New results for scattering on the line

Volume 97A, number 1,2

PHYSICS LETTERS

8 August 1983

NEW RESULTS FOR SCATTERING ON THE LINE D. BOLLE’ Instituut voor Theoretische Fysica, Universiteit Leuven, B-3030 Leuven, Belgium

F. GESZTESY Institut für Theoretische Physik, Universitbt Graz, A-8010 Graz, Austria

and S.F.J. WILK Department of Physics, University of Manitoba, Winnipeg, Manitoba, Canada R3T 2N2 Received 23 May 1983

Low-energy scattering for Schrddinger hamiltonians on the line is discussed, taking into account explicitly the possibility of zero-energy resonances. Analytic expansion are presented for the reflection and transmission coefficient, the phase shift and the resolvent difference. Furthermore, a set of sum rules is obtained, in particular, it is found that Levinson’s theorem completely changes its structure in comparison with its three-dimensional analogue.

The one-dimensional scattering problem on the entire real line has received great attention in the past, especially in connection with inverse scattering methods, which are used extensively in quantum mechanical problems (cf. e.g. refs. [1—31 and references therein) and quantum field theory (cf. refs. [4,5] for a review). Recently, some new rigorous results on this problem have appeared. In particular, we mention here the studies of the ground state of one-dimensional Schrodinger operators with various potentials, including long-range ones, especially in the limit of weak coupling [6—9].Also bounds for the total number of bound states [8,101 as well as for the magnitude of the imaginary parts of resonances [11] have been obtained. Furthermore, the limit situation where some negative eigenvalues approach zero as the coupling constant approaches a critical value has been discussed thoroughly [12,13]. Finally, scaling techniques have been applied to study in detail the limit of one-dimensional short-range interactions converging to point interactions [14] (this reference contains an extensive list of earlier results.) 1 Onderzoeksleider N.F.W.O., Belgium.

30

In this letter, we are concerned with a systematic analysis of low-energy scattering on the line. Like in some of the negative eigenvalues studies mentioned above, we explicitly make a case distinction according to possible zero-energy eigenstates of the underlying hamiltonian. Such an analysis has been carried out very recently for three dimensions (cf. eg. refs. [15, 16]). In one (and two) dimensions there isthe wellknown additional difficulty that the Green function has a square root (logarithmic) singularity in the limit as the energy tends to zero. In the following, we explain how to overcome this difficulty. As a result, we obtain an analytic expansion ink around k = 0 for the reflection and transmission coefficients, the phase shift and the resolvent difference. Furthermore, we present a set of low-energy sum rules. In particular, we find that the structure of Levinson’s theorem for scattering on the line completely changes in comparison with three dimensions. We consider the hamiltonian H = —d2/dx2 +X 0 V(x), —o°
0 031-9163/83/0000—0000/s 03.00 © 1983 North-Holland

Volume 97A, number

PHYSICS LETTERS

1,2

I V(x)I’I~, u(x)= I V(x)11/2 sign V(x),

uu= V,

8 August 1983

satisfies the following properties

(1)

(i)(—d2/dx2

+

X 0V)i~1i(x)=0

the T-operator is defined as

[X0uR0(k)v+1]~, Imk>0,

T(k)

in the sense of distributions

(2)

(ii)~~EL(R),

2(R),

(11)

where the kernel of the free resolvent R0(k) is given by R 0(k, x, y) = i(2k)~exp(iklx —yj). (3)

where

This expression is singular in the limit k —-0. Therefore we define an operator M(k) by the kernel representation

= (u, X~M0Ø)/(v,u) , c2 = ~X~~(v(), 0) (12) Furthermore, we know that 0(x) is unique for our class of potentials [13]. Finally, there exists a reduced

M(k, x, y) = i (2k)~u(x)[exp(ikIx

—

y I)

—

I] v(y).

This operator has the norm convergent expansion [6, 17] M(k)

~

=

(5)

(ik)’~M~ ,

2(R) where theM~are Hubert—Schmidt operators on L with kernels M~(x,y) = —~u(x)[Ix

—

, n+1 /(n + 1)!] v(y).

(6)

Then the T-operator defined by (2) satisfies the Lippmann—Schwinger equation T(k) =

1

,

1

—

A0uR0(k)vT(k) (u, 1 )u +M(k)] T(k). X~[i(2k) —

(7)

A low-energy analysis of e.g. T(k) strongly depends, of course, on the zero-energy spectral properties ofH. Following refs. [12,13] we introduce the functions Ø~as solutions of Q~M 2(R), Q (8) 0QØ= —0, 0 E L Ø=Ø, Q=1—P, where P is the following one-dimensional projector P = (u, u)~(u, )u , (9) if X

(iii) [i~i(x)+c1 —c2x/jxj] EL

.

resolvent [17] T0

=

n-lim (QX0M0Q + 1

+ e)_1 (1

—

F0)

,

(13)

e~O

where P0 denotes the projector onto the one-dimensional eigenspace of QX0M0Q to the eigenvalue —1 [c.f. eq. (8)], P 0 = (~, ~)_1(~, )~, ~ = (sign V)0. (14) Consequently, we have to distinguish the following cases in our low-energy analysis: 1: X0 * 0eq. is not a coupling constant threshold, or,Case equivalently, (8) has no solutions. (There exists no resonance function iii.) Case 2a. —l is a simple eigenvalue of Q?v~M0Q with—1 eigenfunction ~ and c1 =of0,QX c2 *0 [c.f. eq. (12)]; 2b. is a simple eigenvalue 0M0Q with eigenfunction 0 and c1 * 0, c2 = 0; 2c. —1 is a simple eigenvalue of QX0M0Q with eigenfunction 0 and c1 * 0, c2 *~ In the cases 2a—2c there exist precisely one resonance function i/i, given by eq. (10), which is not in 2(R). Note that c L 1 and c2 never vanish simultaneously 2(R) and we would because then ~tiwould belong to L have a zero-energy bound state. However, as shown in resfs. [6,12] this is impossible in one dimension, in

?t

0 * 0 is the “critical value” of the coupling constant. Then it can be shown [13,171 that the function ~x) defined by

contrast with all higher dimensions. All the cases described above are easily realized in the example of an asymmetric square well.

i,ti(x) = —(v, X0M00)/(v, u)

We now turn our attention to the S-matrix and reflection and transmission coefficients. For scattering on the line, it is well-known that the S-matrix is a 2 X 2 matrix with elements S61~2,~ ~2 = ~ given by 5e1e2(k)~eie2 +f1~2(k), (15)

—

~xofdy

Ix

—

ylu(y)O(y),

A

(uO)(x)

=

—0(x),

(10) 31

PHYSICS LETTERS

Volume 97A, number 1,2

where ~ is the Kronecker symbol and f1~2the scattering amplitude

8 August 1983

It is certainly of interest to discuss the first coefficients in detail. For the case of no zero-energy resonance,case 1,we obtain [17] 0~ = 1 S~ 2] [(v,u) + (v, M 2(v, u) 1E2 E1C2 s ~l(1)~ = [2/(v, u) 0u) + (v,M0u) (vM 0T0M0u)] + [(er e2)/(v, u)][(v( ), u) —

f~12(k)=(2ik)_lk(v&~1”(), T(k)u e12k( )) , (16) with T(k) given by eq. (2). As usual S÷~(k) = S_ (k) are the transmission coefficients for left and right in-

—

—

—

cidence, S_+(k) and S~_(k)are the corresponding reflection coefficients.2,S_ (We follow the conventions = ~ =R~,S_~ of ref. [3] withS~. = T =R2). To study the low-energy behavior of these scattering observables, we use that T(k) in all cases can be shown to have the following Laurent expansion in k around k0

— —

), T0M0u) + (v, u)~(v( ), u)(v, M0u)] 2. ), T~()u)+~e1e2(u,u)~(v( ), u) (20)

(v( ~e

1e2(v(

So, up to 0(k2), the transmission coefficients are given by T~(k)=Tr(k)=iks~+0(k2) ilc {—2(v, u)—2 [(o, u) + (v, M 1(u,M 2 (v,M 0u) + (v, u) 0u) 0T0M0u)] =

—

T(k) = ~

(ik)~t~,

(17)

n =q

where q = 0 in case land q = —1 in cases 2a—2c. Furthermore, the coefficients tn satisfy a set of recursion relations. Similar to three dimensions [16], the method to find these relations is based on the Lippmann— Schwinger equation (7), projected onto the one-dimensionaleigenspace ofand Q?~M0Q 1 and its orthogonal, on theto usetheofeigenvalue expansion (17) and some properties of the projectors P and P defined before. For more details we refer to [17]. 0 Here we will only discuss certain matrix elements of the first few coefficients later on. Employing the expansion (17) in eq. (16), we arrive —

at the result:

2) ~(u( ), T~()u) + ~(u,u)_1 (v( ), u)2} + 0(k(21) with the reduced resolvent T 0 given by eq. (13), while the reflection coefficients can be determined from [eq. (20)] 2), R~(k)= —l + jks°2 + 0(k 2). (22) R~(k) 1 + are iksS~ 0(k These =results more+ detailed than the ones available —

in the literature (c.f. e.g. refs. [1—3]).For the cases with a zero-energy resonance, cases 2, we arrive at ~(O) =ö E 1e2

Theorem 1. Let VE C~°(R). Then the S-matrix, S~1~2(k), is analytic in karound k = 0 in all cases and has the following expansion S~1~2(k) = ~

(18)

(ik)ns(n)

n =0

—1 +(1c 2 +Ic 2)~ 11 2i

12

X (tci

2

—

e1c~c2+ e2c1c~ e1e2 —

(23) with c1 and c2 given by eq. (12). Noting that c1 and c2 can be chosen real by choosing the zero-energy eigenfunction, 0~real, the transmission and reflection coefficients, become in these cases 2 + c 2)’(~c 2 1c 2)+0(k), TQ(k)=Tr(k)= (1c11 2I 1I 21 R~(k)(Ic 2+1c 2)’(2c 1I 2+~c 21 2)-l(2cic 1c2)+0(k), Rr(k)=_(Icil 2I 2)+0(k). (24) —

where the coefficientss~2are givenby ~

~n0~eie~

n+1—qn+1--q—pn (~~61)m (62)1 m0

1=0

1u) m!l!

X (v( )m, t,~+i_m_i( ) with q = 0 in case 1 and q = —1 in cases 2. 32

(19)

These results areone-dimensional new. behavior of the phase ö, defined This knowledge allows us to find theshift low-energy as (c.f. ref. [3])

det S(k) = exp [2i6 (k)] .

(25)

In case 1, the results (20) immediately

dE E-N

/

lead to

), u)~] + W2)

c n=-2

(-l)nEn+1/2A2n+l

, + 7r(-1)N+1AW_2

. .. .

-R&/z)]

N-2 -

+ (u, u)-1 (u, Muu)2 - (u, M&M@)] - (u( ), T,-,( )u) + (u, u)-‘(u(

Im Tr[R(m)

0

6 (k) = n(n + f ) + ;k (4(u, u)-2 [(u, u) + (u, Muu)

n=O,+l,

8 August 1983

PHYSICS LETTERS

Volume 97A, number 1,2

,

N>O ,

(26)

Correspondingly, S(k)=rntO(k),

in cases 2 we get from (23) n=O,+l,....

(27)

The second main set of new results concerns sum rules (also called trace relations) involving the trace of the difference between the full and the free resolvent on one side and the negativeenergy bound states and zero-energy resonances on the other side. For V E C;(R) we know that the operator [R(k) -Ro(k)], where R (k) is the full resolvent defined as R(k) = (H - k2)-’ is a trace-class operator on L2(R). Expanding Tr [R - Ro] in k around k = 0, we can prove [17]:

(30)

where - .,j= 1,2 )...) Nb denote the negative-energy G bound state positions and the A’s, for the different cases, are given by expression (29). Here we are especially interested in the sum rule (30) forN= 0, which is Levinson’s theorem. Introducing the phase shift in this formula by the following relation [lg] 2 Im Tr [R (k t i0) - Ro(k t iO)]

(31) we obtain as a:

YYzeorem2. Let VECF(R).ThenTr[R(k)-RO(k)] has a Laurent expansion in k around k = 0 in all cases

Tr[R(k)

-R&)1

= ._$_2

(WA,

,

(33)

Corollary (Levinson’s theorem for scattering on the line). Let V E C;(R). Then 6 (-) - 6 (0) = --n(Nb + A_,)

,

(32)

where A_, is given by where Aq-

A _2 = -;

,

in case 1 ,

2 = f(U, fq+p),

A-2=

A, = $(u,&+34

0,

in cases 2 .

(33)

n+l-q +i

ls

(n+2-z-q)Tr(Mn+2-l-qfl+q),

(29)

n>q-1, withq=Oincase

1 andq=-1

incases2.

Furthermore, integrating Tr [R (k) - Ro(k)] in the complex energy plane z = k2 + ir) = E t iv around the spectrum of the hamiltonian H, we obtain the following sum rules [ 171: T’heorem3. Let VEC!r(R).Then satisfies the trace relations

ImTr[R

-R,-,]

The structure of Levinson’s theorem for scattering on the line is entirely different from its analogue in three dimensions (c.f. e.g. ref. [ 193). Indeed, when there is no zero-energy resonance present (case l), we not only get on the right-hand side of (32) the term proportional to the number of negative energy bound states, -nNb, but an additional factor n/2 appears. In the cases with a zero-energy resonance (cases 2) we simply obtain the term -trNb on the right-hand side of (32). This difference is of course due to the additional Dirichlet boundary condition at the origin when considering Schrodinger operators on the halfline.

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Volume 97A, number 1,2

PHYSICS LETTERS

The results obtained in this letter are valid for a more general class of potentials. Detailed proofs, as well as more coefficients in the various low-energy expansions will appear elsewhere [171. The methods used here also work in two dimensions although the explicit analysis in that case is more complicated [201 because of the logarithmic nature of the Green function singularity and the possible existence of zeroenergy bound states besides zero-energy resonances. The authors would like to thank A. Jensen for interesting correspondence about the use of weighted Sobolev space methods (c.f. ref. [211 for three and more dimensions) to obtain similar results. References [1] K. Chadan and P.C. Sabatier, Inverse problems in quanturn scattering theory (Springer, Berlin, 1977). [2] P. Deift and E. Trubowitz, Commun. Pure Appi. Math, 32(1979)121. [3] R.G. Newton, J. Math. Phys. 21(1980)493. [4) L.D. Faddeev, Soy. Sci. Rev. Cl (1980)107. [5] H.B. Thacker, Rev. Mod. Phys. 53(1981) 253. [6] B. Simon, Ann. Phys. 97 (1976) 279. [7] R. Blankenbeckler, M.L. Goldberger and B. Simon, Ann. Phys. 108 (1977) 69.

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8 August 1983

[8] M. Klaus, Ann. Phys. 108 (1977) 288; Helv~Phys. Acta 52 (1979) 223. [9] S.H. Patil, Phys. Rev. A22 (1980) 1655; A25 (1982) 2467. [10] P.G. Newton, Bounds on the number of bound states for the Schrddinger equation in one and two dimensions, preprint. 111] E.M. Harrell, Commun. Math. Phys. 86 (1982) 221. [12] M. Klaus and B. Simon, Ann. Phys. 130 (1980) 251. [13] M. Klaus, Helv. Phys. Acta 55 (1982) 49. [1415. Albeverio, F. Gesztesy, R. Hqlegh-Krohn and W. Kirsch, On point interactions in one dimension, preprint (1983). [15] S. Albeverio, D. Bol1~,F. Gesztesy, R. H~egh-Krohnand L. Streit, Ann. Phys., to be published. [161 and S.F.J. Wilk, Math.Wilk, Phys. (1983) 1555. [17] D. D. Bolld Boll~, F. Gesztesy and J.S.F.J. in 24 preparation. [18] K.B. Sinha, On the theorem of M.G. Krein, University of Geneva, preprint (1975). [19] R.G. Newton, Scattering theory of waves and particles (Springer, Berlin, 1982). [20] D. Boll~,C. Danneels, F. Gesztesy and S.F.J. Wilk, in preparation. [211 A. Jensen and T. Kato, Duke Math. J. 46(1979)583; A. Jensen, Duke Math. J. 47 (1 980) 57; Spectral properties of Schrddinger operators and time decay of the wave 2(R4). functions; results in L