~1 x/(2m2 } = ~ In I + A T ~ / = . ~ - - and

A

O < - l n l + < l n 2 - 8 9 2

p + , / ( 2 . , 2 ) 2 = m

the inequality (2.24) follows with a suitable c. []

Proposition 2.7: Assume (a)-(e). Let the function ]~ll 2 be continuous in [0, oo) with a bounded derivative in (0, oo). Furthermore, let ]~,(0)] # 0 and [gl = go,.

Czech. J. Phys. B 39 (1989) 1 2 9

Then (2.26)

J. Dittrich, P. Exner: A non-relativistic model... I V

lim 60(it) = 89 (mod r:).

9,--* 0 +

Proof: Equations (2.16b), (2.15) and (2.22) together with Lemmas 2.4 and 2.6 give S o ( i t ) = - i t + O(itlln (it/re)l) + 4re z img2lal(X/(2mit))l z x/(2mit)

- 2 + O(itlln (it/m)l) - 4re z im9=lel(,/(emit))l ~ ,/(2mit)

so So(2 ) tends to - 1 as it ~ 0 + and (2.26) follows. [ ]

3. ASYMPTOTIC COMPLETENESS

Now we would like to complete the p r o o f of Theorem 2.1. In order to do so,"

we need a more complete information about the full resolvent (Hg - z ) - ~ of the Hamiltonian. The obtained formulae will be useful also in the next section.

P r o p o s i t i o n 3.1: Let ~a ~ ~fa, and assume that z belongs to the resolvent sets o f H a and H o, then

(3.1a) Ed(FI" - z ) - 1 Eric d = ( D 0 - z ) - 1 (b d + g 2 r . ( z , H g ) ( 0 2 z , t~d) Olz, where the vectors 0iz are defined by

(3.1b) (Plz = ( t ~ 0 - - Z) - 1 ~ , (P2z = ( / ~ 0 - - 5 ) - 1 ~ "

In the last relation, (D o - z)-1 stands as a shorthand for multiplication by ((p2/2rn) ^- - z ) - 1 - cf. ( I ] . 2 . 9 ) .

P r o o f : The first part of the argument is similar to the p r o o f of Proposition II.3.1, with the roles o f J r , and 24~ interchanged. We start from the second resolvent identity which yields the relations

- : ~ ( D . - ~)-1 E, ee,(/?o - ~ ) - ' E , ,

E , ( D . - ~)-~ e . = - : , ( D . - ~)-~ E . ~ E . ( t I o - z ) - 1 E . - - oF,~(;I, - ~)-1 E~F,.(Do - ~)-~ E . .

where we have used eommutativity of E,, Ea with Do, orthogonality of these projec- tions and the relation E , + E a = I. The last term on the rhs of the first equality is zero in view of the Friedrichs condition; the same is true for the first rhs term of the second equality, because E , V E , = 0. Now one has to substitute from the second relation to the first one, and to multiply the obtained operator identity by (Do - z)

130 Czech. J. Phys, B 39 (1989)

J. Dittrich, P. Exner: A non-relativistic model. : . I V

from the right. It gives

(3.2a) Ea(I~g - z) - 1 E a ( E a ( P I o - z ) E a - g2EdPE,(FIo - z) -~ E , PEd) = E a . This equation is solved by

(3.2b) Ee(/~g - z) -~ E a = {Ea[/~ 0 - z - 92pEu(FIo - z) -~ E , fT]Ea} -1 , where the inverse refers to N(o~a).

Comparing with Proposition II.3.1, the situation is now more difficult, because we are looking for inverse of an infinitedimensional operator. Fortunately, the problem is solvable. The curly-bracket operator in (3.2), which we denote by A z for a moment, acts as

(3.3) (A=~a) (p) = - z ~a(P) - 92 ~(p) (~' ~a) E - z

It differs therefore from the projection of (/4o - z) to • a by a rank-one operator.

Then one has to employ the corollary of the second-resolvent identity which is known as Krein formula [6]; according to it, the difference of the corresponding resolvents is again of rank one. It suggests the following guess,

(3.4) ( E a ( R o - z ) -~ E a ~ a ) ( P ) = ( ( p e / 2 m ) - z) -~ ~/a(P) + + g)(o2z,

where the vectors 0~z are given by (3.1b) (recall that e ~ L2(N3)) and c~(z, g ) i s an unknown complex number. Then we express (Ed(FI o - z ) - 2 E a A ~ a ) (p) using the relations (3.3) and (3.4), the expression

+ l = ( p ) _

( p Z / 2 m ) - z

and the analogous one for 0z= (with z replaced by 2). In view of (3.2b), it must be equal to ~a(P). This requirement leads to a condition which yields

g 2

(3.5a) e(z, 9) =

- -

The same argument applied to (A~Ee(FI o - z ) - ~ E ~ t ~ d ) ( p ) gives g z

(3.5b) c~(z, g) =

The conditions (3.5) are, however, consistent because (Oz~, ~) = (v, (o1~) = - G ( z ) . Combining the relations (3.5) with (II.3.4), we get ~(z, g) = gZr,(z, Hg), i.e., the

desired result. []

Czech, J. Phys, B 39 (1989) 131

J. Dittrich, P. Exner: A non-relativistic model...IV

R e m a r k 3.2" With the help of the relations used in the proof, one can easily write down also the "non-diagonal blocks" of the resolvent. For any t~a e o~r we have (3.6a) E,(/-]ro - z) -~ Ea~ a =

- g r , ( z , Hg)(~2=, ~a).

The result is, of course, a complex number, i.e., an element of o~f, (recall that we work with

~fr~X = C @ Lz(N3)).

On the other hand, for any ~, e ~ f , we have (3.6b)

Ee(Ft o - z) -~ E,~, = - g ~ , r , ( z , Ho) ~o~=.

The relations (3.6) together with (3.1 t and (II.3A) describe the resolvent (/~0 - z) - t completely; after a short calculation, we obtain for any ~ : / ~ : ) the expression (3.7) (~, (/4g - z) -~

~) = (~a, (FIo - z) - ~ ~a) + ru(z, Ho) .

[ 0 . - -

N o w we are ready to deal with the problem left from the preceding section.

P r o o f of the asymptotic completeness - - Theorem 2.1: First of all, we must ( ~ " ) of a dense set in o~. W e t a k e ~a~

estimate Im (~, (/~g - z ) - I ~) for g' = t~ a

C~~ Further we choose a finite interval [a, b ] c 01 ca (0, oo) (cf. Theorem II.3.6) and consider z = 2 + ie with 2 ~ [a, b] and ~ > 0. For the free resolvent, we have

fR e[~a(P)]2 d p . (3.8) I m ( ~ a , ( H o - z ) - ' ~a) = 3 ((p2/2m~ 22~-)z + 82 If we denote e = 2/e, the rhs of (3.8) may be estimated as follows:

f f lc2dk

(2m) 3/2 ./g f

l~a(4(?ms)--k)--[2

dk < 47t(2m) 3/2 4(e) sup ]~a(p)[ 2 (k 2 c~) 2 + 1"

In the last integral, we substitute u ^{= k 2 _} ~ and estimate it as

f ~ x/(u + @_du <_ f ~ \/tuI + x/e du

2(u 2 + 1) - J - m 2( u2 + 1) Together we get

(3.9) 0 < Im

(~d, (fflo -- Z) -~ ~/a) < 8rt2m3/2

sup ]~a(p)] 2 [~/(s) + (89

p e R 3

therefore, the imaginary part (3.8) is a bounded function of z in [a, b] x (0, %]

for any eo > 0. We shall set So = 1 in the following. The relations (3.7) and (3.9) then give

[Im(7~, (/~o - z) -1 ~)1 < 8n2m3/2[ 1 +

(89 l/z]

sup [l~d(p)] 2 q'-

p~R 3

132 Czech. J. Phys. B 39 (1989)

J. Dittrich, P. Exner: A non-relativistic m o d e l . . . I V

Under the assumptions ( a ) - ( c ) , the function

r,(., 1to)is

continuous for a sufficiently small 9, and therefore bounded in [a, b] x [0, 1] (on the real axis, the analytic continuation is used - cf. Theorem II.3.6). Suppose that the function ~1 has a bound- ed derivative in [a, b]. Since ~a e C~~ the same argument as in the proofs o f Lemmas II.3.3 and II.3.4 shows that the function defined on [a, b] x (0, 1] by

z~-* 9 3 (p~2m~) ~ z dp

and extended continuously to I m z = 0 is continuous within

[a,

b] x [0, l], and therefore bounded; the same is true for the other function that appears in the last estimate. Collecting these results, we see that

(3.10) sup I, b Jim (~, (/~r o - 2 - ie) -1 ~')l q d2 <

0 < e < l

for any q > 1. The well-known criterion (cf. Ref. [I.27], Theorem XIII.19) then implies /~ttg((a, b ) ) ~ s o~f'ac(/~g ). Since C~(R 3) is dense in LZ(~3), it follows that Ran/~Hg((a, b)) c o'r176

N o w we would like to extend this result to the whole R + and arbitrary values of g.

According to the assumption (e) and Lemma II.3.4,

r,(., 1to)

may be continued analytically accross B +, with a possible exception of the points 2 e R + at which 2 = E + yz Go(2). These points correspond to eigenvalues of /~g. However, the analyticity assumptions (b) and (e) imply that such points are isolated with no accumulation points except infinity. If none of them is contained in [a, b], the procedure described above may be carried out.

We have required 01 to have a continuous derivative within [a, b]. By assumption,

~'~ is piecewise continuous, i.e., its discontinuity points are isolated and have no accumulation points except infinity. Together we have a closed subset M ⁼ {2j} ^c

c B + such that at its points either r.(.,

Ho)

is not bounded or ~ is discontinuous.

Let 2j, 2j+1 be anytwo neighbouring points of M. Then/~n,((2j, 2j+1) ) out ~ c W,~(R0) since

ff, ng((a,

b ) ) ~ c OUgar for any 2 j < a < b < 2j+1. It means that /~ng(R + \ M) o'er ~ c o~,r and therefore trsing(/-]ro)is void. []

Remark 3.3: (a) The result holds without any limitation on 9. If

Igl

is small

enough,

then Theorem II.3.6 ensures that a .

r, ( , Ho)

has no real-axis poles around E. Such a pole can appear in the strong-coupling case; however, it is always an isolated point which does not contribute ^{t o} 0"sing(Ho).

(b) In some cases,

Hg

has no eigenvalues - cf. Lemma III.2.4. The eigenvalue problem will be further discussed in Section 5.

(c) The smoothness requirement on ~1 does not follow from the assumptions III.2.1, since the latter concern the modulus of vl only.

Czech, J. Phys. B 39 (1989) 133

J. Dittrich, P. Exner: A non-relativistic model.. ,IV

4. SPECTRAL CONCENTRATION

We have seen that the embedded eigenvalue E corresponding to the initial particle disappears once the interaction is turned on. Nevertheless, the system "remembers"

the dissolved eigenvalue: if the coupling is sufficiently weak, the states whose energy support does not contain a small interval around E are nearly orthogonal to the original eigenstate; the weaker is the coupling, the smaller is this interval. This effect is known as spectral concentration; we are going to formulate it now for the model under consideration.

Theorem 4.1: Let ~, fl be positive numbers, fl < 2, and denote (4.1a) A o = (E - cr p, E + oclgllJ).

Suppose the assumptions ( a ) - ( c ) are valid and ~ is continuously differentiable in some neighbourhood of ~/(2mE), then

(4.1b) s-lim EH.(Ao) = e . .

0 ~ 0

Proof: In view of Theorem II.3.6, we can choose a positive go such that the func- tions ru(', Ho) and ~ are continuous in z] o for all 0 such that 0 < Igl < g o The proof presented in the previous section then shows that Ran/~u.(z]o) = ~f',~(//o).

Then we have

= - l i m - l l - I m ( ~ , ( / : / o - 2 - i ~ ) - ' T ) d 2 ,

where the second equality follows from the Stone formula - el. Ref. [-I.27], Theorem VII.13. Now we choose ~a e C ~ ( ~ 3) and calculate the g ~ 0 limit of (4.2). In view of (3.7), we have

l i m ( ~ , E n o ( A o ) ~ ) = l l i m ^__

lim f Im{f.

iel~d(P)]2dp ^{- I -}

, ~ o ~ , ~ o . ~ o . ~ ~ ( ( p ~ ~ L ~ + ~

I f R3 (pe/2rn)O(P) ~a( ) 1 ~ie dP T r , ( Z + i e , Ha) 0 , - - g -- p- (4.3)

• [ ~ . - The estimate (3.9) gives

0 ~ lira lim

g~O e ' - * 0 +

(p2/2~m) : ~ _

i~ dP]} d2"

I A I el~a(p) 12 dp , .3 ((v2/2m) - 4) 2 + d ^d2

<_ lira lira 8n2m 3/2 sup ]~a(p)] 2 {v/e ~- [~-(E 7- celg.~)] 1/2} 2~]gla =

g~O s ~ O ~ p e R 3

1 3 4 Czech. J. Phys. B 39 (1989)

J . Dittrich, P . E x n e r : A non-relativistic m o d e l . . . I V

i.e., the contribution of the first .term of the preceding section: it shows that is bounded. Then we may interchange in this way

(4.4)

= lim 16~7~2m 3/2 sup I<(p)l ~ [89 + ~[af)] i n ]gl p = o ,

g~O p~R a

is zero. We shall use once more the p r o o f the second term in the curly bracket in (4.3) the limit e .~ 0 + with the integral obtained

lim (~,/~Ho(Ao) ~ ) =

g~0

1 l i m f a Im { r , 9 ( H g ) [ - lim fR ~ ( p ) ~ a ~ ) ]

= - 2, 0,, g dp

rc 0-~o , ~-+o+ , ( p 2 / 2 m ) - - ie

• [~.-a lim f~

_{, - o +} 3 ( p 2 / ~ m ) - - - - 2 - - i~

^~.(p)eO,) dp]} d2.

We have seen in the preceding section that the limits in the square brackets are continuous and bounded functions of 2. The same is then true for the product of the two square brackets in (4.4), which we denote for a moment as (r g). Next one

r~(2,

has to express Ha) from (II.3.11) and (II.3.15) and calculate first the limit r~(2, n0) Re if(2, g) d2

lim ~A~ Im =

g--*0

E + ~ l g l a 4rc292rn[~,(x/(2rn2))[ 2 \/(2m2)

= lim o-~o j E_=Iol e [ E - - 2 -b ~ v ) ] 2 ~ 3 ~ - 4 7 z 4 g ~ ( \ / ( 2 m 2 ) ) [ 4 x Re if(2, O) d2.

One substitutes x = g - 2 ( 2 - E) and uses the dominated-convergence theorem, then the limit equals

4rt2mlel(X/(2mE))] e ~/(2mE) Re N(E, 0) dx

x o~ I x - 4 ~ I ( E , v)] 2 + 3 2 = 4 m 3 E l ~ a ( ~ / ( 2 r n E ) ) ] 4 = 71 Re C~(E, 0).

Since ~r 0) =

l < ~,

1 lim~a. Im a

- r . ( 2 , Hg) Re if(2, 9) d2 = I < =

g--*0

(4.5a) Similarly,

(4.5b) lim ~ag Re r.a(2, 17.) Im if(A, g) d2 =

O.-*0

f ~M~-2 - x + 4~t 1(2, v)

= lim g ! [x - 4re 1(2, v)] 5 + 32n4m32]~a(~/(Zm2))]" 9 -1 Im fq(2, g) dx .

o-'o d - ~ [ o l ~ - 2

Realizing that for considered values of 2 the functions I(., v), g ' l lm f#(., g) are

Czech. J. Phys. B 39 (i989) ] 3 5

J. Dittrich, P. Exner: A non-relativistic m o d e l . . . I V

bounded independently of g and

2Ig,(~/(2mO) 14 >__ 89 > 0

for sufficiently small O, the absolute value of (4.5b) can be easily estimated by lim Igl (c, + c 2 In

Id)

^{= 0}

g ~ 0

with suitable constants q , c 2. So (4.5b) vanishes and (4.4), (4.5a) imply (4.6) lim

(~, E.~(Ao) ~) = I~.12 = (~, E.~)

g o 0

w;t r. t t at t is set e.se i .

for a l l ~ = ~a

\ /

together with the polarization identity, we get w-lim ~n,(A,) = E, .

g + 0

At the same time, (4.6) together with the density argument gives lim l[ff, n,(A,) g'[I 2 = []E,g*ll z

0-+0

for all ~' E W. The last two relations imply

s-lira ff, uo(Ao) = E.

g--+0

which is the p-representation form of (4.1b). Hence the theorem is proved. []

5. B O U N D S T A T E S

Let us turn now to the problem of the existence of bound states, i.e., eigenstates of the Hamiltonian Hg.

Proposition 5.1: A bound state ~ = ( ; ) with energy ~' existsiffl) ~EL2(R3), ga~l(p)

(5.1.) ~(p) = ~ _ (pZ/2m i

and

(5.2) e = E + 4zg z f~Oo g s e [ e ' ( p ) [ : - : dp (the last integral exists because ~ e Lz(R3)).

Proof: The assertion follows immediately from the Schr6dinger equation /-'1~ =

= g ~ , . [ ]

2) Common abbreviation iff is used for if and only if.

1 3 6 Czech. J, Phys. B 39 (1989)

J. Dittrich, P. Exner: .4 non-relativistic model...IV

Proposition 5 . 2 : Assume g 4: 0. Let [911 ~ C(0, oo) and d" > 0 is an eigenvalue of Hg, then

(5.3) = 0 .

Moreover, if 19,[ or e, belong to C'(0, oo), then a positive o ~ is an eigenvalue of H, iff the relations (5.2) and (5.3) hold.

Proof: One has only to use Proposition 5.1, and to realize that for t911 e C1(0, oo) or ~1 ~ Cl( 0, oo), the function (5.1) is square-integrable iff (5.3) holds. []

Proposition 5.3: Let g 4= 0, then ~ = 0 is eigenvalue of

H a

iff the function p ~-~

~_+ p - 2 9~(p) e L2(D? +, p2 dp) and

E = 87zg2m I~

191(P)I 2 d p .

Proof is an immediate application of Proposition 5.1. []

Proposition 5.4: (a) There is at most one bound state with negative ehergy.

(b) Let ~ e L2(B+), then a bound state with negative energy exists iff (5.4) E < 8r~g2m .fa ~ [e,(p)l 2 d p .

Proof: If g < 0, then the function (5.1) is in L2(Z~a). The lhs of (5.2) is increasing, while the rhs is non-increasing with respect to d" in ( - o % 0) so equation (5.2) has a t most one solution. Furthermore, both sides of (5.2) are continuous functions of # in ( - m , 0),

lim [ E + 4rcg2 r ~~

[~1(~ 12 p 2 d p ] : E > O ,

and for 91 e

L2(R+),

lim [ E + 4rcg 2 foo

[91(_~12 p2 dp]=E_81zrnt72 ,~

lt)l(p)]2 dp.

A solution 8 < 0 to (5.2) clearly exists iff the last limit is negative. []

Corollary 5.5: Let ~1 e/-2( N+, p2 dp) be a function with non-zero values whose modulus is continuous in [0, oo) and g =t= 0. Then there is a bound state (just one, and with a negative energy) iff

(5.5) g2 > go2 _ E

8zrm I~ ~ I~l(p)[ 2 dp"

Proof: According to Propositions 5.2 and 5.3, there is no non-negative eigenvalue

o f H a

under our assumptions. Since ]91] is continuous, and therefore bounded in a (right) vicinity of zero, 9 t 6 L2(n+). Then a bound state exists iff 9 2 > 9~ 2, due

to Proposition 5.4b. []

C ~ h . J. Phys. a a9 09a9) 137

J. Dittrich, P. Exner: A non-relativistic m o d e l . . . I V

Remark 5.6: We have seen in Proposition 2.5 that the s-wave phase shift changes its behaviour at 2 ~ 0 + for

191

-- 9 , . Since the b o u n d state emerges at the same value o f 9, the validity o f a Levinson-type theorem is indicated - in this connection cf. Ref. [7].

Remark 5.7: The possible bound states with positive energy (eigenvalue o f Hg embedded in the continuous spectrum) discussed in Proposition 5.2 are similar to that appearing in the two-particle problem with rank-one interaction

9v(v, ")

if the function vl has a zero [8].

The authors are indebted to Dr. M. Sotona for Remark 5.7.

Received 29 April 1988

References [1] Dittrich J., Exner P.: Czech. J. Phys. B 37 (1987) 503.

[2] Dittrich J., Exner P.: Czech. J. Phys. B 37 (1987) 1028.

[3] Dittrich J., Exner P.: Czech. J. Phys. B 38 (1988) 591.

[4] Reed M., Simon B.: Methods of Modern Mathematical Physics, IlI. Scattering Theory.

Academic Press, New York, 1979.

[5] Amrein W. O., Jauch J. M., Sinha K. B.: Scattering Theory in Quantum Mechanics. Benjamin, Reading, 1977.

[6] Akhiezer N, I., Glazman I. M.: The Theory of Linear Operator sin Hilbert Space. Nauka, Moscow, 1966; w 106 (in Russian).

[7] Wollenberg M.: Math. Nachr. 78 (1977) 223.

[8] Tabakin F.: Phys. Rev. 174 (1968) 1208.

We do not reproduce the full list of references from the preceding parts of the paper. Here we have referred t o

[I.27] Reed M., Simon B.: Methods of Modern Mathematical Physics, I. Functional Analysis, IV. Analysis of Operators. Academic Press, New York, 1972, 1978.

138 Czech. J. Phys. B 39 (1989}