Báo cáo toán học: "A complete treatment of low-energy scattering in one dimension " ppsx

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1 OPERATOR THEORY Copyright by INcREsT, 1985

A COMPLETE TREATMENT OF LOW-ENERGY SCATTERING IN ONE DIMENSION

D BOLLE, F GESZTESY and S F J, WILK

1 INTRODUCTION

The purpose of this paper is to provide a systematic analysis of low-energy scattering on the entire real line, taking into account explicitly the possibility of zero-energy resonances of the Hamiltonian Such an analysis has been carried out very recently for three dimensions [1], [2], [13] In one (and two) dimensions, this problem is more involved due to the well-known additional difficulty that the free Green’s function has a square root (logarithmic) singularity in the limit as the energy tends to zero

Different aspects of the one-dimensional scattering problem have received much attention in the past, especially in connection with inverse scattering techniques, which are used extensively in quantum mechanical problems (cf [14], [16], [34] and the references therein), and quantum field theory (cf [17}, [46] for a review) More recently, new rigorous results have appeared ([3], [7], [15], [19], [20], [26 — 30], (33], [26], [38], [39], [41], [47] In particular, one has studied the ground-state properties of one-dimensional Schrédinger operators with various potentials, includ- ing long-range ones, especially in the limit of weak coupling [7], [26], [27], [38], [39], [41] Also bounds for the number of bound states [26], [27], [36] as well as for the imaginary parts of resonances [20] have been obtained Other results are concerned with the limit situation where some negative eigenvalues approach zero as the coupling constant approaches a “‘critical value’’ [29], [30] Furthermore, scaling techniques have been applied to analyse in detail the limit of one-dimensional short-range interactions converging to point interactions [3] We remark that the latter paper contains an extensive list of earlier one-dimensional results which are not explicitly mentioned here

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4 D BOLLE, F GESZTESY and S F § WILK

essentially two cases i.e H has no zero-energy resonances (=- generic case)and H has a zero-energy resonance with multiplicity one This study is based unon and extends some of the results of Klaus [29] and Klaus and Simon [30] in the sense that the class of potentials V can be extended from C#(R) to those satisfying (I~ ix}2)VƠ â LYR)

Section 3 describes in detail the low-energy behavior of the transition operator T(k), assuming roughly exponential fall off for V at infinity (t establishes

recursion relations for the coefficients in the Taylor expansion for 7(k) (generic

case) or the Laurent expansion for 7(k) (other cases) Analogous resuits for the resolvent and the evolution group of general elliptic differential operators have been obtained by Murata [33] (thereby extending the work of Jensen [22], [23] and Jensen and Kaio [24))

In Section 4 we present Taylor expansions for the reflection and transmission coefficients In the generic case, we thus obtain results that are more detailed than the ones available in the literature (cf e.g [14], [16], [34]) For the other cases, the results are new

Section 5 derives two sets of trace relations involving the continuous spectrum, ic negative energy-moments of the trace of the difference between the full and free resoivent, on one side, and the point specirum, i.e negative-energy cound states and zero-energy resonances, on the other side Such trace relations for positive-energy moments were initially introduced by Gelfand and Levitan [{8] (For a list of further references we refer to [Li], [13].) As a special case of these relations we obtain Levinson’s theorem for scattering on the line We find that its structure completely changes in comparison with three dimensions

If one is interested in asymptotic expansions of the scattering parameters instead of analytic ones, Section 6 briefly indicates how the exponential fall off conditions can be relaxed

A brief outline of this analysis has appeared before [10] A similar analysis for two dimensions which is technically more complicated because of the logarithmic nature of the free Green’s function singularity and the possible existence of zero- -energy bound states besides zero-energy resonances, is in preparation {9}

2, ZERO-ENERGY PROPERTIES OF A

in this section we study the one-dimensional Schrédinger operater, allowing the possible occurrence of zero-energy resonances

We define the Schrédinger Hamiltonian H in L?(R) as the form sum

H:- Ay + A0V, Ao € R\ {0}, (2.1)

i - on Ø(H,):: HĐ1XR)

Gx?

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LOW-ENERGY SCATTERING IN ONE DIMENSION 5

Throughout this paper we assume the potential V(x) to be real and satisfy (2.2) \era + Ix!2)IV(x)| < 00, \exven #0

Đ Introducing

(2.3) ex) = (V(x), u(x) = (V(x) P? sign VQ), ue = V, the transition operator 7(k) in L?(R) is defined as

(2.4) T(&) =: ( + 2awRa(k)9)"!— Emk >0, kz0,k?¿>,(H), where R„(&) denotes the free resolvent

(2.5) Rạ(k) = (Hạ — &?) "1, Imk>0

with kernel

(2.6) Rolk, x, ») = (1/2k)ellx=zl,

In order to isolate the first order pole in R,(k) as k—> 0 we follow ref [41] and decompose

(2.7) uR (Ao = (2k)(o,-)u + Mik, Imk = 0, k £0, where M(k) © 4.(L7(R)) for all Imk > 0 If

(2.8) Ề evil ¥(x)| < co for some ø> 0,

R

then Äf(k) is analytic with respect to k in the region Imk > — a/2, and

(2.9) M(k) = Š ik)", »

n=0

where the W, are Hiibert-Schmidt operators with kernels

l2 —_ ylatl

(2.10) M(x, y) = — 27x) = Gay 93 n—0,1,2

T1):

The expansion (2.9) converges in #,-norm Defining

(2.11) P=(v,u)4*(0,.u, Q—1—P,

we thus obtain for 7(k)

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6 D BOLLE, F GESZTESY and S F J WILK

Obviously the low-energy behavior of 7(k) strongly depends on the zero-energy behavior of H As a first step in our analysis of possible zero-energy resonances of H, we state a slightly extended version of Lemma 7.3 of [30]

Lemma 2.1 Let V satisfy condition (2.2) Assume that —1 is an eigenvalue of 2490M,Q and let

Y == {9 € LR) 2.0M,Q9 = — 9},

%ˆ = {yc #'|(0,(Z — 2sMq)"!x) = 0 for some |z) > l|2yMa|} Then

(i) W is independent of z

Git) dim¥#’ =dim¥Y or dim¥W = dimy¥ — 1

(ili) (2 — oP — 15My)~*y = (2 — 46M, )—y for all yEW, cE C (iv) AgMox = — x for all yew

(V) ƒ @ạc 7# then

2gMoy0o = — 0Ø +- (0, 8)~1!2s(, Mgoa)

(v) ƒ xe then (0, — 2sMạ)"1y) = 0 is equivalent to (v, Moz) == 0 Con- sequently

W = {ye V|(v, Moy) = 0}

Proof Let oe C, then for |z| large enough (compared to jo! and J'2sÄfạt) one derives

(z — oP — igMy)-* = (2 — 4oMy)* +

(2.13)

+ 1 — ø(0, (2 — ÂaMạ)~'w)((0, 1) = ( — AsÄMa)~1P(Œ — 2aÄ#q)—,

Since the spectrum of 4,M, is a compact subset of the real line (which follows from (signV) M, = M#(signV) cf e.g [2}, [30]), equality (2.13) extends to all Z¢ L(AyMp) U {z! (v, u) = ơ(e, (2 — 49My)~1u)} Expanding (2.13) with respect to o finally yields [30]

(z — oP — 4My)—* = QŒ — 24QM,O@)~1@ —

(2.14)

— Ø~Ì(®, (z — 2sMq)~1)~*(o, #)®(Z — 2sMạ)~1P(z — 2yMg)~} + O(ø~*) for |o| large enough

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= 0 forall y e W (iii) directly follows from (2.13) Taking the limit ¢ > 00 in (2.14) we obtain from (iii)

s-lim (z — oP — 4,M,)-*y = Q(z — 2¿QMạO)~!1Qy =

ane

= (+ 1)'Qx=G@—AM) x, xe

Noting that Py = 0, Qy = x we get (iv) If gE V NW and no = AyMoGy + we have that On) = 0 and ny = (0,1)~*{o, ạ)u Thus

(X, No) = Ag(v, Moo) + (¥; Po) = Ao(v, MoPo)

which proves (v), (vi) and (i) ⁄

REMARK 2.1 Lemma 2.1 (i) — (iv) coincides with Lemma 7.3 of [30], where it was used in the context of coupling constant thresholds (cf [26], [27], [29], [30], [32], [38], [39], [44])jfor Schrưdinger Hamiltonians in two dimensions (We have repro- duced here a full proof for the convenience of the reader.)

Next we give

LEMMA 2.2 Let V satisfy condition (2.2) Assume that

4QM,09 =— 9, ye LXR),

and define the zero-energy resonance function by

(2.15) UO) =~ (0,1)-"ale, Mop) — 2-*el db'x — y20)90

R Then

G) we LR) and Hw =0 in the sense of distributions

(ii) ¿1GR)

(iil) u(x) = — o(x) ae

(iv) Ứ + (ø, w)~*Ao(v, Mop) — 2-*ysign(-)((-)v, ø) e L2(R), W(+ Co) = — (v, u)~*A,(v, Mo) + 2-7%9((-)v, 9)

(V) w is unique and thus the nonzero eigenvalues of 44QM,Q are simple Proof Obviously we Li(R) From (v, @) = 0 we infer

x W(x) = —(, u)~"o(v, Mop) + 2-"y-

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Taking e.g x > 0 the estimate

ad !

Áo dyỢy — x)e0)@0),<2iÀi

x

° i2

dy yeO) /2Q}, < 212g lol ar? voy)]

.v Qo erm B proves that W(-+ 00) = — (b, M)~12a(p, Äạ@) + 2~*2a((-)p, @)

A similar argument holds for (— oo) Thus ye L&R) Moreover multiplying (2.15) with u(x) yields (iii) by Lemma 2.1 (v), equality (2.10), and the compactness of M, Since by (2.15) w is locally absolutely continuous we get

(2.17) W(x) = is\ dy v()e(y)

This shows that W’ is also locally absolutely continuous Differentiating (2.17) once again we finally obtain

(2.18) Ủˆ(X) =: —- 2aÐ(X)@(x) =' 2,V(X)/(x) ae This proves statement (i) of the lemma

Next, we define

(2.19) Bix) = Wx) + (0,1) “Mol 0, Mg) — 2~12o sign(x)((-)e Ø)

Using equality (2.15) in (2.19) and the fact that ‘x{v,m): O we arrive et (e.g x > 0)

Ủ(x) = — rah ay (y — x) ey”) e()

x

Employing (iii) and we L*(R) we obtain

WX) < Fo Liles 7? \ dy VOY,

and similarly for + < 0 This completes the proof of (iv)

Finally fet us assume that ý € L*°(R) Then, because of (iv), we have (v, Af,g)= = ((-)v, g) = 0, and by (iii) and (2.16) we obtain the equation

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LOW-ENERGY SCATTERING IN ONE DIMENSION Ụ

Iterating (2.20) (this type of Volterra operator is quasinilpotent) yields e.g for x>0O

lứ@)| < [ol dy;lyy — x|V0x) Ai — VO)

x yy

\ dy,lyy — Yyal VOI WO)’ < < |2qI Il lool!) [2 \aronivoni]’ nai

This means that w = 0 such that (ii) is proved Uniqueness of ý and hence (v) foi-

lows in exactly the same way from equality (2.16) Đ

The converse of Lemma 2.2 is contained in

LEMMA 2.3, Let V satisfy condition (2.2) Assume that we L°(R) and Hy = 0 in the sense of distributions Define

(x) = u(x) {( 2\ 02/09) de \ dx'dy’V(x")ix! — vow} ~

R Rề

(2.21)

— 29h ay uQr)|x — y!VYŒ)Ĩ@)

R

then o € LR), 440M,C9 = — ¢ and again (ii) — (v) of Lemma 2.2 hold

Proof Since the norm of the second term on the right-hand-side of (2.21)

is bounded by JiW|8.[|(-)°I;|IV|# +- 3I|V|L|(-)VIỂ] we get pe L2(R) Next we

introduce the function W(x) satisfying

W(x) = — 27-"(v, w)~12s \ dx'dy’V(x')ix’ — "VOY WO”) +

R?

(2.22)

+ 225 dy|x — y]Y(y)@)

R such that

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Since ¥ is locally absolutely continuous we obtain

W'(x) = 2-Mg \ dyV(yQ) — 2-My \ dyÝ0)@0)

and differentiating once again

P(x) = AVOW(X) = W(X) ae

Consequently

(2.23) YX) =v) +exr+d

for some constants c and d From equalities (2.17), (2.18), (2.22) and (2.23), and we L(R) we get

x¬+œ lim x“!W(x) =c= + 24, dy Viv) = 0

R

Moreover a direct calculation shows that Pg = 0 and thus

(, 9) =— \= V@)#@) =0,

R

which proves đ = 0 Thus #(x) = w(x) and after multiplying equality (2.22) with u(x) we get

(2.24) — 0(Y) = #(%)Ú(x) = — (0, w)~}2g(0, Mo)u(x) + 2qCMao@)(3)

Applying Q on both sides of (2.24) (observing Ou = 0, Og = ¢) finally yields

—9 = 1,.0M,Q¢9

From here one can follow the proof of Lemma 2.2 A

REMARK 2.2 Different proofs of most of the results of Lemmas 2.2 and 2.3 under the assumption V € Cf°(R) have appeared in [29] (cf also [16])

With the help of Lemmas 2.1 —2.3 we are able to distinguish the following cases in the zero-energy behavior of H If the potential V obeys condition (2.2), then we have

Case I — 1 is not an eigenvalue of 4,QM,Q (i.e H has no zero-energy resonance)

Case iI — | is a simple eigenvalue of 4,0M,Q, 7,>0M,Q9) = —@, for some @o € L*(R) (i.e HW has a zero-energy resonance) and

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LOW-ENERGY SCATTERING IN ONE DIMENSION il or b) ec, ¥0, cg =0 or c) ¢, #0, ce #0 where

(2.25) Cy = (v, u)~“"(v, My Go), Cg = 2-((-)0, Go)

Note that in the Cases II a) —c) we have (v, go) = 0 Furthermore, in these cases there exists precisely one resonance function w, ¢ L°(R) (up to multiplicative constants) given by equality (2.15) Since H has no zero-energy bound states (or equivalently (v, Mj@o) and ((-)v, @ 9) do not vanish simultaneously) and nonzero eigenvalues of A,QM,Q are simple, the above list of cases is complete It is trivial to realize all Cases I, II a) — cc) in the example of an asymmetric square well

3 LOW-ENERGY BEHAVIOR OF 7(k)-RECURSION RELATIONS

We discuss in detail the low-energy behavior of the transition operator 7(k) for the different cases presented in Section 2 In particular we establish recursion relations for the coefficients in its Laurent series around k = 0

We start with

Lemma 3.1 Let e€ C\{0} small enough Then the norm convergent expansion

G.1) (1+ A/0M0 + 3)! = =2 +S (— oT m=0

holds Here P, denotes the projection onto the (at most one-dimensional) eigenspace of 2,QM)Q to the eigenvalue — 1

Py = 0 in Case I,

Po == (Pos Po) (Po>-)Po in Cases II a) — c), where

AoOMaQøo = —ọa, Po € LR), Polx) = sign Vx)po(x) Ty denotes the corresponding reduced resolvent viz

(3.2) Ty = n-lim (1 + 144QM,2 + £)-'O,, Og=1—Py

670

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12 D BOLLÉ, F GESZTESY and S$ F ) WILK Assume (4,.0M,0 + 1)*¢ =0 and deũne ƒ = (2¿@QMQỢ + l)s Then (2a4Ø@Mfa¿O 1)f = 0 and consequently

(, f) = (4¿0*M‡O* + lẽ, (220MạO + 1)g) =:

= ( (2¿O@M¿O + 1)°g) = 0, where

f= (AoQ*MQ* + Dg, g = (sign Vg

Furthermore

0 = —Œ, uOMạOƒ) = — (đ 22QuHg 10O[) =:

ters

= — }(Ho 'uQ*f, Ho vOf) = — to He Of

implies vOf = 0 and hence f = 0 (since f = —/,QuHs 1vOf) Thus the eigenvalue t of 2490M,Q has algebraic and geometric multiplicity equal to one r2

Next we collect some relations which turn out to be useful in the sequel: PP, == PoP 0, OP =: PạO = Pạ,

PQ, = QAP = P, QQ = HO == Qo — P= O- Py,

Py -+ P+ QQ, 1,

as toa C2 sa?

OT) = TO =T)— P, PTạ = T,P= P, PTạQ: 0,

AgPoMoPo = — Po 4oPoMoQo = (Qo; Po) cE (U.P s

12

(Po, Mi@o) == 2 ]ea]® We then prove

THEOREM 3.1 Assume (v,u) 4 0 and e Ve LYR) for some a> 0 Then Ftk} has the following norm convergent Laurent (resp Taylor) expansion aruunad

3.4) Tk) = Š' (IM}', Hàng

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Proof Case 1 From equalities (2.4) and (2.7) we get T(kK) = [1 + GAp(v, /2k)P + 2gM(k)]~! =

_ [; de [ -— 1As(®,w)P }o -t [ —_— 1Ãa(b, )P —

| 2k + lÀs(b, 1) 2k ~+ lÀa(0, 1)

=[l + 2Ø@Mẹ + O(]~![@ + O(9]

This proves analyticity of 7(k) around k == 0 since

(1 + 29QM,)~* = 1 — AL + 22@Ma@)7'10M¿

exists

Cases {1 a) — c) Take e € C\{0} small enough Then we have

(1+ 29 QMy + 6)“ = (1 + AgQM0Q + Ul + Ag OM aPC + 499M 0O + 8)-]? =

= l2 + ` (— Te | I — Tnhh

& m=0 l+e

where we have used Lemma 3.1 and the relation

(: + Tin] = ụ — 6)

l+s

After some straightforward calculations using some of the equalities in (3.3) this leads to

(3.5) (1+ OM, + a)-t = —0PoMo, ®t O(1)

Next, we consider the operator 7(k) (see equality (2.4)) which can be written as

Tứ) = {[H + (12, w)/2k) PIL + CL + (2s, 0)/2k)P)”1AsM(k)]}~" =

_ | 1+ 20M9 + 2kAyPM(k) Ị 0 2kP —

H

2k — 12g(b, 1) 2k + idg(v, u) }

Afier some manipulations we get by expanding M(k) (see equality (2.9)) to order k?

[ 2ik 2ikM,

| + ApQM gy — mm “ iy too 12) \ 4

tt QM cw) [im OM ( y + O69 Ì | x

«(Ue 0m 2? (9 2H) Ao(v, U) 2a(Đ, 0)

TK) =

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or using equality (3.5) with e == —2ik(A9(v, u))~1 we arrive at

mm —1 1 ue -1

T(k) ma { + | + (Phu, | 00) l + (B Po» )Po | *

(Pos Po) (Qos Po)

(3.6) 2ã(0, t) 2ik x | 2 pM, + OQ) —- “=—|, aig Me FOF | (2 2g(b, ) } where B= 22%(v, u) i — MoPMy _ ae | 2 (t, u) (v, u)

Finally, we calculate the inverse appearing in equality (3.6) Employing

3.7) Bu Py 0 ee + tal, (Gos Po) (Pa, Po)

we get

(B*@o,+)Yo F 1 a

j + BG Wo] yp BG, Jay

(Go, Po) (||? + |ea|®)2ã, 2) ° °

Inserting this into equality (3.6) and calculating the terms up to O(1) we obtain

T(k) = [2ikAg(\e1|® + eel? X@o 5+) + OU),

where we have used again equality (3.7) and the relation 7,P)M)Q = —Pp (see equality (3.3))

This completes the proof of Theorem 3.1 GZ

Assuming the potential condition (2.8) and of course (v, uv) # 0 throughout the rest of this section we now derive a systematic way to calculate the coefficients t, in the Laurent series (3.4) for T(k)

We start from the integral equation satisfied by T(A), viz

j

(3.8) T(k) = 1 — Ag(i(v, u)/2k)P + M(K)IT(K)

Following [13] by defining

(3.9) P(k) = PoT(k), Q(k) = QoT(k), Imk > —a/2,k #0

equality (3.8) leads to the following set of coupled equations

21k2s(Øo, @ạ)~ˆ|cal3⁄P(k) = Py — 2g(Øo, Øạ)~1c‡(Q(k)*e, -}@ạ — (3.10)

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and

(3.11) OK) = Q — (iAg/2k)(Q(K)* 0, «uu — 29 QoM(k)P(K) — 2oQ.M(K)Q(A) where

(3.12) MK) = ¥iK"M,, j= 12,

n=j Rewriting equality (3.11) using

[1 + (iAg/2k) (v, «Ju + 29 QMyQ + s]~” = = £~!Pạ + Tạ{I + [Ao(v, u)/(2ik — Ag(v, u))}P} + Of),

one obtains

O(k) == Tyf{1 + [Ag(v, )/(2ik ~ Ag(v, #))]}P} [Òa — ^QsM%(k)P(k) — (3.13) — 2sOsM¿POŒ) — 2,0yM®S2)Q()] —

— [2lk/Q1k — 24, 1))][(0a Po) *eyAg(P(K)* Go, + Ju + 4¿PMsoo@)I

Inserting (3.13) into the second term on the right-hand-side of (3.10) leads to, after some calculations,

PŒ) = (2aIk)~'(0a, @q)e[Pa — AoPoMO(K)P(k) — ApPpM(K)Q(K)] +

(3.14) + [2cc#/(Ag(v, u) — 2i)\({1 — 2uMŒ(&)P(k)— 2sMŒ)Q(k)Y*9,-)@y—

— [4 ik cle,|?/(Ag(v, u) — 2ik)| P(x),

where

(3.15) c= [2(le,? + le|?)]}-*

Equalities (3.13) and (3.14) may then be rewritten as

(3.16) P(k) = Polk) — Py(k)P(k) — PAK) O(K),

and

(3.17) O(k) = Alk) — Qi(K)P(kK) — OAA)OK)

where the explicit expressions for Po(k), , Qo(k) can be easily read off from equa-

lities (3.13) and (3.14)

Next, from Theorem 3.1 we infer the existence of the norm convergent expansions

co oO

(3.18) PŒ)= Yo CK)", OK) = FUN"

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In order to calculate the coefficients p,, g, in all Cases I, II a) c) we expand

(3.19) Polk) = Š (PA, eo 1 (3.20) Puk) = Ÿ) (iky"B,, Fool (3.21) Pak) = Š (9Ð, „=0

and insert these expansions (3.18)—(3.21) into equality (3.16) We get

p-4 = A-1,

(3.22)

Pa = ~ » B, 141P 1-1 — Ÿ Đa ấn: H2 0,

0

where

AL; = 28 *€(Øa, - 0a

(3.23)

4; =: ccŸ[2/2u(6, 8)|"?!(6.:)0ạ, n >0,

B, = cic, |7[2/Ao(v, u))"Po => c(0a; M,,+190)Po ~r

(3.24)

+ Age? > {2/2a(b, 0} !(b, Äf,¿a@as)Pạ, n3 1,

Re

(3.25) D,= (Qos Po)ePoMy 41 Oo rect 2 [2/zote, wrt “1({M Qy}* ‘ty )@y, 1 >0

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LOW-ENERGY SCATTERING JN ONE DIMENSION 17

we obtain from equality (3.17)

n

~ K n

(3.29) = F,— ae -141Pi-1 ~ bee

where we do not need to write down at this moment the known explicit expressions for the F,, K, and L,, The reason is that since g, occurs also on the right-hand-side of (3.29) we have to bring the term Lg, = AoToQMy Pq, to the left and invert (1 + 49%) QM,P) Doing this we finally obtain from (3.29)

qo = Fy — Kyp-1, (3.30) n —1 “À* n1+1Pi—1 — ST Lys n>], I=0 ¡=0 where + = ToQ, (3.31) Tn = —[2/Ag(v, z)]*q — ^*gTqOM,)P, n > 1,

Kì = ÀsToOM,LPạ — 2( Qo; Go) ~*(0, #)—ˆey(Øạ, ©)[Í — Ao TQM lu,

(3.32) K,= ÂqToOM,P\ — Ag(@o» Po) [2/Ao(v, 0)]"€i(@o, -)[Í — ÂyTe@Mg]u —

a-1

— ro ` [2/2a(, 8)|—!( — AsTaOMQ)PM,PQ, n >2,

f=1

Ly = AgT>QMoQ »

n—1

(3.33) Ly = Ao ToOM,Qo — Ao ¥; [2/Ao(2, 1)Ì"~!(1 — AyToQM,)PM,Q,), n > 1 [=0

Equalities (3.12) and (3.30) represent the final result They allow us to compute all p, and g, recursively Below we list a few coefficients explicitly and state certain matrix elements needed for later purposes:

Case I

(3.34) ?y=0, n> —],

@.35) b= = TQ,

== —AgT)OM,T,O — [2/Ag(v, WILP — ApT>QMyP — ApPMyT,Q + (3.36)

+ AGT)OMoPMoT QI,

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18 D BOLLE, F GESZTESY and S F J WILK (3.37) (v, fot) = (0, Jott) = 0,

(3.38) (v, tu) == (tv, qu) = — 2/24,

(v, to) = (v, Gott) = — [2/Zo(v, u)P{(v, CL + ÀaÄMfq)u) —

(3.39)

— Ag(v, MyT>QMqu)] Cases II a) — c)

(3.40) Das = C10 os or

đo = TạQ — c€z(0ạ,-)Tạ(-)M + (e, 0)~ces((-)0, w) (0a, -)M +

(3.41)

+ [2/2a, #)Ìee(0a,-) (Ì ~— 27aOM,)u, (3.42) PoP ~= 225 "(0, w)~1ecŸ(b, -S0a

(3.43) pnu =0, n> — Ì, (3.44) (v, t_yu) == (v, p-yu) = 0,

(3.45) (v, to) == (v, dou) = 0,

(3.46) (v, td) = (v, qu) = — 4/9 lejcol?

4 S-MATRIX, REFLECTION AND TRANSMISSION COEFFICIENTS

in this section we apply the preceding results to get low-energy expansions for the on-shell scattering matrix on the line and thus for the reflection and transmission coefficients Throughout we assume condition (2.8) and (v, v) # 0 on V

Combining Jost functions techniques (cf e.g [34]) with Fredholm methods (cf e.g equality (2.24)), one arrives at the following expression for the one-dimensional on-shell scattering amplitude

(4.1) feye,(k) = (21k)~}2s(®ÿ (6y, k), T(k)®g (s;, k)), Imk > — a2, k 40,

where

t; = 3: ], j= 1,2,

and

Trang 17

LOW-ENERGY SCATTERING IN ONE DIMENSION 19 The on-shell scattering matrix is then given by (cf e.g [16], [34])

(4.3) Seek) = bee, +fee(k), Imk > —a/2, k #0, or equivalently,

(4.4) 5 = (ie ml Imk > —a]2, Rk) TK) k #0

Here the matrix elements

(4.5) S44(k) = Tk) = Tk) = S (h)

denote the transmission coefficients and

(4.6) S_.(k) = R(), Si -(k) = RA)

represent the reflection coefficients for left and right incidence respectively Clearly S(k) is analytic in Imk > —a/2, k # 0

Employing the low-energy expansion (3.4) for T(k) in equality (4.1) we arrive

at

THEOREM 4.1 Assume (v, u) # 0 and e@'1V € L(R) for some a > 0 Then the on-shell scattering matrix S(k) is analytic with respect to k in Imk > —a/2 In particular we have the following Taylor expansion in k around k = 0

(4.7) Sie) = Ÿ (5U),

n=0

where

nt+1-—q nt+l-q-l

(4.8) 52, = Sede, + 2a YS (— aM mY ACI tana ten I")

with q = 0 in Case I and q = —1 in Cases Il a) — c) Moreover, the leading coefficients in Case I read

(4.9) sae, = = dee, —_ 1,

se 2 = 2A Mv, u)-[(v, u) + Ao(v, Mou) -+Ag(v, u)-"(v, Mou)? —Ag(v, MoT>Myu)}+ (4.10) + (uv, u)~*(e, — e9)[(-)v, wu) — Ag((-)v, ToMgu)+ Ag(v, u)-7*((- Ju, uv, Myu)]—

Trang 18

and in Cases Il a) — c)

0

(4.11) See, = Oee, — | + [ái — eychee + enced — &y89!¢9|*)/fle!? + 1c!)

where T, and M, are given by equalities (3.2) and (2.10) respectively, and the c,, C2 are given by equality (2.25)

Proof Because of equalities (3.4) and (4.1) we know that See Ak) has a Laurent expansion of the type

(4.12) f#„(@ = Š (9/0

nes —14-q

The coefficients fee, ” , are found by expanding 7(k) and đặ(z,, k) in & The result is

t+t1l-q nn 1l-q-l

(4.13) fae =2! ` py (—ei)'@;)"Œ! m)~*((-)0, fx++~¡~ mÁ* )”E)

In Case I (Le g = 0), insertion of expressions (3.34)—(3.39) in equality (4.13) 1mmediately gives (4.9) —(4.10) Similarly, in Case II (1e g =— l), inserting expressions (3.40) — (3.46) in equality (4.13) leads directly to equality (4.11) Y So, up to O(k?), the transmission and reflection coefficients in Case I are given by

T14) = T'),= i39} + O(#),

(4.14) ROS 1 + iks®, + O(Kk%), RO oI + i&sÐ) + O(?),

where the Stee can be read off from equality (4 10) These results are more detailed than the ones available in the literature (cf e.g [14], [16], [34]) In Cases II a) — c)

the transmission and reflection coefficients are °

leal® — leal® , '(& ————— + O#*), 7= 74) Thác T00) 2c¡cŸ _ FO eit lee | = 2c¡c% R&) = — — ¬+- O(#) = |cal? + |eal? “

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5 TRACE RELATIONS

This section investigates the low-energy behavior of the trace of the difference between the full and the free resolvent This is then used to derive so called trace relations (or sum rules) involving moments of the phase shift derivative (or time delay) on one side, and bound state and zero-energy resonance contributions on the other side Such (positive moment) trace relations were initially introduced by Gelfand and Levitan [18] For a list of further references we refer to [11]—[13] Here we are interested in proving the zero-th and negative moment relations allowing the possible occurrence of a zero-energy resonance As a special case we obtain Levinson’s theorem for scattering on the line

In order to be able to apply the results of Section 3 we assume condition (2.8) and (ø, ) # 0 on the potential First we discuss

LEMMA 5.1 Jn all Cases I, II a) — c), Tr[R(kK) — R,(k)] has the following Laurent expansion in k around k =90

Cc (5.1) Tr{R(K) — RK = YY Gk)", ti>q—2 where , A,-2 = 4-12,(v, ty +14), (5.2) n+1-—@ A, = 47 Ag(¥, ths) + 2-1, ` (n+2—1— q)TT[ỆM,¿s-i-ai+ál 2 2q—1 ¡=0

with q == 0 in Case I and q = —1 in Cases Wa) — c) Proof Define

(5.3) G(k) = kuR,(k)v

Then, by mimicking the proof of Proposition, 5.6 in [43] one gets

(5.4) IG@)|k < eax eP(—Im 4) Hm ALCL + |x|**9)| V(x)| < 00,

R

for Ink > ~-a/2 and some c, 6 > 0 (@(s) denotes the step function: @(s) = 1, s > 0; 6(s) = 0, s < 0) Similarly, defining G’(k) to be the operator with kernel

(5.5) G'(k, x, y) = — 2~(x)|x — y| e#*=>io(y)

one obtains

IG'@lh < cl ax e2X-Im iim All + |x)>*9)|V(x)] < 00,

R

(5.6)

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Applying this method of proof once again one finally arrives at

6.7) 'GO—G) _ GK)! <ck — KIỆM 2% ImBitmaisi (I 41x34) V(x) <o0,

| k—ky ul

R

c,6 >0, Imk > — a/2, Imk, > — a/2

for some k, Imk > —a/2 depending on k and ky (The estimate (5.7) also proves that wR,(k)v is analytic in trace norm around any k # 0, Imk > — a/2 and has a Laurent expansion around & = 0 convergent in trace norm.) Consequently

Tr[R(k) — Ry(K)] == — Ay Tr R,(K)oT(QuR,(K)] = — Jy TrluR(K)vT(K)] =

(5.8) =— 2/08971T | ~ấ_ 8/99] ru} _

=: 4-1, (1k)-%(v, T(k)u) — 2-Mok-! Tr IR Mi] rel, Im k> —a/2, k £0,

where we have used the cyclic properties of the trace and equality (2.7) Insertion of equalities (2.9) and (3.4) into (5.8) yields, after interchanging }) and Tr (which is allowed by the above arguments since k Ea mee | T(k) is analytic around k = 0 in trace norm)

THRŒ) — Rạ(k)] = 4-14 Ÿ (0œ, øys) + n=q—-3

c ntl-@q

$ 2-H GO EU Am ::q—1 I=0

fork # 0, |k| small enough Since (0, f,4) == 0 by equalities (3.37) respectively (3.44),

‹5.1) follows Z

Next we establish the high-energy behavior of Tr[R(&) — Ro()]

Lemma 5.2 fa all Cases 1, If a) — c) there exists a ky > 0 and ac>QOQ (depending on ky) such that

(5.9) RE) — Roll < elki-2, [kl > ky > 0, Imk > ~ a/2

Proof From the estimate

Trang 21

we obtain

(5.11) |} TA) = || + ApuRy(A)v)-1|| < const, |k] & ky > 0, Imk > — a/2 On the other hand equalities (5.4) and (5.6) imply that

12) JIzR(&)0||¡—=(|k|2~1||G')—k~1ŒG(& || <e|k|~®, — |k| >kạ>0, Imk>—aj2 Thus equality (5.9) follows from

| RA) — RolAdll, < [ol IT) I leRool, & # 0, Imk > — a/2 N

Lemma 5.3 In all Cases I, Il a) — c) there exists a kj > 0 and ac>O (depending on ky) such that

(5.13) [Tr [R(k) — R,(k)]| < elk|-8, [k; > ky > 0, IMk > O Proof Equality (3.8) implies

(5.14) Tr[R(A) — Ro(K)] = — ApTr[uRR(A)v) + ART r[uR3(k)ouR, (k) eT (k)]

By equalities (5.10) and (5.12) the second term on the right-hand-side of (5.14) is O(|k|-5) as |k|~> co, Imk 2B 0 The first term can be treated as follows Writing uRf(k)u = (2k3)~1GŒ(k) — (2k?)~1G'(k) (cf equalities (5.3) and (5.5)) we get

Tr[u Rấ(k)ø] = (1/4k3) dxf(x) — (1/2k??Tr[G'(k)] =

(5.15)

= (/4k°) Jarre) k #0, Imk>0

R

since for Imk > 0

le Ro(AV|[E = |LR¿(Œ&)»|lễ = (1/4|k|?Im &)||W||i < œ

Trace norm continuity of uRg(k)v with respect to k in Imk > 0 (cf the proof of Lemma 5.1) then proves equality (5.15) for all Imk => 0

For asymptotic expansions of R(k) as |k|—> co we refer to [50] and references therein

Trang 22

Lemma 5.4 Define the phase shift d5(k) by

(5.16) đet S(k) = e?2Œ, k > 0

wift lim Š(k) -= 0 Then ð(k) is continuously differentiable in k > 0 and

k¬œ

(5.17) Im Tr[RŒ&) — Rạ(#)] == — (i/4k)Tr[S*Œ)S'Œ&)] = (1J2k)ð'&), k > 0

Proof Since R(k) — Ro(k) is trace class, Krein’s theorem [31] (cf also [6],

[21], [45]) implies

&(p)

where € is real and (1 + }-%ée L'(R) Secondly, writing S(A) = e!4®, it also implies that

(5.19) Tr A(k) = — né(k) = 5(K), &k > 0

Then, from equalities (4.1), (4.3) and (5.11) we get the high-energy estimates (5.20) idet SQ) — 1] < ck-1, Jd) < ck, kek, >O0

for some ¢c, c’ > 0 depending on ky Next, by hypothesis (2.8) the đợ(e,,k) are infinitely many times strongly differentiable with respect to k Furthermore, since uR,(k)v, Imk > — a/2, k # 0 is analytic in trace norm (cf the proof of Lemma 5.1), T(k) and, using (4.1), also S(k) are norm analytic in k >0 Thus 6(k) is C? in k > 0 Moreover, an estimate similar to (5.20) shows that

d 2

Lá ts +}9,< O(k-*) and hence

(5.21) lỗ Œ)J = 2-3 (5 ees} <c'k-*, k>ek,>O

Next, integration -by parts in the integral from 0 to oo in (5.18) using (5.20) and (5.21) yields

0

é(?Xp° — k*)

lim Im Tr[R(&k + 1£) — Rạ(& -+ iz)] = — lim 4e dp

Him Im Tok + i) — Ro(k + i] = — lim \ pip Tế

Trang 23

Dominated convergence in the first term on the right-hand-side of (5.22) (observing k > 0) and a standard 6-function computation ({40], p 128, using the fact that (2p)-!Tr A’(p) is continuous and bounded in a small k-dependent neighbor-

hood of p =k > 0) then proves equality (5.17) Z

Given the above results we can now state the main theorems of this section THEOREM 5.1 Assume (v, u) # 0 and el |V e L(R) for some a> 0 Then the following trace relations (sum rules) hold

nea —2 2\ dk k-2N+1 fim Tr[R(k) — R,(k)] — Š (— 1ÿk#+1 4a] = (5.23) N, =— 7Ø W (—z)7N + m(—l)Y~!A4,y.ạ, N=0,1,2, jel

where — xj, j= 1, ,.N, denote the (negative and simple) eigenvalues of H, and the A,, are defined by equality (5.2)

Proof Following [8], [37], we introduce the function

(5.24) Fy(k) = 2k-2N+1 {mre® — Rạ(K)] — “SGA, } N=0,l,

n:¬g—2

where g = 0 in Case I and g =—1 in Cases II a) — c) Clearly Fy(k) is analytic in the open upper k-half plane with possible poles on the positive imaginary axis Since (1 + |-{)V € L1(R) these poles are finite in number [26], [34], [42] and non-de- generate (e.g by well-known Volterra integral equation arguments similar to that in the proof of Lemma 2.2) Moreover H has no nonnegative eigenvalues and purely absolutely continuous spectrum on [0, co) [48], [49]

In order to derive (5.23) we apply contour integration techniques and inte- grate Fy(k) along the following paths: From — R + ie to R + ie avoiding the origin by a semi-circle C, , = {ne” | @€ [x — arcsin(e/y), arcsin(/n)]}, along the semi- -circle Cz , == {Re'’ | 0 ¢ [arcsin(e/R), x — arcsin(e/R)]}, and finally encircling all bound state energy positions ix, clockwise C,, = tre /|0;e [0, 2z], r; suffci- ently small},1 <j < N,

We then study the different contributions From C R,e We get by Lemma 5.3

Row end R00

(5.25) “oe n=q-2

lim lim \ dk Fy(k) = — Jim ail aocre9- aN+2 y ” GRei”y'4, = =

Reg 9

Trang 24

Since R(k) has a first-order pole at k = ix, we get from C,,

Ny Ấy

(5.26) ` ? dk Fy(k) = 271 ` (—z?)-*

= ° j=1

To get the contribution from C,, , we insert (5.1) into (5.24) to obtain

770 0, Là n=2N-—1

(5.27) lim lim dk Fy(k) = lim 2i\ d0(nei%)-24 +2 Š (ine')"A,„ = 0

Ihe ”

Finally we calculate the contribution form C,, p= {k + ie | —R<k< y or

n<k< R}:

lim lim lim \ dk Fy(k) =

Roc) , £0, Ce, 4, R R

= lim lim ima R242 TLC + ie) — Ry(k + i2)] — R00 770, 640,

1

(5.28)

+ R

= lim lim 4i ae fim TRC) — Rạ()] — R¬œ 730, 1 2N-2 ~'¥ fie + ionray) n=q~2 — s TH, =Ái dkk-9**! [ImT4R4) — Rạ(k)] — Meso N-2 ~F Cua}, mn=:—2

where we have used dominated convergence, the low-energy behavior of Tr[R(&)— — R,(k)] in (5.1) and the high-energy bound (5.13) Noting then that

Ny

\ ak Fy) =o, C= Cav {UCr, | UCU Cane

c „1

Trang 25

The sum rule for N =O in (5.23) corresponds to Levinson’s theorem for scattering on the line:

COROLLARY Š.l Let (v, u) 4 0 and assume e4|:|\V € L1(R) for some a > 0 Then (5.29) 6(0,) = x(M, + 4.;), 4(00) = 0

where

—1/2 in Case I,

A_» =

0 in Cases ILa) — c)

‘Proof Insertion of (5.17) into the left-hand-side of (5.23) and taking N = 0 immediately yields (5.29) By equalities (5.2) and (3.38) we obtain in Case I

A_, = 471A,(v, tu) = — 1/2

In Cases II we employ equalities (5.2), (3.46) and the relation (go, M1») = 2)c2!* {see equalities (3.3)) to get

As = 4-1},(v, tu) + 2-1 Ag Tr[M,t_4] =

= —cleq|* + clea|? = 0 | Z

The structure of Levinson’s theorem for scattering on the line is completely different from its analogue in three dimensions Indeed, when there is no zero-energy resonance present (Case I), we not only get on the right hand side of (5.29) the term proportional to the number of negative-energy bound states, zN,, but an additional factor — n/2 appears In the case of a zero-energy resonance (Cases JI), we simply obtain the term 2N, This difference is of course due to the additional Diri- chlet boundary conditions at the origin when considering Schrédinger operators on the half line A one dimensional Levinson’s theorem has been studied recently for scattering by a local impurity in a periodic potential [35] and in the context of (1-[-1)-dimensional field theoretical models ({5] and references cited therein)

Our second main result of this section reads

THEOREM 5.2 Assume (v,u) #0 and el:|Ve LR) for some a> 0 Then

ne=—TL

2\ ack {Re THR) — RO — “Fe (HK D# 4u) =

(5.30)

N

= in Y (— )-M-¥2 + n(—1)@Agy-1, M=0,1,2, ,

Trang 26

28 D BOLLE, F GESZTESY and S, F J WILK

where —xi,j = 1, .,.N,, denote the eigenvalues of H, and the A, are defined by equality (5.2)

Proof Analogous to that of Theorem 5.1 after replacing Fy(k) by

(5.31) Gy(k) = 2k-2! {riRe) — Rạ(k)] — " (⁄4,} M=0,1,2,

1:-g—5

Combining Corollary 5.1 and Theorem 4.1 we obtain the following low-energy behavior of the phase shift in Case I

Ok), = MN, 1/2) + {245 (v, we) 7[(v, w)t-Ag(v, Mou) + Ag(v, )~ (0, Maw}° —

(5.32)

— Ag(v, MoToMou)] 2-*Ao((-)v, To(-)u) + 2-1g(v, w)~1((-)o, u)?}k + OCR)

For the Cases If a) — c) we get

(5.33) 5(k) = 2N, + O(k),

where N, denotes the number of (negative and simple) eigenvalues of H

6 POSSIBLE GENERALIZATIONS

1f one is interested in asymptotic expansions (instead of Laurent expansions) of the various quantities discussed before, Condition (2.8) on the potential, which roughly implies exponential fall off at infinity, can be relaxed considerably In particular we shall now briefly indicate how the condition (1 + |-|?)Ve L*(R) for sui- table p > 2 (depending on the order of the asymptotic expansions involved) can be shown to suffice to derive most of the results of this paper

Assume (v,u) #0 and

(6.1) | exc + |x|?"+?")|V(x)| < co for some ¢ > 0, m = 1,2,3,

R

then, by dominated convergence, M(k) has the asymptotic expansion

m1

Trang 27

which is valid in Hitbert-Schmidt norm (i.e lim k1-*-™{jo(k™+®-J)j|, = 0) Simi-

k-»0

larly T(k) has the asymptotic expansion

m-1

T (k) = ` (ik)"t,, + o(k"+*-1), m > lin Case I, and

"0

¬0, (6.3)

m-3 „

+ n=—1

which are valid in norm

Given the expansion (6.3) one infers e.g

m—2

— tưn cứ mte-

SN x (ik)"see, + O(K”“**~?), mm >2 in Case I, and

(6.4)

— Sh Ĩ nạ) e— _

See, OOS, x (ik) Sees + o(k"+°-4), om 2B 4 in Cases If a) — c)

Analogously equalities (4.14) (with O(&?) replaced by o(k)) are true if condition (6.1) with m = 3 holds In the same way equalities (4.15) (with O(k) replaced by o(1)) are valid with m == 4 in (6.1) In addition for k # 0, |k| small enough

m4

THR(K) — RK, == YL (ky, + o(k"**-4), mm > 2 in Case I, and

+0, n—=9

(6.5)

mà—6

Tr[R(&k) — R,(k)] S ` (iky’A, + o(k™+*-*%), m > 3 in Cases La) —c)

T=:—8

Finally, in order to indicate that the trace relations (5.23) and (5.30) hold under considerably weaker assumptions on V it suffices to discuss Levinson’s theorem i.e N = 0in (5.23) Following the proof of Theorem 5.1 step by step shows that in Case I

6(0,) = 2(N, — 1/2), ð(ce) =0

if we take m == 2 in (6.1) Similarly, in Cases IT a) — c), one gets ð(0,) = zW,, ð(co) =0

by taking m = 4 in (6.1) (It is reasonable to expect that these conditions on V in the case of Levinson’s theorem may be improved.)

Trang 28

gratefully acknowledges the hospitality extended to him by the Zentrum fiir interdisziplinare Forschung der Unversitắt Bielefeld and the financial support by the Alexander von Humboldt Foundation

The first author is a Onderzoeksleider N.F.W.O., Belgium The second author is an Alexander

von Humboldt research fellow

REFERENCES

1 ALBEVERIO, S.; BOLLÉ, D.; GESzTEsy, F.; HOEGH-KROHN, R.; STREIT, L., Low-energy para~ meters in nonrelativistic scattering theory, Aun Physics, 148 (1983), p 308 ALBEVERIO, S.; GESzTESY, F.; HGEGH-Kroun, R., The low-energy expansion in noaretati-

vistic scattering theory, Ann Inst H Poincaré A, 37(1982), p 1

3 ALpeverio, S.; Geszresy, F.; HOEGH-KRouN, R.; KirscH, W., On point interactions in one

dimension, J Operator Theory, 12(1984), 101- -126

4 ALBEVERIO, S.; GESZTESY, F.; HOEGH-KROHN, R.; StreIT, L., Charged particles with short range interactions, Aan Inst H Poincaré A, 38(1983), p 263

5 BrRG, B., S-matrix theory of the massive Thirring model, Lecture Notes in Physics, 126

(1980), p 316

6 BIRMAN, M S.; KRLÏN,M.G., On the theory of wave operators and scattering operators,

Soviet Math Dokl., 3(1962), p 740

7 BLANKENBECLER, R.; GOLDBERGER, M L.; Simon, B., The bound states of weakly coupled long-range one-dimensional quantum Hamiltonians, Anu Plysics, 108(1977), p 69 8 BorLé, D., Higher-order Levinson°s theorems and the high-temperature expansion of the

partition function, Ann Physics, 121(1979), p 131

9 Botié, D.; DANNEELS, C.; GeszTesy, F.; WILK, S F J., in preparation

10 BoLLé, D.; GESzTEsy, F.; Wik, S F J., New results for scattering on the line, Phys Lett As

97(1983), p 30

11 BoLLt, D.; Ossporn, T A., Spectral sum rules for the three-body problem, Phys Rev A,

26(1982), p 3062

12 BoLLE, D.; Smeesters, H., Trace formulas for time delay and the second virial coefficient, Piys

Lett, A, 62(1977), p 290

13 EoLLé, D.; Wnx, S F J., Low-energy sum rules for two-particle scattering, J Math Phys., 24(1983), p 1555

14 CHADAN, K.; SABATIER, P C., Jnverse problems in quantum scattering theory, Springer, New York, 1977

15 Davies, E B., Hypercontractive and related bounds for double well Schrédinger namilto- nians, preprint King’s College, London

16 Deirr, P.; TRUBOWITZ, E., Inverse scattering on the line, Comm Pure Appl Math., 32(1979), p 121

17 FADpEEv, L D., Quantum completely integrable models of field theory, Sov Sci Rev Cl, 1980, p 107

18 GELFAND, I M.; Levitan, B M., A simple identity for the eigenvalues of a second-order differential equation, Dokl Akad Nauk SSSR, 88(1953), p 4

19 HarreLt, E M., Double wells, Comm Math Phys., 75(1980), p 239

20 HarReLL, E M., General lower bounds for resonances in one dimension, Com Math Phys, , 86(1982), p 221

Trang 29

LOW-ENERGY SCATTERING IN ONE DIMENSION 31 21 22 23 24 25 26 27 28 29 30 31 32 33 36 37 38 39 40 41 42 43 44 45 46 47

Jaucu, J M.; SINHA, K B.; Misra, B N., Time delay in scattering processes, Helv Phys

Acta, 45(1972), p 398

JENSEN, A., Spectral properties of Schrédinger operators and time-decay of the wave fun-

ctions Results in Z°(R”), m > 5, Duke Math J., 47(1980), p 57

JENSEN, A., Spectral properties of Schrédinger operators and time-decay of the wave func-

tions Results in £°(R*), preprint, Univ of Kentucky

JENSEN, A.; KaTo, T., Spectral properties of Schrédinger operators and time-decay of the wave functions, Duke Math J., 46(1979), p 583

Kato, T., Perturbation theory for linear operators, Springer, New York, 1980

Kaus, M., On the bound state of Schrédinger operators in one dimension, Ann Physics,

108(1977), p 288

Kaus, M., A remark about weakly coupled one-dimensional Schrodinger operators, Helv

Phys Acta, 52(1979), p 223

Kaus, M., Some remarks on double-wells in one and three dimensions, Amn Inst H Poincaré A, 34(1981), p 405

Kraus, M., Some applications of the Birman-Schwinger principle, Helv Phys Acta, 55(1982),

p 49

Ktaus, M.; Simon, B., Coupling constant thresholds in nonrelativistic quantum mechanics I: Short-range two-body case, Ann Physics, 130(1980), p 251

Krein, M G., Perturbation determinants and a formula for the traces of unitary and

self-adjoint operators, Soviet Math Dokl., 3(1962), p 740

LAKAEV, S N., Discrete spectrum and resonances of a one-dimensional Schrédinger oper-

ator for small values of the coupling constants, Theoret.and Math Phys., 44(1980),

p 810

Murata, M., Asymptotic expansion in time for solutions of Schrédinger-type equations, J Functional Analysis, 49(1982), p 10

NEWTON, R G., Inverse scattering I: One dimension, J Math Phys., 21(1980), p 493

35 NEWTON, R G., Inverse scattering by a local impurity in a periodic potential in one dimension, J Math Phys , 24(1983), p 2152

Newton, R G., Bounds on the number of bound states for the Schrédinger equation in one and two dimensions, J Operator Theory, 10(1983), p 119

Ossorn, T A.; BoLLé, D., An extended Levinson’s theorem, J Math Phys., 18(1977), p 432 Patit, S H., 7-matrix analysis of one-dimensional weakly coupled bound states, Phys Rev

A, 22(1980), p 1655

PaTit, S.H., Wave functions for weakly-coupled bound states, Phys Rev A, 25(1982), p 2467

Simon, B., Quantum mechanics for Hamiltonians defined as quadratic forms, Princeton, Univ

Press, 1971

Simon, B., The bound state of weakly coupled Schrffdinger operators in one and two dimensions, Ann Physics, 9711976), p 279

Simon, B., An introduction to the self-adjointness and spectral analysis of Schrédinger operators, Acta Phys Austriaca Suppl., 17(1977), p 39

Simon, B., Trace ideals and their applications, Cambridge, Univ Press, 1979

Simon, B., Schrédinger semigroups, Bull Amer Math, Soc., 7(1982), p 447

Sinna, K B., On the theorem of M G Krein, preprint, Univ of Geneva, 1975

THACKER, H B., Exact integrability in quantum field theory and statistical systems, Rev Modern Phys., 53(1981), p 253

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32 D BOLLE, F GESZTESY and S F J, WILK

48 WEIDMANN, J., Zur Spektraltheorie von Sturm-Liouville-Operatoren, Math Z., 98(1967),

p 268

49 WEIDMANN, J., Absolut stetiges Spektrum bei Sturm-Liouville-Operatoren und Dirac-Systemen, Math Z., 180(1982), p 423

50 Witk, S F J.; Fusrwara, Y.; Osporn, T A., N-body Green’s functions and their semiclas- sical expansion, Phys Rev A, 24(1981), p 2187

D BOLLE F GESZTESY

Instituut voor Theoretishe Fysica, Zentrum fiir interdisziplinare Forschung,

Universiteit Leuven, Universitat Bielefeld,

B-3030 Leuven, D-4800 Bielefeld 1,

Belgium West Germany

Permanent address: Institut fiir Theoretische Physics,

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