Table Of Content(cid:13)c 1993 KORENBLIT S.E.
THE FROISSART-GRIBOV REPRESENTATION
FOR JOST FUNCTIONS OF DIRAC OPERATOR
IN ARBITRARY DIMENSION SPACE
9 INSTITUTE OF APPLIED PHYSICS, IRKUTSK STATE UNIVERSITY,
0 RUSSIA
0
2
Abstract
n
a
J AdynamicschemebasingonequationforT-matrixmomentumtransferspectralden-
3 sityandintegralrepresentationforJostfunctionisproposedforlocalDiracHamiltonians
in arbitrary N- dimension spaces and for Schrodinger one with singular or nonlocal gen-
] eralized Yukawa-typepotentials.
h
p Ageneralizationoftheoff-shell-JostfunctionmethodforthatHamiltoniansand
- universal renormalization procedure of Jost function calculation for singular and
h
nonlocal potentials is proposed.
t
a
m
1 Introduction
[
1 It is well known, that determinant
v
1 d(W)=det G (W)G−1(W) =det[I G (W)V]=det[I+G (W)V]−1, (1)
4 0 V − 0 V
3
0 withGreenfun(cid:2)ction(resolvent)(cid:3)GV(W)=[W HV]−1,accumulateallobservable
−
. information about spectra of the stationary Hamiltonian H = H +V in most
1 V 0
0 economical form [1]:
9
0 nmax W 1 ∞ δ(ε)
: d(W)= 1 n exp dε (2)
v − W −π (ε W)
i nY=1(cid:18) (cid:19) (cid:26) Z0 − (cid:27)
X
whichmakesitveryconvenientforsolvingbothdirectandinversescatteringprob-
r
a lems [2, 3, 4], and for finding a different sums on spectra H [5]. It arise also
V
in one-loop calculations for different quantum effects in external fields V(~x) or
in the semiclassical quantization of field theory near nontrivial classical solutions
[6,7,8]. However,derivationofthis determinantusually imply solutionoftwodif-
ferenteigenvalueproblemsforfindingcharacteristicsofdiscreteW andcontinuous
n
δ(ε) spectra separately. That makes their calculation and utilization much more
complicated. This circumstancestimulatessearchofanotherwaysforconstruction
determinant does not requiring any information about H eigenvalues.
V
On the other hand, in the case of spherically symmetrical Hamiltonian V(~x)=
V(r), using Green function’s partial expansion onto irreducible representations of
rotation group SO(N), for instance, for Dirac operator:
1
<~xG(N)(Wζ(ib))~y>=
| V | (ry)(N−1)/2·
(3)
∞ Π (~n,~ω) Gζ11 iΠ (~n,~ω)(~σ~ω) Gζ12
κξ κξV κξ κξV
·JNX=λNξX=±1 −i(~σ~n)†Πκξ(~n,~ω) Gζκξ2V1 (~σ~n)†Πκξ(~n,~ω)(~σ~ω) Gζκξ2V2 !
1
where: ~x=r~n; ~y=y~ω,andΠ (~n,~ω)isprojectorontosubspacewithfixedorbital
κξ
l(N) and total J angular momentum (see Appendix); one can formally factorize
ξ N
d(W) into infinite product:
∞
d(N)(Wζ(ib))= Fζ (b) ∆(N,JN) (4)
Dir κξ
JNY=λNξY=±1h i
where dimension of the SO(N)-representation for this case is 1:
(J +λ )!
Tr Π =∆(N,J )=2[(N−1)/2] N N ; (5)
{ κξ} N (J λ )!(N 2)!
N N
− −
and the following notations are accepted hereafter:
1 1 N 1
a = (3 N); λ = a = 1; κ κ =ξ J + ;
N N N ξ N
2 − 2 − 2 − ≡ 2
(cid:18) (cid:19)
ξ
L =J + =l(N) a ; ξ = 1. (6)
ξ N 2 ξ − N ±
with the numbers l(N) =0,1,2,.., and J =λ ,λ +1,λ +2,... defining eigen-
ξ N N N N
valuesofsquaredorbitalandsquaredtotalangularmomentumrespectively[9](see
Appendix)
1 1 1 2 1
(L L) l(N) l(N)+2λ ; (J J) J + (N 1)(N 2).
2 · ⇒ ξ ξ N 2 · ⇒ N 2 − 8 − −
(cid:16) (cid:17) (cid:18) (cid:19)
ThepartialdeterminantsorJostfunctionsFζ (b)aredefinedbythesameformulae
κξ
(1),(2) with partial Green function which matrix elements are Gζij (b;r,y), and
κξV
with scattering phase δ (ε) [10] respectively. In contrast with d(W) [1], they
κξ
are well defined for arbitrary local potential which is less singular at r = 0, than
appropriate free Hamiltonian H , and disappear sufficiently fast at r [2, 3].
0
→∞
There is still one problem on this way, of finding an integral representation
determineda generalformofJostfunction’s l(N),J - dependence. Suchrepresen-
ξ N
tation may be useful both for field theoretical calculations mentioned above and
forReggephenomenology[12]. Therewasanattemptmadein[11]forSchrodinger
case with N=3. It led to representation with two variable’s weight function which
satisfies to complicate nonlinear integral equation and does not have any known
physical meaning [2, 11].
A quite different integral representation for Jost function (matrix) was es-
tablished recently in [13, 14, 15] for the Dirac operator with N=3, and for the
SchrodingeroneinarbitraryN-dimensionspaceandinmodelwithNstronglycou-
pled channels. It play a role,analogousto Froissart-Gribovrepresentationfor par-
tial amplitudes, but define the Jost functions in all analytical region over complex
variablesJ andbintermsofquadraturesfromhalf-off-shellT-matrixspectralden-
N
sityovermomentumtransfer,withenergeticvariables,analyticallycontinuedfrom
the continuum to the bound state region, and provides a group-theoretical inter-
pretationdirectlyfortheJostfunction. TogetherwithlinearVolterra-typeintegral
equationforthespectraldensitythisrepresentationformsadynamicscheme,from
1Theminimalgamma-matrixrepresentationisassumed.
2
which all Jost functions (matrices) are found via solution of one regular problem,
which has nothing to do with eigenvalue one.
Presentworkgivesageneralizationofthis schemefor awide classofoperators,
including N-dimension regular Dirac operator, singular or nonlocal Schrodinger
operators, and the last with relativistic corrections to potential.
2 Equation for spectral densities
The aim of this section is to derive equations for T-matrix spectral densities over
momentum transfer, constituting the foundation for the dynamic scheme in ques-
tion, and to elucidate their analytical properties.
We define a family of Dirac operators in R : µ,ν =0,1,2,...N,
N
H =(~Γ P~)+Γ m; Γ ,Γ =2δ ; P~ = i~ ; (7)
0 0 µ ν µν N
· { } − ∇
a)H =H +IV; b)H =H +Γ V; (8)
V 0 V 0 0
with local Yukawa-type potentials
4π ∞ r a
V(r)= dν Σ(N)(ν) χ (νr), (9)
Ω πar 2ν −a
N Zµ0 (cid:16) (cid:17)
or in momentum representation [16] 2:
2 ∞ Σ(N)(ν)
<~qV ~p>= dν +(subtractions). (10)
| | πΩ [ν2+(~q ~p)2]
N Zµ0 −
The normalization conditions are:
1
<~q~p>=δ (~q ~p); <~x~p>=exp( iπa )ei(~p·~x)(2π)−N/2;
N N
| − | −2
and the following notations are used hereafter: Ω =2πN/2/Γ(N/2);
N
N 2; ~q=q~τ, ~p=p~v;
≥
2
χ (βr)=( βr)1/2K (βr); χ (βr)=e−βr; (11)
l π l+12 0
where K (z) are McDonald function [17]. Choosing for Γ-matrices the following
m
representation [18]:
O σ I O
(~Γ) Γ = k , Γ = ; (12)
k ≡ k σ† O 0 O I
(cid:18) k (cid:19) (cid:18) − (cid:19)
where matrices σ for k,j =1,2,...N satisfy to conditions:
k
σ σ† +σ σ† =σ†σ +σ†σ =2δ , (13)
j k k j j k k j jk
we have a complete set of eigenfunctions for operator (7):
H (~p) u (~p,[λ])=wζ(p) u (~p,[λ]);
0 ζ ζ
2Heresubtractions leadto ultralocal terms (∆)nδN(~x)in(9) correspondingto regularization
of the potential in the sense of distributions. For Dirac Hamiltonian such singular potential is
unstable with respect to particle creation, and we assume the absence of subtractions for that
case.
3
ζ ε(p)+mζ
u (~p,[λ])= w (~v);
ζ ε(p) mζ (~σ~v)† [λ]
(cid:18) p− (cid:19)
u (~p,[λ])pu†(~p,[λ])=ε(p)+ζH (~p);
ζ ⊗ ζ 0
X[λ]
u† (~p,[µ]) u (~p,[λ]) =2ε(p) δ δ , (14)
ζ” · ζ ζ”ζ [µ][λ]
where following defi(cid:16)nitions are used: (cid:17)
ε(p)=+ p2+m2; wζ(p)=ζε(p); Wζ(ib)=ζ m2 b2; ζ,ζ = 1. (15)
− ±
Spinorsw (~pn)withquantumnumbers[λ]ongrouppSO(N)realizeitsspinorrepre-
[λ]
sentationof half dimension than u (~p,[λ]), andsatisfy to the following conditions:
ζ
w† (~n) w (~n) =δ ; w (~n) w† (~n)=I (16)
[µ] · [λ] [µ],[λ] [λ] ⊗ [λ]
(cid:16) (cid:17) X[λ]
We consider also Schrodinger operators for N=3 with relativistic correction to
potential V(r) (here σ are Pauly matrices):
1,2,3
H =P~2(2m)−1+U(~x), (17)
V
1
U(~x)=V(r) (2m)−1 (~σ P~), (~σ P~),V(r) (18)
− 2 · ·
and with nonlocal interaction: h h ii
U(~x)=V (r)+(2m)−2(P~2V (r)+V (r)P~2). (19)
1 2 2
Using definitions (14) and the Lippman-Schwinger (LS) equation
T(W)=V+VGc(W)T(W), (20)
0
with the help of free Green function’s decomposition:
−1
Gc(Wζ;k) = (Wζ +H (~k)) (Wζ)2 k~2 m2 i0 =
0 0 − − −
= 1 huζ′(~k,[λ])⊗u†ζ′(~k,[λ]i),
2ε(k)ζ′X=±1X[λ] Wζ −ζ′(ε(k)−i0)
h i
we define for Hamiltonian (7),(8) T-operator acting on spinors (16):
w† (~τ) (~q,ζ” T(Wζ)~p,ζ) w (~v)=u† (~q,[µ])<~qT(Wζ)~p>u (~p,[λ]), (21)
[µ] | | [λ] ζ” | | ζ
with symmetry properties:
†
(~q,ζ” T(Wζ)~p,ζ)=( ~q,ζ” T(Wζ) ~p,ζ)= (~p,ζ T(Wζ)~q,ζ”) . (22)
| | − | |− | |
(cid:16) (cid:17)
It is not difficult to see from (10),(20), that it possess spectral representation:
(N) 1 ∞ dν
(~q,ζ” T (Wζ(ib))~p,ζ)=
| | πΩ m [ν2+(~q ~p)2] ·
N Z0 −
(N)
D (1)(ν; ip,b2, iq)ζ +(~σ ~τ)(~σ ~v)†
· ζ”ζ − − · · ·
(cid:20)
(N)
D (2)(ν; ip,b2, iq)ζ +(subtractions); (23)
· ζ”ζ − −
(cid:21)
4
where for the Born term:
(N) (1) (1)
D (2)(ν; ip,b2, iq)ζ Σ(N)(ν) A(2)(q,p)=Σ(N)(ν)
ζ”ζ − − ⇒ ζ”ζ ·
ζ”ζ (ε(q) mζ”)(ε(p) mζ) 1/2 =Σ(N)(ν) A((21))(p,q)=
· 1 ± ± ζζ”
(cid:26) (cid:27)
(cid:2) ε(p)+mζ 1/2 η(cid:3)ζ”(q) −1
=Σ(N)(ν)ζ q (24)
(cid:20)ε(q)+mζ”(cid:21) ( (cid:16) ηζ(p)(cid:17)
ε(p)+mζ 1/2 (1)
Σ(N)(ν)ζ q M(2)(q,p);
≡ ε(q)+mζ” ζ”ζ
(cid:20) (cid:21)
ηζ(p) (wζ(p) m)/p p/(wζ(p)+m); η−ζ(p)= (ηζ(p))−1. (25)
≡ − ≡ −
It is convenient to pass to the quantities, depending from quantum numbers ζ”,ζ
only via sheet’s indices of the functions wζ(p),wζ”(q) i.e. via brunch indices
ζ = sgn(Re w(p)) of analytic functions w(p) = p2+m2 1/2,w(q). This may
be achieved by putting in accordance with (24) for i=1,2:
(cid:0) (cid:1)
(N)
D (i) (ν; ip,b2, iq)ζ = (26)
ζ”ζ − −
1/2
ε(p)+mζ (N)
=ζ q D (i) (ν; ip,b2, iq)ζ =
ε(q)+mζ” ζ”ζ − −
(cid:20) (cid:21)
ε(q)+mζ” 1/2 (N)
=ζ” p D (i) (ν; iq,b2, ip)ζ;
ε(p)+mζ ζζ” − −
(cid:20) (cid:21)
then the symmetry (22) takes the form: i=1,2
(N) ηζ”(q) (N)
D (i) (ν; iq,b2, ip)ζ = D (i) (ν; ip,b2, iq)ζ (27)
ζζ” − − ηζ(p) ζ”ζ − −
Now followingto Fubini and Stroffolini[19] andto [13, 14]we calculatediscon-
tinuity overmomentum transfert= (~q ~p)2 fromboth sidesofLSequation(20)
− −
with the help of the arbitrary dimension’s relation:
dΩ (~n) Ξ[M](~n) ( 1)l4πλ+1 ∂ ∂
N l = − Ξ[M] ~τ +~v
[X (~τ ~n)][Y (~v ~n)] 2lΓ(l+λ) l ∂X ∂Y ·
Z − · − · (cid:18) (cid:19)
∞ dZ W(X,Y,Z) l−aN
; (28)
·ZZ+(X,Y) [W(X,Y,Z)]1/2[Z−(~τ ·~v)](cid:18) Z2−1 (cid:19)
W(X,Y,Z)=X2+Y2+Z2 2XYZ 1;
− −
Z (X,Y)=XY (X2 1)(Y2 1) 1/2;
±
± − −
for sphericalfunction Ξ[M](~n) ongroupSO(cid:2) (N) [20], andcam(cid:3)e to the followingsys-
l
temofequationforT-matrixspectraldensityovermomentumtransferforoperator
(8a):
(N) (1) (1)
D (2)(ν; ip,b2, iq)ζ Σ(N)(ν) M(2)(q,p)= (29)
ζ”ζ − − − ζ”ζ
5
(N 2) ∆(q2,p2, ν2) a ν ν−γ ω+(ν;µ,γ;q,p)
= − − dγΣ(N)(γ) dµ dk2
2mπ ν2 ·
(cid:20) (cid:21) Z0 Z0 Zω−(ν;µ,γ;q,p)
(ω+ k2)(k2 ω ) −12−a k gζ′( ik;b)ζ
−
· − − − ·
′
(cid:2) (cid:3) ζX=±1
(1) (2) Z Y X (N) (1)
·("Mζ(2”ζ)′(q,k)+Mζ(”1ζ)′(q,k)(cid:18) νZν2µ−−1 γ(cid:19)# D (ζ2′)ζ(µ;−ip,b2,−ik)ζ
(1) Z X Y (N) (2)
+Mζ(2”ζ)′(q,k)(cid:18) νZν2γ−−1 µ(cid:19) D (ζ1′)ζ(µ;−ip,b2,−ik)ζ)
which is independent fromsubtractions in (23). Here we put µ =0 for simplifica-
0
tion of the formulas and the following notations are accepted hereafter:
′ ( 1) 1 Wζ(ib) ′ ′ −1
gζ ( ik;b)ζ = − 1+ = 2wζ (k) Wζ(ib) wζ (k) ;
− (k2+b2)2 wζ′(k)! −
h (cid:16) (cid:17)i
q2+k2+γ2 p2+k2+µ2 q2+k2+ν2
X = ; Y = ; Z Z(qpν)= ;
γ µ ν
2qk 2pk ≡ | 2qp
2ν2ω+(ν;µ,γ;q,p)=ν2(ν2 µ2 γ2)+q2(ν2+µ2 γ2)+
− − − −
+p2(ν2 µ2+γ2) ∆(ν2,µ2,γ2) ∆(q2,p2, ν2) 1/2;
− ± −
∆(a,b,c)=(a+b c(cid:2))2 4ab; (cid:3) (30)
− −
For case (8b) we must change M(2) M(2). For the same function V(r) with
⇒ −
different dimensions of r and r its solutions are connected by simple Weyl’s
(N) (D)
integral transformation: i=1,2
(D) Γ(N/2) d n ν (ν2 γ2)aD−aN+n−1 (N)
D (i)(ν; )= 2ν dγ − D (i)(γ; ), (31)
·· Γ(D/2) dν2 Γ(a a +n) ··
(cid:18) (cid:19) Zµ0 D− N
whereintegernumbernisrestrictedonlybyconvergenceconditionofthisintegral:
n max[(D N)/2;0]. This transformation is identical with Schrodinger case,
≥ −
and may be checked by the same way [14].
For the Hamiltonian (18) the formulas may be simplified by choosing helicity
representationfor the spinors w(~n): (~σ ~n)w (~n)>=2λw (~n)>. Then, instead
λ λ
· | |
(24), we have:
<~q,µU~p,λ>=∗ (1/2)(R ) 1 q2+p2−2µ2λ2qp 1 <~qV ~p>; (32)
| | D λµ ~τ~v − 2(2m)2 2m | |
(cid:20) (cid:21)
where potentialV(r) (10)for N=3 alsois assumedto be regular(without subtrac-
tions). Separating in (20),(23) spin-rotation matrix
∗ (1/2)(R )=<w (~τ)w (~v)>,
D λµ ~τ~v µ | λ
one can decompose the spectral density matrix onto the sum of two orthogonal
projectors with coefficients D(1),(2)[13]:
D ( )=D(1)( )+2µ2λ D(2)( ); (33)
µλ
··· ··· ···
6
2(Π+) =(I +σ ) =1,(for all µ,λ); 2(Π−) =(I σ ) =2µ2λ;
1 µλ 1 µλ 1 µλ − 1 µλ
Then discontinuity calculation like above give the same system like (29) for N=3
with changed normalization
q (N) q
D (i) (ν; ip,b2, iq)ζ D(i)( ); M(i)(q,p) A(i)(q,p);
2m ζ”ζ − − ⇒ ··· 2m ζ”ζ →
and with substitutions:
′
gζ ( ik;b)ζ (k2+b2)−1; (34)
− →
′
ζX=±1
1
A(1)(q,p)=1 (q2+p2)(2m)−2; A(2)(q,p)=qp(2m)−2. (35)
− 2
Spectral density’s equation for Hamiltonian (19) has more simple form [14] with
substitution:
Σ(3)(ν) Σ (ν)+Σ (ν)(q2+p2)(2m)−2 (36)
1 2
⇒
and in particular case V = V appears from last system, if one put on them
1 2
D(2) =A(2) =0 eliminating all dependence from spin.
Let now shortly consider analytic properties of the spectral density over en-
ergetic variables q,p. It may be shown [21], that due to volterrian property of
eq.(29) providing convergence of its iteration serie, the spectral density possess
analytic continuation to the domain [14]
p=i̺, q =iu, k =iα; ̺>0; 0<ν <u ̺;
−
∆(q2,p2, ν2) 1/2 = eiπ ∆(u2,̺2,ν2) 1/2; (37)
−
(cid:2) ω−+(ν;µ,γ;q(cid:3),p)= eiπ Λ(cid:2)+−(ν;µ,γ;u,̺(cid:3));
where we put:
Λ+(ν;µ,γ;u,̺)=Λ0(ν;µ,γ;u,̺) 1 ∆(ν2,µ2,γ2) ∆(u2,̺2,ν2) 1/2;
− ± 2ν2
(cid:2) (cid:3)
2ν2Λ0(ν;µ,γ;u,̺)=ν2(µ2+γ2 ν2)+u2(ν2+µ2 γ2)+
− −
+̺2(ν2 µ2+γ2), (38)
−
and that continued functions satisfy to system (29) continued to this domain (see
bellow).
3 Generalizations of the off-shell Jost function method
for Dirac operator
Let us now turn to generalizations of the off-shell Jost functions method. Such
preliminaries is necessary to establish the relation in question between Jost func-
tion and T-matrix momentum transfer spectral density. There are two ways to
introduce off-shell Jost functions (OSJF). The first one derive it only for local po-
tential from solution of nonhomogeneous radial Schrodinger (or Dirac) equations
i.e. off-shell Jost solution (OSJS). The second one relate OSJF with half-off-shell
partial amplitude. Both this ways are equivalent obviously for local nonsingular
7
potentials, successfully added each other for singular and nonlocal potentials. Al-
thoughthe ideasofthis methodisnotnew[22,23],weoutlinehereitsmainpoints
inmodifiedform,convenientforouraimstogetitsgeneralizationoncomplexvalue
of total angular momentum J and demonstrate its applicability for a wide class
N
of operators.
We begin with the second way [14] introducing off-shell partial amplitudes by
expansion of T-matrix (23):
(N) 2(qp)a ε(q)+mζ” 1/2
(~q,ζ” T (Wζ(ib))~p,ζ)= ζ”
| | − πq ε(p)+mζ ·
(cid:20) (cid:21)
∞
Π (~τ,~v) Tζ”ζ(q,p;b2)ζ. (39)
· κξ κξ
JX=λNξX=±1
For sufficiently large value of J it possess a Froissart-Gribov integral representa-
N
tion:
4πe−iπa ∞ (N)
Tζ”ζ(q,p;b2)ζ = dν Qa (Z ) D (1)(ν; iq,b2, ip)ζ+
κξ −4mΩ πa Lξ ν ζζ” − −
N Zµ0 (cid:20)
+Qa (Z )(DN) (2)(ν; iq,b2, ip)ζ ∆(q2,p2, ν2) −a/2, (40)
L−ξ ν ζζ” − − −
(cid:21)
(cid:2) (cid:3)
(Z isdefinedin(30))whichforBornterm(see(24))takeplacewithoutrestriction.
ν
The OSJF Fζ (̺, ik)ζ is introduced as two variable’s function analytic in the
κξ −
domain
̺,ζ;k,ζ : Re̺>0, ̺ [m,+ ), ζ = 1; Imk <µ , ik [m,+ ), ζ = 1
0
6∈ ∞ ± | | ± 6∈ ∞ ±
(cid:16) (41(cid:17))
which decompose the partial half-off-shell amplitude
Tζ”ζ(q,k;b2)ζ =Tζ”ζ(±)(q,k)
κξ b=0∓ik κξ
(cid:12)
(cid:12)
according to [22]: (cid:12)
k Lξ −1
Tζ”ζ(±)(q,k)= Fζ”(iq, ik)ζ Fζ”( iq, ik)ζ 2iFζ ( ik) . (42)
κξ q κξ − − κξ − − κξ ∓
(cid:18) (cid:19) h ih i
It means, that Jost function is simply related with OSJF:
Fζ ( ik, ik)ζ =Fζ ( ik). (43)
κξ ∓ − κξ ∓
However, inversion of the decomposition (42) now is not as straightforward as for
Schrodinger case [14]. One can see, that all mentioned above properties of OSJF
hold for the following ansatz:
Fζ (̺, ik)ζ Zζ (̺2,k2)ζ =Fζ ( ik)1 ∞ds2 s Lξ
κξ − − κξ κξ ∓ π k ·
′ ′ ′ Z0 (cid:16) (cid:17)
gζ ( is;̺)ζ Nζ(̺, is)ζ Tζ ζ(±)(s,k); (44)
· − ξ − κξ
′
ζX=±1
8
where we introduce the notation:
1−ξ 1−ξ
Wζ(ib)+m 2 ηζ(i̺) b 2
Nζ(̺,b)ζ = = ; Nζ(̺,̺)ζ =1. (45)
ξ "wζ(i̺)+m# "ηζ(ib) ̺# ξ
and meromorfic on two-sheet’s Remanian surface (41) unknown function of ̺2,
disappearing in difference (42) and satisfying to condition:
Zζ ( k2,k2)ζ =1. (46)
κξ − loc
(cid:12)
(cid:12)
Due to this condition the asatz (44) for ̺ =(cid:12) ik convert in accordance with (43)
∓
to general representationfor Jostfunction. The last follows directly from abstract
definition(1)withthehelpofknownreasoning[24],usingdecompositionofpartial
Green function (3) into Volterrian and separable parts (see Appendix), and the
relation for physical solution of radial Dirac equation (see bellow) which reads:
(κ=κ ),
ξ
′
V(r) 1 ∞ φζ (s,r)
ψ(±)ζ(k,r)= ds2 κ0 Tζ”ζ(±)(s,k). (47)
−ηζ(k) κV π 2wζ′(s)ηζ′(s) κ
Z0 ζ′X=±1
Substituting the partial LS-equation (which is a Fredholm-type equation)
Tζ”ζ(±)(q,k) Tζ”ζ(q,k)= 1 ∞ds2 gζ′( is; ik)ζ
κ − κ0 −π − ∓ ·
Z0 ζ′X=±1
Tζ”ζ′(q,s) Tζ′ζ(±)(s,k), (48)
· κ0 κ
to the right hand side of ansatz (44), and using its particular form for ̺ = ik
∓
(clf.(43)) in the first of appearing items, one has for this r.h.s. the expression:
Zζ ( k2,k2)ζHζζ(̺,k) Fζ ( ik)1 ∞ds2 s Lξ gζ′( is; ik)ζ
κξ − κξ − κξ ∓ π k − ∓ ·
Z0 (cid:16) (cid:17) ζ′X=±1
′ ′ ′
Tζ ζ(±)(s,k) Hζζ (̺,s) Nζ( ik, is)ζ Hζζ(̺,k) ; (49)
· κξ κξ − ξ − − κξ
h i
where the auxiliary kernel is introduced:
Hζζ(̺,k)= 1 ∞ds2 s Lξ gζ′( is;̺)ζNζ(̺, is)ζ′ Tζ′ζ(s,k). (50)
κξ π k − ξ − κξ0
Z0 (cid:16) (cid:17) ζ′X=±1
The relation which following [21] from formula (70) (see bellow) for
Re j > 1 a ; T(u̺ν)= u2+̺2 ν2 /2u̺:
N
− − | −
∞ (cid:0)dα Pa(T(u(cid:1)αν)) u j
j | =
Zu+ν [∆(u2,α2,ν2)]a/2(α2+k2)(cid:16)α(cid:17)
∞ ds2 Qa(Z(sk ν))e−iπa s j
= j | ; (51)
Z0 2πk[∆(s2,k2,−ν2)]a/2(s2+u2)(cid:16)k(cid:17)
9
and easily verifying formulae
′ 1(1−ξ)
1 Wζ(k) wζ (s) m 2
1+ ± =1;
ζ′X=±12(cid:18) wζ′(s)(cid:19) Wζ(k)±m!
′ 1(1−ξ)
1 1+ Wζ(k) wζ (s)±m 2 ηζ′(s) ±1 = k ξ ηζ(k) ±1; (52)
ζ′X=±12(cid:18) wζ′(s)(cid:19) Wζ(k)±m! h i (cid:18)s(cid:19) (cid:2) (cid:3)
allow to rewrite the auxiliary kernel (50) as:
Hζζ(̺,k)= ∞ dα ̺ Lξ gζ′(α; ik)ζNζ′(α, ik)ζKζ′ζ(α,̺); (53)
κξ α − ξ − κξ
Z̺+µ0 (cid:16) (cid:17) ζ′X=±1
where the new Volterrian kernels are introduced for the case (8a):
4π u−̺ Σ(N)(ν)
K (u,̺)= dνPa(T(u̺ν)) ; (54)
j ΩNπa Zµ0 j | [∆(u2,̺2,ν2)]a/2
′ iu 1
Kζ ζ(u,̺)= K (u,̺)+ηζ(i̺) K (u,̺) . (55)
κξ 2m ηζ′(iu) Lξ L−ξ
(cid:20) (cid:21)
Here for the case (8b) the second term has opposite sign (clf. remark after (30))
and the branch wζ(p) takes value at p = i̺+0,̺ > m : wζ(i̺) iζ ̺2 m2.
⇒ −
This choice is conventionaland does not affect on sum over the sheets ζ = 1, for
p ±
whichkinematicalcuts ̺>mdisappears. Substitutionofthe(53)andrepeating
±
use of ansatz (44) under the α-integral, converts the relations (44), (49) to the
following Volterra-type equation for OSJF:
Fζ (̺, ik)ζ Zζ (̺2,k2)ζ = ∞ dα ̺ Lξ gζ′(α; ik)ζ
κξ − − κξ α − ·
Z̺+µ0 (cid:16) (cid:17) ζ′X=±1
′ ′ ′
Kζ ζ(α,̺) Fζ (α, ik)ζ Zζ (α2,k2)ζ +Nζ (α, ik)ζ . (56)
· κξ κξ − − κξ ξ −
(cid:20) (cid:21)
A natural choice of OSJF’s normalization now is given by the relation:
Zζ (̺2,k2)ζ =Nζ(̺, ik)ζ, (57)
κξ lok.nonsin. ξ −
(cid:12)
(cid:12)
where function in the right hand s(cid:12)ide obviously satisfy to all conditions (41),(46)
written out for the left one. It transforms the equation (56) for b = ik to the
−
following form:
Fζ (̺,b)ζ Nζ(̺,b)ζ = ∞ du ̺ Lξ gζ′(u;b)ζ
κξ − ξ u ·
Z̺+µ0 (cid:16) (cid:17) ζ′X=±1
′ ′
Fζ (u,b)ζ Kζ ζ(u,̺).
′κξ ′ κξ (58)
·( Nζ (u,b)ζ aζ ζ(u,̺;b2)ζ.
ξ κξ
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