Multichannel generalization of eigen-phase preserving supersymmetric transformations

0.4 MB·English

by  Andrey M. Pupasov-Maksimov

Multichannel generalization of eigen-phase
preserving supersymmetric transformations
January 18, 2013
3 Andrey M Pupasov-Maksimov 1,2
1 1 UniversidadeFederaldeJuizdeFora,JuizdeFora,MG,Brazil
0 2 OOOExpertEnergo,Moscow,Russia
2
E-mail: [email protected]
n
a
J Abstract. We generalize eigen-phase preserving (EPP) supersymmetric
(SUSY) transformations to N > 2 channel Schr¨odinger equation with equal
7
thresholds. It is established that EPP SUSY transformations exist only in the
1
case of even number of channels, N =2M. A single EPP SUSY transformation

provides an M(M −1)+2 parametric deformation of the matrix Hamiltonian
] withoutaffectingeigen-phaseshiftsofthescatteringmatrix.
h
p
-
h
at 1. Introduction
m
In this paper we study N channel radial Schr¨odinger equation with equal thresholds.
[
Suchequationmaydescribescatteringofparticleswithinternalstructure,forinstance,

1 spin [1, 2, 3]. Supersymmetric (SUSY) transformations allow analytical studies
v
of Schr¨odinger equation with a wide class of interaction potentials [4, 5, 6, 2].
9
In particular, the inverse scattering problem [8, 9] for two-channel Schr¨odinger
9
1 equation with equal thresholds may be treated by combined usage of single channel
4 SUSYtransformations[10]andeigen-phasepreserving(EPP)supersymmetric(SUSY)
. transformations[11]. Thesetransformationsconservetheeigenvaluesofthescattering
1
0 matrix and modify its eigenvectors (coupling between channels), in a contrast to
3 phase-equivalent SUSY transformations which do not modify scattering matrix at
1 all [12, 13, 14, 15].
:
v In [3] the two-channel neutron-proton potential was reproduced by a chain of
i SUSY transformations, where the coupled channel inverse scattering problem were
X
decomposed into the fitting of the channel phase shifts [10, 16] and the fitting of the
r
a mixing between channels. The fitting of the mixing between channels was provided
by the EPP SUSY transformations.
Thispaperextendsthetwo-channelEPPSUSYtransformationstohighernumber
of channels.
The paper is organized as follows. We first fix our notations and recall basics
of SUSY transformations [6, 17, 18, 19]. Given the explicit form of a second-order
SUSY transformation operator we study the physical sector of SUSY transformations
between real and symmetric Hamiltonians. We analyze the most general form of a
second-order SUSY transformation for the case of mutually conjugated factorization
energies. Then we discuss applications of SUSY transformations to the scattering
problems and calculate how S-matrix transforms.
Multichannel generalization of EPP SUSY transformations 2
We start section 3 re-examining the conservation of its eigenvalues in the
two-channel case. There is the following asymptotic condition for EPP SUSY
transformations. The term with first-order derivative in the operator of EPP SUSY
transformation vanishes at large distances.
To generalize EPP SUSY for arbitrary number of channels we study the matrix
equation which comes from this asymptotic condition. We find that EPP SUSY
transformations may exist for 2M-channels only and obtain their general form. A 4-
channel example explicitly shows how EPP SUSY transformations acts. We conclude
with a summary of the obtained results and discussions of possible applications.
2. Second order SUSY transformations
2.1. Definition of SUSY transformations
SUSY transformations of stationary matrix Schr¨odinger equation are well known
[17, 19]. In this subsection we just fix our notations.
Consider a family of matrix Hamiltonians H={H }
a
d2
H =−I +V (r), (1)
a Ndr2 a
where I is the N ×N identity matrix, V (r) is the N ×N real symmetric matrix
N a
potential. A multi-index a parameterizes a family of potentials V (r). A matrix
a
Hamiltonian defines the system of ordinary differential equations
H ϕ (k,r)=k2ϕ (k,r). (2)
a a a
on N ×N matrix functions ϕ (k,r).
a
A (polinomial) SUSY transformation of (2) is a map of solutions
L : ϕ (k,r)→ϕ (k,r)=L ϕ (k,r), (3)
ba a b ba a
provided by a differential matrix operator
dn dn−1 d
L =A +A +...+A +A , (4)
ba ndrn n−1drn−1 1dr 0
where A , j = 0,...,n, are some matrix valued functions. This differential matrix
j
operator obeys the intertwinning relation
L H =H L . (5)
ba a b ba
Theintertwinningrelation(5)definesboththeoperatorL andthetransformed
ab
Hamiltonian H . We will consider a family of Hamiltonians H[H ] = {H |L H =
b a b ba a
H L }relatedwiththegivenHamiltonianH bytransformationoperator(4). Inthe
b ba a
next subsection we present explicit form of the second order transformation operator
and the transformed Hamiltonian.
2.2. Second-order SUSY algebra of matrix Schr¨odinger equation
Given initial matrix Hamiltonian H , we choose two N ×N matrix solutions
0
H u =E u , j =1,2, (6)
0 j j j
with E , E called factorization constants. These functions determine a second order
1 2
operator L [u ,u ]:
20 1 2
L f(r)=(cid:2)I ∂2−V +E +(E −E )(w −w )−1(w −∂ )(cid:3)f(r), (7)
20 N r 0 1 2 1 1 2 1 r
Multichannel generalization of EPP SUSY transformations 3
where
w (r)=u(cid:48)(r)u−1(r), w2+w(cid:48) +E =V , j =1,2. (8)
j j j j j j 0
Functions w (r) are called superpotentials. We also introduce a second-order
j
superpotential
W (r)=(E −E )[w (r)−w (r)]−1. (9)
2 2 1 1 2
The more symmetric and compact form of formula (7) reads
(cid:20) (cid:18) (cid:19)(cid:21)
E +E w +w
L f(r)= −H + 2 1 +W 1 2 −∂ f(r). (10)
20 0 2 2 2 r
Operator L and Hamiltonian H obey the following algebra
20 0
L H =H L , H =H −2W(cid:48), (11)
20 0 2 20 2 0 2
L† L =(H −E )(H −E ), L L† =(H −E )(H −E ).(12)
20 20 0 1 0 2 20 20 2 1 2 2
The new (transformed) potential is expressed in terms of second-order
superpotential W as follows
2
V =V −2W(cid:48). (13)
2 0 2
The transformation operator has a global symmetry
L [u ,u ]=L [u (r)U ,u U ], detU (cid:54)=0. (14)
20 1 2 20 1 1 2 2 1,2
The second order SUSY transformations of an initial Hamiltonian H form the
0
family H [H ] = {H |L H = H L }. We will work only with Hamiltonians from
2 0 2 20 0 2 20
H [H ] and we omit subscripts in the notation of transformation operator L →L.
2 0 20
2.3. Restrictions to the SUSY transformations
We restrict our consideration only to second order SUSY transformations with
mutually conjugated factorization energies E = E∗ = E. Potentials V and V are
1 2 0 2
supposed to be real and symmetric. Hence the transformation functions u and u
1 2
have to be mutually conjugated, u = u∗ = u. The symmetry of V demands the
1 2 2
symmetry of superpotentials (8), wT = w = w, wT = w = w∗. Defining the
1 1 2 2
Wronskian of two matrix functions as
W[u ,u ](r)≡uT(r)u(cid:48)(r)−uT(cid:48)(r)u (r) (15)
1 2 1 2 1 2
=uT(r)(cid:2)w (r)−wT(r)(cid:3)u (r). (16)
1 2 1 2
we see that the symmetry of superpotential w implies a vanishing self-Wronskian
W[u,u]=0 of transformation functions [2].
Wepresentthesecond-ordersuperpotentialW intermsofthematrixWronskian
2
for further needs,
W (r)=(E −E )u (r)W[u ,u ]−1(r)uT(r). (17)
2 1 2 2 1 2 1
To specify acceptable choice of transformation solutions explicitly, we choose the
basisinthesolutionspace. Naturalbasisfortheradialproblem, r ∈(0,∞), isformed
by the Jost solutions f(±k,r) with the exponential asymptotic behavior
f(k,r →∞)→I eikr. (18)
N
Let us expand the transformation functions in the Jost basis
u(r)=f (−K,r)C +f (K,r)D, (19)
0 j 0
Multichannel generalization of EPP SUSY transformations 4
where K = k +ik , K2 = E, k > 0. Complex constant matrices C and D should
r i i
providevanishingself-wronskianW[u,u]=0. Thewronskianoftwosolutionswiththe
same k is a constant. For instance, W[f(−k,r),f(k,r)] = 2ikI . Then, calculating
N
W[u,u] we get a constraint on the possible choice of matrices C and D,
DTC =CTD. (20)
Matrices C and D have an ambiguity due to symmetry (14). Rank of matrix
C, rankC = M ≤ N, determines the structure of transformation operator. The sum
of ranks rankC + rankD ≥ N, otherwise operator L is undefined. Using (14) we
may transform C to the form, where only first M columns are non-zero and linearly
independent. Reorderingchannels(bypermutationsofrowsinthesystemofequations
(2))wecanputnontrivialM×M minorofC intotheupperleftcorner. Then,C and
D obey the following canonical form,
(cid:18) I 0 (cid:19) (cid:18) X −QT (cid:19)
C = M , D = , (21)
Q 0 0 I
N−M
whereX =XT isasymmetricM×M complexmatrix,andQis(N−M)×M complex
matrix. This canonical form is a gauge which fixes ambiguity (14) of transformation
solutions.
2.4. Application to the scattering theory
In concrete physical applications of SUSY transformations we may further restrict
the class of Hamiltonians. In particular, in scattering theory [1] we work with the
radial problem, r ∈ (0,∞). The interaction potentials decrease sufficiently fast at
large distances and may contain centrifugal term
lim r2V(r)=l(l+I ), l=diag(l ,...,l ), eilπ =±I . (22)
N 1 N N
r→∞
The physical solution has the following asymptotic behavior
ψ(k,r →∞)∝k−1/2(cid:2)e−ikreilπ2 −eikre−ilπ2S(k)(cid:3), (23)
where matrix coefficient S(k) is the scattering matrix.
Scattering matrix is related with the Jost matrix
S(k)=eilπ2F(−k)F−1(k)eilπ2 , (24)
where the Jost matrix reads
F(k)= lim(cid:2)fT(k,r)rν(cid:3)[(2ν−1)!!]−1. (25)
r→0
Diagonal matrix ν indicates the strength of the singularity in the potential near the
origin
V(r →0)=ν(ν+I )r−2+O(1). (26)
N
Knowledge of the Jost solutions allows one to define scattering matrix. Super-
symmetrictransformationsofHamiltonianandsolutionsinducethetransformationof
scattering matrix. Formal approach to the calculations of the S-matrices was devel-
oped in the work of Amado [20].
Let us consider how the Jost solution transforms asymptotically,
(Lf )(k,r →∞)= (27)
0
(cid:20) (cid:18) (cid:19) (cid:21)
E +E w +w
= −k2+ 2 1 + W 1 2 (r →∞)−ikW (r →∞) exp(ikr).
2 2 2 2
Multichannel generalization of EPP SUSY transformations 5
Assume that there exists the following limit
(cid:20) (cid:21)
E +E w +w
U (k)= lim −k2+ 2 1 +W 1 2 −ikW . (28)
∞ r→∞ 2 2 2 2
Then the transformed Jost solution reads
f (k,r)=(Lf )(k,r)U−1(k), (29)
2 0 ∞
Making similar manipulations with the physical solution (23) we establish the
form of transformed S-matrix
S2(k)=eilπ2U∞(k)e−ilπ2S0(k)e−ilπ2U∞−1(k)eilπ2 . (30)
In the case of our second-order SUSY transformation, the transformed S-matrix
depends on the factorization energy E and parameters Q, X through the matrix
multipliersU (k)andU (k)−1. Ingeneral,thisdependencemaybeverycomplicated.
∞ ∞
Moreover,thescatteringmatrixS mayhaveunphysicallowandhighenergybehavior.
2
SUSY transformations that deform the scattering matrix in a simple way are
useful tools to solve inverse scattering problem. In the two-cannel case there is a
special kind of deformation, when U (k) becomes an orthogonal matrix [11]. We call
∞
such deformations as eigen-phase preserving transformations.
3. Eigen-phase preserving SUSY transformations
3.1. Two channel case
Letusanalyzeconditionsthatmakeatwo-channelSUSYtransformationbeaneigen-
phase preserving one [11]. In this case parameters of the transformation, Q=q, X =
x, are just some numbers. Matrix U (k) depends on q only and becomes orthogonal
∞
when q = ±i. The determinant of u vanishes at large distances, detu(r → ∞) → 0
with such choice of q. Let detu(r → ∞) (cid:39) (cid:15), then superpotential w diverges as
w(r → ∞) (cid:39) (cid:15)−1 and two-fold superpotential W vanishes as w (r → ∞) (cid:39) (cid:15). As a
2 2
result, the limit (28) contains only even powers of k
(cid:20) (cid:21)
E +E w +w
U (k)= lim −k2+ 2 1 +W 1 2 . (31)
∞ r→∞ 2 2 2
The cancelation of odd powers of k is a necessary condition to provide EPP SUSY
transformations. In the next subsection we establish the most general form of matrix
Q which leads to the vanishing limit
lim W =0, (32)
2
r→∞
for the case N >2.
ParameterxisalsoshouldbefixedtoprovidedetW[u,u∗](cid:54)=0forallr >0which
leads to a finite V .
2
3.2. Asymptotic SUSY transformation at large distances for arbitrary N
The transformation function u (19) has the following asymptotic behaviour at large
distances
u(r →∞)→u (I +Λr−1+o(r−1)), u =Ae−iKrΣ, (33)
∞ N ∞
Multichannel generalization of EPP SUSY transformations 6
where
(cid:18) I −QT (cid:19) (cid:18) I 0 (cid:19)
A= M , Σ = M . (34)
Q I M,N−M 0 −I
N−M N−M
For each concrete EPP transformation N and N −M are fixed, therefore we will use
notation Σ instead of Σ .
M,N−M
The two-fold superpotential behaves asymptotically as
lim W =W =(E −E∗)u∗ W[u ,u∗ ]−1uT (35)
2 2,∞ ∞ ∞ ∞ ∞
r→∞
=2iE A∗eiK∗rΣW[u ,u∗ ]−1e−iKrΣAT. (36)
Im ∞ ∞
Using asymptotic (33) we see that this limit is a constant matrix
W =2iE A∗eiK∗rΣ(cid:104)uT (u∗ )(cid:48)−(cid:0)uT (cid:1)(cid:48)u∗ (cid:105)−1e−iKrΣAT (37)
2,∞ Im ∞ ∞ ∞ ∞
=2iE A∗eiK∗rΣ(cid:104)e−iKrΣATA∗(cid:16)eiK∗rΣ(cid:17)(cid:48)−(cid:0)e−iKrΣ(cid:1)(cid:48)ATA∗eiK∗rΣ(cid:105)−1e−iKrΣAT
Im
=2E A∗(cid:2)K∗ATA∗Σ+KΣATA∗(cid:3)−1AT .
Im
We introduce auxiliary matrix W
∞
(cid:18) k (I +QTQ∗) ik (QT −(Q∗)T) (cid:19)
W :=K∗ATA∗Σ+KΣATA∗ =2 r M i . (38)
∞ ik (Q−Q∗) −k (QQ†+I )
i r N−M
Limit (32) leads to the following matrix equation
W =0⇒A∗W−1AT =0, detW (cid:54)=0, (39)
2,∞ ∞ ∞
which provides asymptotic cancelation of k in (28). In the two channel case these
equations fix Q uniquely. When N >2, these equations determine a set of Q values.
Equation (39) may be satisfied if and only if matrix A is singular. Matrix W
∞
is invertible, rankW = N. Let rankA = n, then rankA∗ = rankAT = n and
∞
rank(W−1AT) = dimImg(W−1AT) = n. The dimension of kernels dimKerA =
∞ ∞
dimKerA∗ = dimKerAT = N − n. Equation (39) implies that Img(W−1AT) ⊂
∞
KerA∗,hencen≤N−n. Thereforeequation(39)hassolutionsonlyifn≤ 1N. From
2
the other hand, from explicit form of matrix A, (34), its rank n ≥ max(M,N −M).
That is, (39) has solutions if and only if
N
rankA= , N =2M. (40)
2
From here it follows that for odd number of channels equation (39) has no solutions.
Consider 2M ×2M matrix A
(cid:18) I −QT (cid:19)
A= M , (41)
Q I
M
with rankA=M. Two its rectangular sub matrices have the same rank
(cid:18) I (cid:19) (cid:18) −QT (cid:19)
rank M =rank =M. (42)
Q I
M
We can take first M columns of A as linearly independent, then from (40), (41) and
(42) follows that there exists M ×M matrix Z, such that
(cid:18) I (cid:19) (cid:18) −QT (cid:19)
M Z = =M. (43)
Q I
M
Solving this equation we obtain Z =−QT and QQT =−I .
M
Multichannel generalization of EPP SUSY transformations 7
Let us extract i from Q,
Q=±iB, BTB =BBT =I , (44)
M
andsubstituteQinthisforminto(39). FirstofallweinvertmatrixW . Thismatrix
∞
can be factorized in two ways
1
W =
2 ∞
(cid:18) k BT(B+B∗) −k (BT +B†) (cid:19) (cid:18) k (B†+BT)B∗ −k (BT +B†) (cid:19)
r i = r i =
−k (B+B∗) −k B(B†+BT) −k (B+B∗) −k (B+B∗)B†
i r i r
(cid:18) k BT −k I (cid:19)(cid:18) (B+B∗) 0 (cid:19)
r i M =
−k I −k B 0 (B†+BT)
i M r
(cid:18) (B†+BT) 0 (cid:19)(cid:18) k B∗ −k I (cid:19)
r i M .
0 (B+B∗) −k I −k B†
i M r
We note that
(cid:18) (B†+BT) 0 (cid:19)(cid:18) (B+B∗) 0 (cid:19)
W W∗ =4(k2+k2) . (45)
∞ ∞ r i 0 (B+B∗) 0 (B†+BT)
Therefore the inverse matrix reads
1 (cid:18) k B† −k I (cid:19)(cid:18) (BT +B†)−1 0 (cid:19)
W−1 = r i M = (46)
∞ 2|K|2 −kiIM −krB∗ 0 (B+B∗)−1
1 (cid:18) k B†(BT +B†)−1 −k (B+B∗)−1 (cid:19)
r i
2|K|2 −k (BT +B†)−1 −k B∗(B+B∗)−1
i r
Let us introduce notations for auxiliary matrices
(cid:18) (BT +B†) 0 (cid:19)
B˜ = ,
0 (B+B∗)
(cid:18) k B† −k I (cid:19)
B = r i M .
k −k I −k B∗
i M r
Then W =E A∗B B˜−1AT/(k2+k2).
2,∞ Im k r i
Using the following matrix identities
BT(BT +B†)−1 =(B∗+B)−1B∗, B†(BT +B†)−1 =(B∗+B)−1B, (47)
(BT +B†)−1BT =B∗(B∗+B)−1, (BT +B†)−1B† =B(B∗+B)−1, (48)
(cid:18) (BT +B†)−1 0 (cid:19)(cid:18) I iBT (cid:19)
B−1AT = M = (49)
0 (B+B∗)−1 −iB I
M
(cid:18) I iB∗ (cid:19)(cid:18) (BT +B†)−1 0 (cid:19)
M ,
−iB† I 0 (B+B∗)−1
M
(cid:18) B† −iI (cid:19)
A∗B =K∗ M , (50)
k −iI −B∗
M
we see that
A∗B B˜−1AT =
k
(cid:18) B† −iI (cid:19)(cid:18) I iB∗ (cid:19)(cid:18) (BT +B†)−1 0 (cid:19)
K∗ M M =0.
−iI −B∗ −iB† I 0 (B+B∗)−1
M M
Multichannel generalization of EPP SUSY transformations 8
Thus (44) gives solutions of (39).
Nowwecancalculatetheasymptoticformofatransformationoperatorexplicitly.
To calculate asymptotic of W w we use the symmetry of superpotential w =u(cid:48)u−1 =
2
(uT)−1(uT)(cid:48),
lim W w =W w =2iE u∗ W[u ,u∗ ]−1uT (uT )−1(uT )(cid:48) =
2 2,∞ ∞ Im ∞ ∞ ∞ ∞ ∞ ∞
r→∞
−iE
−2iKE A∗W−1ΣAT = ImA∗B B˜−1ΣAT =
Im ∞ K∗ k
(cid:18) −iB† I (cid:19)(cid:18) (BT +B†)−1 0 (cid:19)
2E M
Im −I −iB∗ 0 (B+B∗)−1
M
Matrix W is real, therefore
2,∞
w +w∗
Ω=W ∞ ∞ =Re(W w )=
2,∞ 2 2,∞ ∞
(cid:18) i(BT −B†) 2I (cid:19)(cid:18) (BT +B†)−1 0 (cid:19)
2k k M
r i −2I i(B−B∗) 0 (B+B∗)−1
M
The matrix U defined in (28) reads
∞
U (k2)=(cid:0)−k2+k2−k2(cid:1)I +Ω, (51)
∞ r i N
Matrix Ω is real, orthogonal (up to a normalization), ΩTΩ = 4k2k2I , and
r i N
antisymmetricΩ=−ΩT. Toestablishitsorthogonalityandantisymmetryoneshould
use relations (47), (48). With these two properties of Ω the matrix U (k2) becomes
∞
proportional to the orthogonal matrix
U (k2)UT(k2)=(cid:0)(−k2+k2−k2)2+4k2k2(cid:1)I (52)
∞ ∞ r i r i N
ThatistheJostsolutionsatlargedistancesarerotatedbyorthogonalmatrixU
∞
(Lf)(k,r →∞)→U exp(ikr), (53)
∞
In this case the S-matrix transformation (30) is just an energy-dependent
orthogonal transformation,
S (k)=R (k2)S (k)RT(k2), (54)
2 S 0 S
with the orthogonal matrix, RTR =I ,
S S N
RS =eilπ2U∞e−ilπ2 (cid:2)(−k2+kr2−ki2)2+4kr2ki2(cid:3)−1/2 . (55)
That is we obtain desired generalization of two-channel EPP SUSY transformations.
The above analysis is valid for an arbitrary M × M symmetric matrix X.
Transformed S-matrix S depends on matrix Q only. Therefore X might provide
2
additional M(M +1)/2 parametric deformation of potential V without affecting the
2
S-matrix. From the other hand, possibility of such deformations contradicts to the
uniquenessoftheinversionofthecompletesetofscatteringdata. Therefore,theremay
exist only one matrix X corresponding to one physical potential V . The EPP SUSY
2
transformation should be uniquely determined by the factorization energy, M ×M
complex orthogonal matrix B and a sign factor. In the next subsection we show how
to fix matrix X and prove that the corresponding potential V is regular for all r >0.
2
Multichannel generalization of EPP SUSY transformations 9
3.3. Eigen-phase preserving SUSY transformation near the origin
To analyze the properties of EPP SUSY transformation in the vicinity of r = 0 we
will use the solution
i
ϕ (k,r)= [f (−k,r)F (k)−f (k,r)F (−k)], (56)
0 2k 0 0 0 0
vanishing at the origin
(cid:18) rν1+1 rνN+1 (cid:19)
ϕ (k,r →0)→diag ,..., , (57)
0 (2ν +1)!! (2ν +1)!!
1 N
where F (k) is the Jost matrix (25). We rewrite transformation solution in the basis
0
(ϕ (K,r),f (K,r)) expressing f (−K,r) from (56)
0 0 0
2K
f (−K,r)= ϕ (K,r)F−1(K)+f (K,r)F (−K)F−1(K), (58)
0 i 0 0 0 0 0
and substituting in (19)
2K
u(r)= ϕ (K,r)F−1(K)C+f (K,r)(D+s C). (59)
i 0 0 0 0
where
(cid:18) s sT (cid:19)
s0 =F0(−K)F0−1(K)= s1 s2 , eilπ2s0eilπ2 =S0. (60)
2 3
We can transform matrix (D+s C) to the form of matrix D (without affecting C)
0
multiplying u from the right
(cid:18) I 0 (cid:19) (cid:18) X˜ ∓iBT (cid:19)
(D+s C) M = , (61)
0 −(s ±is B) I 0 I
2 3 M M
where
X˜ =X+s ±i(sTB+BTs )−BTs B, (62)
1 2 2 3
Then the transformation solution reads
2K (cid:18) I 0 (cid:19) (cid:18) X˜ ∓iBT (cid:19)
u(r)= ϕ (K,r)F−1(K) M +f (K,r) . (63)
i 0 0 ±iB 0 0 0 I
M
Consider the case X˜ = 0. In this case the potential V is regular for all
2
r > 0. Let us prove this. According to the Wronskian representation of the
second-order superpotential W (16), the potential V will be regular if and only
2 2
if detW[u,u∗](r)(cid:54)=0.
The derivative of the Wronskian W[u,u∗] reads
W[u,u∗](cid:48)(r)=(E −E∗)uT(r)u∗(r). (64)
ByconstructionW[u,u∗]isananti-Hermitianmatrix, i.e.W[u,u∗]=−W†[u,u∗]. We
represent transformation solution in a block-diagonal form
(cid:18) (cid:19)
u u
u(r)= 11 12 , (65)
u u
21 22
with M ×M matrix blocks. When X˜ = 0 these blocks obey the following boundary
conditions: u (0) = 0, u (0) = 0, u (∞) = 0, u (∞) = 0. As a result the
11 21 12 22
Wronskian
(cid:18) uT uT (cid:19)(cid:18) u(cid:48)∗ u(cid:48)∗ (cid:19) (cid:18) u(cid:48)T u(cid:48)T (cid:19)(cid:18) u∗ u∗ (cid:19)
W[u,u∗]= 11 21 11 12 − 11 21 11 12 , (66)
uT uT u(cid:48)∗ u(cid:48)∗ u(cid:48)T u(cid:48)T u∗ u∗
12 22 21 22 12 22 21 22
Multichannel generalization of EPP SUSY transformations 10
hasvanishingblocksattheorigin,W˜ (0)=0,andatinfinityW˜ (∞)=0. Boundary
11 22
behavior of W˜ is not determined.
12
Now we can calculate diagonal blocks of the Wronskian integrating its derivative
W[u,u∗](cid:48)(r) (cid:18) W˜(cid:48) W˜(cid:48) (cid:19) (cid:18) uT u∗ +uT u∗ uT u∗ +uT u∗ (cid:19)
= 11 12 = 11 11 21 21 11 12 21 22 . (67)
(E −E∗) W˜(cid:48)† W˜(cid:48) uT u∗ +uT u∗ uT u∗ +uT u∗
12 22 12 11 22 21 12 12 22 22
Integration of (67) with the established boundary conditions yields
 r 
(cid:82) (cid:0)uT u∗ +uT u∗ (cid:1)dt W˜
11 11 21 21 12
W[u,u∗](r)=(E −E∗) 0 ∞ . (68)
 W˜† −(cid:82) (cid:0)uT u∗ +uT u∗ (cid:1)dt 
12 12 12 22 22
r
Assumethatthereisapointr wheredetW[u,u∗](r )=0. Hence,matrixW[u,u∗](r )
0 0 0
has at least one zero eigenvalue
W[u,u∗](r )(cid:126)v =0. (69)
0
Let us represent N dimensional eigen-vector (cid:126)v as two M dimensional vectors (cid:126)v and
u
(cid:126)v and rewrite (69) as a system of equations
d
W˜ (cid:126)v +W˜ (cid:126)v =0, (70)
11 u 12 d
W˜† (cid:126)v +W˜ (cid:126)v =0. (71)
12 u 22 d
The first term of the scalar product ((cid:126)v ,W˜ (cid:126)v )+((cid:126)v ,W˜ (cid:126)v ) = 0, (((cid:126)a,(cid:126)b) = a∗bj) is
u 11 u u 12 d j
positive
((cid:126)v ,W˜ (cid:126)v )=
u 11 u
(cid:90)r0 (cid:90)r0
((cid:126)v , dt(cid:0)uT u∗ +uT u∗ (cid:1)(cid:126)v )= ((u∗ (cid:126)v ,u∗ (cid:126)v )+(u∗ (cid:126)v ,u∗ (cid:126)v ))dt>0,
u 11 11 21 21 u 11 u 11 u 21 u 21 u
0 0
therefore ((cid:126)v ,W˜ (cid:126)v ) = n < 0 is real and negative. Now calculating scalar product
u 12 d u
((cid:126)v ,W˜† (cid:126)v )+((cid:126)v ,W˜ (cid:126)v )=0 with negative second term
d 12 u d 22 u
((cid:126)v ,W˜ (cid:126)v )=
d 22 d
∞ ∞
−((cid:126)v ,(cid:90) dt(cid:0)uT u∗ +uT u∗ (cid:1)(cid:126)v )=−(cid:90) (cid:16)(u∗ d(cid:126) ,u∗ (cid:126)v )+(u∗ d(cid:126) ,u∗ (cid:126)v )(cid:17)dt<0,
d 12 12 22 22 d 12 u 12 d 22 u 22 d
r0 r0
weobtainacontradiction,((cid:126)v ,W˜† (cid:126)v )=n∗ =n >0. Thiscontradictionprovesthat
d 12 u u u
Wronskian W[u,u∗] have only non-zero eigenvalues for all r >0. As a result W[u,u∗]
isinvertible,andhencebothW andV areregular(finite)forallr >0. Anynon-zero
2 2
X˜ will lead to the potential V which is singular in some point r .
2 0
This prove completes our construction of multi-channel EPP SUSY transforma-
tions. In the next subsection we present an illustrative example.