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2 2
— —
’ (ψ+ ) + W ψ+ ψ+ + W ψ+ (4.29)


where the dependence on (x, k) of ψ+ , ψ+ and W has been sup-
pressed. Simplifying the right hand side one ¬nds

1— — —
E(k)Ψ(x, x, k) = ’ ψ+ ψ+ + ψ+ (ψ+ ) + 2(ψ+ ) ψ+
4
d —
+2 (W ψ+ ψ+ )
dx
1d d — —
=’ (ψ+ ψ+ ) + 2W ψ+ ψ+
4 dx dx
1d d —
=’ + 2W (ψ+ ψ+ ) (4.30)
4 dx dx


© 2001 by Chapman & Hall/CRC
From (4.26) one thus has [6]
d 1d d
[K+ (x, x, β) ’ K’ (x, x, β)] = + 2W K+ (x, x, β)
dβ 4 dx dx
(4.31)
d∆β
The above identity greatly facilitates the computation of dβ . Indeed
inserting (4.31) in the right hand side of (4.23) we can project out
the β-dependence of ∆β in a manner

d∆β 1 d d
= dx + 2W K+ (x, x, β) (4.32)
dβ 4 dx dx
which can also be expressed, using (4.24), and (4.6a) as
d∆β 1 d +’
e’βEk
= dx 2Ek ψk ψk (4.33)
dβ 2 dx k =0

Let us consider now the solitomic example W (x) = tanh x cor-
responding to which the supersymmetric partner Hamiltonians are

1 d2 1
1 ’ 2 sec h2 x
H+ = ’ + (4.34a)
2
2 dx 2
1 d2 1
H’ = ’ + (4.34b)
2 dx2 2
If one employs periodic boundary conditions over the internal [’L, L]
the associated wave functions of H± turn out to be of the following
two types
1
ψ’ = N cos k1 x (4.35a)
N
1
ψ+ = {k1 sin k1 x + tanh x cos k1 x} (4.35b)
2
k1 +1
2
ψ’ = N sin k2 x (4.36a)
N
2
ψ+ = {’k2 cos k2 x + tanh x sin k2 x} (4.36b)
2
k2 +1
These wave functions are subjected to

k1 > 0 : k1 sin k1 L + tanh L cos k1 L = 0 (4.37)
k2 > 0 : sin k2 L = 0 (4.38)


© 2001 by Chapman & Hall/CRC
i i
which have been obtained from the considerations ψ± (L) = ψ± (’L),
i = 1, 2 and E(ki ) = 1 (1 + ki ), i = 1, 2. The energy constraints
2
2
i i
from the Schroedinger equations H+ ψ+ = E(ki )ψ± , i = 1 and 2. It
is worth mentioning that (4.35) and (4.36) are consistent with the
intertwining conditions (4.6).
d∆
We can now use the formula (4.33) to calculate dββ correspond-
ing to the wave functions (4.35) and (4.36). It is trivial to check that
d∆
as a result of the associated boundary conditions dββ = 0. [Note that
in addition to (4.35b) and (4.36b) H+ also possesses a normalizable
zero-energy state »sech2 x, » a constant, but it does not contribute
to the sum in (4.33)].
We now pass on the limit L ’ ∞ when one recovers the con-
1
tinuum states. Employing the usual normalization N ’ √π , the
d∆β
derivative is found to be

β 2
e’ 2 (k +1)

d∆β 1
dk √
= [cos kx {k sin kx + tanh x cos kx}
dβ 2π k2 + 1
0

1 + k 2 + sin kx {’k cos kx + tanh x sin kx}
x=+∞
1 + k2
x=’∞
’β ∞
1e βk2
2
dk [tanh x]x=+∞
e’
= 2
x’∞
2 2π 0
’β
1e 2

= (4.39)
2 2πβ
which, clearly, is β-dependent.
The above discussions give us an idea on the behaviour of a reg-
ulated Witten index. Actually the evaluation of the index depends
a great deal on the choice of the method adopted and in ¬nding a
suitable regularization procedure. Also the behaviour of ∆β depends
much on the nature of the spectrum; a purely continuous one extend-
ing to zero may yield fractional values of ∆β . In this connection note
that if we use the representation (4.32) we run into the problem of
determining exactly the heat kernels. There is also the related issue
of the viability of the interchange of the k and x limits of integra-
tion. For more on the anomalous behaviour of the Witten index and
its judicious computation using the heat kernel techniques one may
consult [6].


© 2001 by Chapman & Hall/CRC
4.5 Index Condition
We now analyze the Fredholm index of the annihilation operator b
de¬ned by [23,24]

δ ≡ dim ker b ’ dim ker b+ (4.40)

where b and b+ are, respectively, the annihilation and creation oper-
ators of the oscillator algebra given by (2.6) and (2.7). In (4.40) dim
ker corresponds to the dimension of the space spanned by the linearly
independent zero-modes of the relevant operator. Since b|0 >= 0 it
is obvious that dim ker b = {|0 >} while dim ker b+ is empty. Thus
δ = 1.
Apart from (4.40) we also have [24]

dim ker b+ b ’ dim ker bb+ = 1 (4.41a)

where dim ker b+ b represents the number of normalizable eigenkets
|ψn > obeying b+ b|ψn >= 0. To avoid a singular point in the index
relation it is useful to restrict dim ker b+ b < ∞. A deformed quantum
condition often leads to the existence of multiple vacua when δ may
become ill-de¬ned.
It is interesting to observe that for a truncated oscillator one
¬nds in place of (4.41a)

dim ker b+ bs ’ dim ker bs b+ = 0 (4.41b)
s s

where bs and b+ are the truncated vertions [25] of b and b+ de¬ned in
s
an (s + 1)-dimensional Fock space. To establish (4.41b) we transform
the eigenvalue equations

b+ bs φn = e2 φn (4.42a)
s n

to the form
bs b+ χn = e2 χn (4.42b)
s n

by setting χn = e1 bs φn with en = 0. We thus ¬nd the normal-
n
izability of the eigenfunctions φn and χn to go together. For a
¬nite-dimensional matrix representation we also have T r(b+ bs ) =
s
+ ). In this way we are led to (4.41b).
T r(bs bs
When applied to SUSYQM δ can be related to the Witten in-
dex which counts the di¬erence between the number of bosonic end


© 2001 by Chapman & Hall/CRC
fermionic zero-energy states. This becomes transparent if we focus
on (4.41a). Replacing the bosonic operators (b, b+ ) by (A, A+ ) ac-
cording to (2.34) in terms of the superpotential W (x), the left hand
side of (4.41a) just expresses the di¬erence between the number of
bosonic and fermionic zero eigenvalues. We thus have a correspon-
dence with (4.8).
In the following we study [26] the Fredholm index condition
(4.40) for the annihilation operator of the deformed harmonic os-
cillator and q-parabose systems in a generalized sense to show how
multiple vacua may arise. We also look into the singular aspect of δ
and point out some remedial measures to have an ambiguous inter-
pretation of δ.


4.6 q-deformation and Index Condition
Interest in quantum deformation seems to have started after the work
of Kuryshkin [27-30] who considered a q-deformation in the form
AA+ ’ qA+ A = 1 for a pair of mutually adjoint operators A and
A+ to study interactions among various particles. Later Janussis et
al. [31], Biedenharn [32], Macfarlane [33], Sun and Fu [34] and sev-
eral others [35-48] made a thorough analysis of deformed structures
with a view to inquiring into plausible modi¬cations of conventional
quantum mechanical laws. Recent interest in quantum deformation
comes from its link [49,50] with anyons and Chern-Simmons theo-
ries. The ideas of q-deformation has also been intensely pursued to
develop enveloping algebras [51] quasi-Coherent states [52], rational
conformal ¬eld theories [53] and geometries possessing non commu-
tative features [54].
Let us consider the following standard description of a q-deformed
harmonic oscillator
AA+ ’ qA+ A = q ’N , q ∈ (’1, 1) (4.42)
with A and A+ obeying
[N, A] = ’A, [N, A+ ] = A+ (4.43)
As the deformation parameter q ’ 1, A ’ b and we recover the
familiar bosonic condition bb+ ’ b+ b = 1 for the normal harmonic
oscillator.


© 2001 by Chapman & Hall/CRC
The operators (A, A+ ) may be related to the bosonic annihila-
tion and creation operators b and b+ by writing

A = ¦(N )b, A+ = b+ ¦(N ) (4.44)

where N is the number operator b+ b and ¦ any function of it.
Exploiting the eigenvalue equation ¦(NB )|n >== ¦(n)|n >, the
representations (4.44) lead to the recurrence relation

(n + 1)¦2 (n) ’ qn¦2 (n ’ 1) = q ’n (4.45)

The above equation has the solution

1 q n ’ q ’n
¦(n) = (4.46)
n + 1 q ’ q ’1

which implies that ¦(N ) is given by

[N + 1]
¦(N ) = (4.47)
N +1
with
q x ’ q ’x
[x] = (4.48)
q ’ q ’1
From (4.44) and (4.48) the Hamiltonian for the q-deformed har-
monic oscillator can be expressed as
ω
Hd = A, A+
2
ω
= {[N + 1] + [N ]} (4.49)
2
Note that the commutator [A, A+ } reads

A, A+ = [N + 1] ’ [N ] (4.50)

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