H has no zero-energy resonances =- generic caseand H has a zero-energy resonance with multiplicity one.. t establishes recursion relations for the coefficients in the Taylor expansion fo
Trang 11 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
Different aspects of the one-dimensional scattering problem have received much attention in the past, especially in connection with inverse scattering techni- ques, 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 pro- perties 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-dimen- sional 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
Let us now give a short description of the results obtained in this paper In Section 2 we study the occurrence and properties of zero-energy resonances of the one-dimensional Schrédinger Hamiltonian H This leads to a classification of
Trang 24 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)V¥ © 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 spec- trum, 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 posi- tive-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 logarith- mic 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}
i - on Ø(H,):: HĐ1XR)
Gx?
Ny =
Trang 3LOW-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
Ñ 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
(2.7) 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,
we thus obtain for 7(k)
(2/12) Fk) = [Lb + (12a(6, w)/2k)P + ApM(K))-3, TImk 3 0, k0, ke š,(H).
Trang 46 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
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
+ 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
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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
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)
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 Schrödinger 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
(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
Trang 7LOW-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)
\ 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-
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} ~
(2.21)
— 29h ay uQr)|x — y!VYŒ)Ó@)
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
Ox) = — u(x)¥(x) ae .
Trang 810 D BOLLE, F GESZTESY and S F J WILK
Since ¥ is locally absolutely continuous we obtain
W'(x) = 2-Mg \ dyV(yQ) — 2-My \ dyÝ0)@0)
and differentiating once again
(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
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
a) cy := 0, C2 #0
Trang 9LOW-ENERGY SCATTERING IN ONE DIMENSION il
(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 cons- tants) given by equality (2.15) Since H has no zero-energy bound states (or equi- valently (v, Mj@o) and ((-)v, @ 9) do not vanish simultaneously) and nonzero eigen- values 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
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
Proof In principle one could follow the proof of Lemma 3.1 in [2] step by step but we prefer another argument based on (25, p 180] (cf also (22], Theorem 4.3) It turns out that (3.1) holds if we can show that (2,QM,Q + 1)?g = 0, g € L>(R), implies that (A,0M,O + l1)g = 0
Trang 1012 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) = — (ñ 22QuHg 10O[) =:
= — }(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:
Trang 11LOW-ENERGY SCATTERING IN ONE DIMENSION 13
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 —
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)-]? =
(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)]}~" =
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?
Trang 1214 D BOLLE, F GESZTESY and S F J WILK
or using equality (3.5) with e == —2ik(A9(v, u))~1 we arrive at
T(k) ma { + | + (Phu, | 00) l + (B Po» )Po | *
(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))
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)
—AgPy\M@(K)P(kK) — 2aPyM)(k)Q(&)
Trang 13LOW-ENERGY SCATTERING IN ONE DIMENSION 15
[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
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-
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In order to calculate the coefficients p,, g, in all Cases I, II a) c) we expand
Trang 15LOW-ENERGY SCATTERING JN ONE DIMENSION 17
we obtain from equality (3.17)
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 —