Bargmann potentials and Marchenko transformations

8 Pages·0.16 MB·

by  S. A. Zaitsev

Bargmann potentials and Marchenko transformations.
S. A. Zaitsev1
Department of Physics, KhabarovskState Technical University,
Tikhookeanskaya 136, Khabarovsk 680035,Russia
Abstract
The Marchenko phase-equivalent transformation of the Schr¨odinger equation for two
coupled channels is discussed. The combination of the Marchenko transformations valid
in theBargmann potential case is suggested.
3
0
0 Introduction
2
Letthe scatteringdatafitthenecessaryandsufficientconditions(see,e.g. [1])ofthe existence
n
of the solutions of the Marchenko equation
a
J
∞

0 K (r, r′)+Q (r, r′)+ K (r, t)Q (t, r′)dt=0,
αβ αβ αγ γβ
2 1 Qαβ(r, r′)= 21π ∞ dk[δαβ −Sαβ]eik(Rrr+r′)+AαAβe−κ(r+r′), α, β =1, 2. (1)
−∞
v
R
7
Then the elements of resulting potential matrix
6
0
d
2 [V (x)] =−2 K (x, x) (2)
1 0 αβ dx αβ
2
for some ε (0<ε<1) satisfy the condition [2]:
0
/ ∞
h
p x1+θ [V (x)] dx<∞ (−ε<θ <ε). (3)
0 αβ
-
t Z0 (cid:12) (cid:12)
n (cid:12) (cid:12)
(cid:12) (cid:12)
a The purpose of the work is to construct the potential (3) which describes the interaction in a
u
system of two coupled channels of the angular momenta ℓ = 0 and ℓ = 2 starting from the
q 1 2
: Bargmann potential (2).
v
The transformations of the coupled Schr¨odinger equation
i
X
Ψ ′′−V (x)Ψ +k2Ψ =0, (4)
r 0 0 0 0
a
into
Ψ ′′− V (x)+ 6 P Ψ +k2Ψ =0, P = 0 0 (5)
1 1 x2 1 1 0 1
(cid:18) (cid:19) (cid:18) (cid:19)
havebeendetailedinthebook[2],wherethematrixphase-equivalenttransformations(MPET)
associated with the scattering data of the three different types have been suggested. The
conditions [1] corresponds to the case a) in chapter VII of [2], however the method of chapter
VIII in Ref. [2]) cannot be used in our case, since generally the necessary condition for its
implementation
[F(0)] =0, (6)
21
where F(k) is the Jost matrix
∞
1 0
[F(k)] =I+ K (0, t)eiktdt, I = , (7)
αβ αβ 0 1
Z (cid:18) (cid:19)
0
does not hold. As a consequence V (x) has the singularity x−2 at the origin or at infinity.
1
Below are given the auxiliary (direct and inverse) Marchenko transformations which in
combination with MPET furnish the potential which is regular at the origin and decreases at
infinity faster than x−2.
1SupportedinpartbytheRussianFoundationofBasicResearchunderthegrantNo02-02-17316andState
ProgramRussianUniversities
Bargmann potential properties
We proceedfromthe assumptionthatthe elements ofthe potentialmatrix (2)differ fromzero
at the origin:
[V (0)] 6=0. (8)
0 α,β
In that case it can been shown that the regular solution G (x, k) of Eq. (4) has the behavior
0
near the origin:
x+λx3+o(x3) νx3+o(x3)
G (x, k)= , (x→0), (9)
0
νx3+o(x3) x+µx3+o(x3) !
where
[V (0)] −k2 [V (0)] −k2 [V (0)]
0 11 0 22 0 12
λ= , µ= , ν = . (10)
6 6 6
The asymptotic behavior of G (x, iσ), σ ≥0 is given by the following expression [2]:
0
eσx
G → F(iσ), (σ >0); G →xF(0)+o(x1−θ), (σ =0), (11)
0 0
2σ
where tilde denotes transpeosition. e
In the following, the functions
Y (x)=G (x, iσ)P, σ >0, Z (x)=G (x, 0)P (12)
0 0 0 0
are used. We also will keep track of the bound state wave function Φ (x)≡Ψ (x, iκ), κ>0:
0 0
u x+u x3+o(x3)
1 2
Φ (x)= , (x→0), (13)
0
v x+v x3+o(x3) !
1 2
Φ (x)→e−κx A1 , (x→∞) (14)
0 A
2
(cid:18) (cid:19)
transformations.
First transformation
The intentofthis transformationisto obtainthe Jostfunctionthatsatisfiesthe condition(6).
Here, the matrix-function
x
Y =Y J−1, J =I+ Y (t)Y (t)dt (15)
1 0 1 1 0 0
Z
0
f
is constructed in such a way that the matrix
Ψ (x, k)=Ψ (x, k)+Y (x)W Y (x), Ψ (x, k) (σ2+k2)−1, (k 6=iσ) (16)
1 0 1 0 0
n o
obeys the Schr¨odinger equation f
Ψ ′′−V (x)Ψ +k2Ψ =0 (17)
1 1 1 1
with the transformed potential V (x):
1
′
V (x)=V (x)−2 Y (x)Y (x) . (18)
1 0 1 0
h i
Here, W X(x), Y(x) denotes the Wronskian: W X,fY =X Y′- X ′Y.
The ansymptotic beohavior of the functions Y0 andnY1(x)o[2]:
e e e e
1
Y →eσxCP, C = F(iσ), (19)
0
2σ
e
Y →2σs−1e−σxCP, sP =PCCP, (20)
1
provides the modification of the initial S matrix and Jost matrix [2]:
e
S(k)=T(k)S(k)T(k), F(k)=F(k)T(k), (21)
where b b
T(k)=I−2σ(σ+ik)−1s−1CPC. (22)
The parameter σ value is defined by the expression:
e
[F(0)T(0)] =0, (23)
21
that is equivalent to
ζ2−2ζ ζ−1=0, (24)
0
where ζ(σ)=F (iσ)/F (iσ), ζ =ζ(0).
22 21 0
The solution Z (x) of Eq. (17) asymptotic behavior is given by the expression:
1
Z (x)→ F(0) xI+o(x1−θ) P, (x→∞). (25)
1
22
(cid:16)h i (cid:17)
One can see from the regular sbolution Φ (x, k) of Eq. (4) transformation
0
x
Φ (x, k)=Φ (x, k)−Y (x) Y (t)Φ (t, k)dt (26)
1 0 1 0 0
Z
0
e
that Φ (x, k) and Φ (x, k) have the same behavior at the origin and V (0)=V (0).
0 1 1 0
Second transformation
Now that the Jost function F(k) meets the condition F(0) =0, MPET [2] may be imple-
21
mented to Eq. (17). Here, the solution Z (x) of Eq. (1h7) anid matrix-function
1
b b
x
Z =Z J−1, J =I−P − Z (t)Z (t)dt (27)
2 1 2 2 1 1
Z
0
f
are used for the transformation construction [2]:
Ψ (x, k)=Ψ (x, k)−Z (x)W Z (x), Ψ (x, k) k−2. (28)
2 1 2 1 1
n o
Ψ (x, k) obeys the equation f
2
6
Ψ ′′− V (x)+ P Ψ +k2Ψ =0, (29)
2 2 x2 2 2
(cid:18) (cid:19)
with the potential:
′ 6
V (x)=V (x)+2 Z (x)Z (x) − P. (30)
2 1 2 1 x2
h i
The behavior of Φ2(x) and Y2(x) at the origin: f
u x+u ′x3+o(x3) 0 o(x3)
1 2
Φ (x)= , Y (x)= , (31)
2 2
v ′x3+o(x3) ! 0 µ′x3+o(x3) !
2
can be deduced from the regular solution transformation formula [2]:
x
Φ (x, k)=Φ (x, k)+Z (x) Z (t)Φ (t, k)dt. (32)
2 1 2 1 1
Z
0
e
Reference to Eq. (27) shows that
0 −3ν+o(1)
Z (x)= , (x→0), (33)
2  0 − 3 +O(1) 
x2
 
so that the second term in the right-hand side of (30) reads
−18ν2x2+o(x2) −6ν+o(x)
′
2 Z (x)Z (x) = , (x→0). (34)
2 1  −6ν+o(x) 6 +O(1) 
h i x2
f  
From (34) it follows that V (x) is regular at the origin and [V (0)] =0.
2 2 12
Z (x) behaves at infinity like [2]
2
−1
Z (x)→ −3 F(0) x−2I+o(x−2−θ) P, (x→∞). (35)
2
22
(cid:18) h i (cid:19)
It follows from (25) and (35) thatb[2]
′ 6
2 Z (x)Z (x) → I+o(x−2−θ) P, (x→∞). (36)
2 1 x2
h i (cid:18) (cid:19)
One can see from (29), (30),e(34) and (36) that MPET increases the second channel angular
momentum to 2.
Third transformation
Consider the following transformation:
Ψ (x, k)=Ψ (x, k)+Y (x)W Y (x), Ψ (x, k) (σ2+k2)−1,
3 2 3 2 0
(37)
n ′ o
V (x)=V (x)−2fY (x)Y (x) ,
3 2 3 2
h i
where f
∞
Y =Y J−1, J =I−P − Y (t)Y (t)dt. (38)
3 2 3 3 2 2
Z
x
f
Fromtheorem6.2.1inthebook[2]itfollowsthatthistransformationisasymptoticallyinverse
of the first one, since Y (x)= −Y (x) and −J (x)=J−1(x) at infinity.
3 0 3 1
Notice that the condition (3) is not affected by each of the three transformations.
Other angular momenta
Note that the regular solution G (x) of
3
6
′′
G − V (x)+ P G =0 (39)
3 3 x2 3
(cid:18) (cid:19)
then can be used to equally increase the angular momenta in both channels. Indeed, MPET
corresponding to the transformations [3], can be written in the forms:
Ψ (k, x)=−k−2G−1(x)W G (x), Ψ (k, x) ,
4 3 3 3
(40)
−n1 ′ ′ o
V (x)=V (xe)−2 G e(x)G (x) ;
4 3 3 3
h i
Ψ (k, x)=Ψ (k, x)+k−2Xf(r)W Gf(x), Ψ (k, r) ,
4 3 3 3
(41)
n ′ o
V (x)=V (x)−2 X(x)Ge(x) ,
4 3 3
h i
e
where
x −1
X(x)=G (x) G (t)G (t)dt . (42)
3 3 3
 
Z
0
MPET(40)and(41)increaseeachofℓ ,ℓ by1eand2,correspondingly,providedthatG −1(x)
1 2 3
and X(x) exist.
Example
As anexample the potential is calculatedcorrespondingthe simplest S matrix coupling S and
D states [5]:
1 2χ2 k2 k+iφ k+iκ 0 2χ2 −k2
S(k)= k−iφ k−iκ . (43)
k4+4χ4 −k2 2χ2 ! (cid:16) (cid:17)0(cid:16) (cid:17) 1 ! k2 2χ2 !
Notice that the transformation V → V have been carried out using MPET, i. e. the
0 3
solution Ψ (x, k), (k >0) has asymptotic behavior identical to that of Ψ (x, k):
3 0
Ψ (x, k)→ i e−ikxI−eikxS , (x→∞). (44)
0
2
(cid:8) (cid:9)
Thus in order that the asymptotic properties of Ψ (x, k) correspond to the potential V [4]:
3 3
Ψ3(x, k)→ i e2iπL e−ikxI−e−iπLeikxS , (x→∞), (45)
2
(cid:8) (cid:9)
where
0 0
L= , (46)
0 2
(cid:18) (cid:19)
the matrix
S =e−2iπLSe−2iπL (47)
must be used as S matrix in the Marchenko equations (1). In this instance (43) is used as
S, since, for example, all values −iResS (k) are negative and hence the corresponding
α,β
k=iκ
asymptotic normalization constants A , A are positive.
1 2
For the parameters χ = 0.26 fm−1, φ = 0.944 fm−1 and κ = 0.232 fm−1 [5] Eq. (24)
gives σ =0.2053483144fm−1. The initial Bargmann potentials [V (x)] and resulting ones
0 α,β
[V (x)] are presented in Figures 1-3 by dotted and solid lines correspondingly. Figures 4,
3 α,β
5 exhibit the S and D components of the bound state (with the energy −κ2) wave functions:
Φ (x)(dottedline),Φ (x)(solidline). NotethatΦ (x)intheinteriordomainmaybemodified
0 3 3
by, for example, the multichannel phase-equivalent transformation [6]. The asymptotic forms
of Φ (x) and Φ (x) D components are represented by dotted and solid lines in Fig. 6.
0 3
References
[1] K. Chadan and P. C. Sabatier. Inverse problem in quantum scattering theory (2nd ed.,
Springer-verlag,New York, 1989.)
[2] Z. S. Agranovich and V. A. Marchenko, The Inverse Problem of the Scattering Theory
(Gordon and Breach, New York, 1963).
[3] V. A. Marchenko. Sturm-Liouville operators and applications (Berkhauser, Basel, 1986).
[4] R.G.Newton.ScatteringTheoryofWavesandParticles(McGraw-Hill,NewYork,1966.)
[5] R. Newton, T. Fulton. Phys. Rev. 107, 1103, 1957.
[6] A. M. Shirokov, V. N. Sidorenko. Phys.Atom.Nucl. 63 (2000) 1993; Yad.Fiz. 63 (2000)
2085.
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