Characterization of the QWN-conservation operator and applications

Abstract

Based on the finding that the quantum white noise (QWN) conservation operator is a Wick derivation operator acting on white noise operators, we characterize the aforementioned operator by using an extended techniques of rotation invariance operators in a first place. In a second place, we use a new idea of commutation relations with respect to the QWN-derivatives. Eventually, we use the action on the number operator. As applications, we invest these results to study three types of Wick differential equations.

Full text

Chaos, Solitons and Fractals 84 (2016) 41–48
Contents lists available at ScienceDirect
Chaos, Solitons and Fractals
Nonlinear Science, and Nonequilibrium and Complex Phenomena
journal homepage: www.elsevier.com/locate/chaos
Characterization of the QWN-conservation operator and
applications
Hafedh Rguigui∗
Department of Mathematics, High School of Sciences and Technology of Hammam Sousse, University of Sousse, Rue Lamine Abassi, 4011
Hammam Sousse, Tunisia
article info
Article history:
Received 5 June 2015
Accepted 28 December 2015
Keywo rds:
Space of entire function
QWN-conservation operator
Number operator
Wick derivation
abstract
Based on the finding that the quantum white noise (QWN) conservation operator is a Wick
derivation operator acting on white noise operators, we characterize the aforementioned
operator by using an extended techniques of rotation invariance operators in a first place.
In a second place, we use a new idea of commutation relations with respect to the QWN-
derivatives. Eventually, we use the action on the number operator. As applications, we in-
vest these results to study three types of Wick differential equations.
© 2016 Elsevier Ltd. All rights reserved.
1. Introduction
The number operator (called also conservation opera-
tor) was initiated by Piech in [16] as an infinite dimen-
sional analogue of a finite dimensional Laplacian on an ab-
stract Wiener space. Such operator is very important in
quantum physics because it coincides with the Hamilto-
nian of the harmonic oscillator. It has been extensively
studied in [12,13] and the references cited therein. In par-
ticular, Kuo [11] formulated the number operator as con-
tinuous linear operator acting on the space of test white
noise functionals. Let S(R)and S(R)be the Schwarz
space consisting of rapidly decreasing C∞-functions and
the space of the tempered distributions, respectively. Based
on the test function space Fθ(S
C(R)) of holomorphic func-
tions with θ-exponential growth and its topological dual
space F∗
θ(S
C(R)),see for more details [6], the number op-
erator Nhas the following expressions:
N=R2
τ(s,t)a∗
satdsdt =R
a∗
sasds (1)
∗Tel.: +21 697810448.
E-mail address: hafedh.rguigui@yahoo.fr,hafedhrguigui@yahoo.fr
where atand a∗
tare the annihilation operator and creation
operator at the point t∈Rand τis the usual trace. In [7],
using the Fock expansions of operators in terms of integral
kernel operators and rotation-invariance, the characteriza-
tion theorem for number operator is given.
Later on, in Ref. [15], by using the new idea of QWN-
derivatives pointed out by Ji–Obata in [8], for continuous
linear operators B1and B2from S
C(R)into itself, the QWN-
analogous NQ
B1,B2is introduced as QWN counterpart of the
conservation operator in (1).WeprovedthatNQ
B1,B2has
functional integral representations in terms of the QWN-
derivatives {D−
t,D+
t;t∈R}and a suitable Wick product 
on the class of white noise operators. Moreover, for B1=
B2=I,the QWN-conservation operator NQis given by
NQ=R2
τ(s,t)(D+
s)∗D+
tdsdt +R2
τ(s,t)(D−
s)∗D−
tdsdt
(2)
=R
(D+
s)∗D+
sds +R
(D−
s)∗D−
sds.(3)
In this paper, our goal is to characterize the QWN-
conservation operator NQby using of the same extended
idea of rotation invariant operator, the chaos decomposi-
tion and the notion of Wick derivation on operators. Then,
http://dx.doi.org/10.1016/j.chaos.2015.12.023
0960-0779/© 2016 Elsevier Ltd. All rights reserved.

42 H. Rguigui / Chaos, Solitons and Fractals 84 (2016) 41–48
we use the number operator to characterize such opera-
tors. Finally, by means of the QWN-derivatives {D−
t,D+
t;t∈
R},we characterize NQ.
The paper is organized as follows. In Section 2,we
briefly recall well-known results on nuclear algebra of
entire holomorphic functions and the Fock expansion of
white noise operators. In Section 3, we characterize the
Wick derivation. In Section 4, we characterize the QWN-
conservation operator NQ.InSection 5,weusethesechar-
acterization theorems to study three types of Wick differ-
ential equations.
2. Preliminaries
Let L2(R)be the real Hilbert space of square integrable
functions on Rwith norm |·|0. It is well known (see Refs.
[12,13]) that the Schwarz space S(R)and its topological
dual space S(R)can be reconstructed using L2(R)and
the harmonic oscillator A=1+t2−d2/dt2. More precisely,
they are given by
S(R)=projlim
p→∞
Sp(R),S(R)=indlim
p→∞
S−p(R),
where, for p≥0, Sp(R)is the completion of S(R)with
respect to the norm |.|p=|Ap.|0and S−p(R)is the topo-
logical dual space of Sp(R). Denoting by SC,Sp,C,S−p,C
and S
Cthe complexifications of S(R),Sp(R),S−p(R)and
S(R),respectively, where the complexification of any real
locally convex space Xis given by
XC=X+iX.
Throughout the paper, we fix a Young function θand θ∗
its polar function given by
θ∗(x)=sup
t≥0
(tx −θ(t)),x≥0,
see Refs. [6] and [14]. The test space Fθ(S
C)and the space
Gθ∗(SC⊕SC)are defined as follows:
Fθ(S
C):=projlim
p→∞;m↓0f∈H(S−p,C);
sup
z∈S−p,C
|f(z)|e−θ(m|z|−p)<∞,
Gθ∗(SC⊕SC):=ind lim
p→∞;m,m→0f∈H(Sp,C⊕Sp,C);
sup
ξ,η∈Sp,C
|f(ξ,η)|e−θ∗(m|ξ|p)e−θ∗(m|η|p)<∞
where H(B)denotes the space of all entire functions on
a complex Banach space B,seeRefs.[2–4,6,14,15,17,18].In
all the remainder of this paper we denote by Fθthe test
space Fθ(S
C)and its topological dual space is denoted
by F∗
θ.
2.1. White noise operators
The space of continuous linear operators from a nuclear
space Xto another nuclear space Yis denoted by L(X,Y)
and assumed to carry the bounded convergence topology.
For z∈S
Cand ϕin Fθ,the holomorphic derivative of ϕat
x∈S
Cin the direction zis defined by
(a(z)ϕ)(x):=lim
λ→0
ϕ(x+λz)−ϕ(x)
λ.(4)
We can che ck that a(z)∈L(Fθ,Fθ)and a∗(z)∈L(F∗
θ,F∗
θ),
where a∗(z) is the adjoint of a(z)withrespecttothedu-
ality between F∗
θand Fθ.Now,ifz=δt∈S(R)we sim-
ply write atinstead of a(δt)andthepairatand a∗
tare
called the annihilation operator and creation operator at
the point t∈R.
It is well known that, for each ξ∈SC,the exponential
function
eξ(z):=ez,ξ,z∈S
C,
belongs to Fθand the set of such test functions spans a
densesubspaceofFθ. The Wick symbol of a white noise
operator ∈L(Fθ,F∗
θ)is by definition [13] aC-valued
function on SC×SCdefined by
ω()(ξ,η)= eξ,eη e−ξ,η,ξ,η∈SC.(5)
By a density argument, every operator in L(Fθ,F∗
θ)is
uniquely determined by its Wick symbol. Moreover, we
have the following characterization theorem.
Theorem 2.1 (See Ref. [10]).The Wick symbol map
yields a topological isomorphism between L(Fθ,F∗
θ)and
Gθ∗(SC⊕SC).
In Ref. [10], it is shown that Gθ∗(SC⊕SC)is closed
under pointwise multiplication. Then, for any 1,2∈
L(Fθ,F∗
θ),there exists a unique ∈L(Fθ,F∗
θ)such that
ω() =ω(1)ω(2). The operator will be denoted
12anditwillbereferredtoasthe Wick product of
1and 2. It is noteworthy that, endowed with the Wick
product ,L(Fθ,F∗
θ)becomes a commutative algebra. It is
a fundamental fact in QWN theory (see [13] and [10])that
every white noise operator ∈L(Fθ,F∗
θ)admits a unique
Fock expansion
=
∞

l,m=0
l,m(κl,m),(6)
where, for each pairing l, m ≥0, κl,m∈(S⊗(l+m)
C)
sym(l,m)
and l, m(κl, m ) is the integral kernel operator character-
ized via the Wick symbol transform by
ω(l,m(κl,m))(ξ,η)=κl,m,η⊗l⊗ξ⊗m,ξ,η∈SC.(7)
Which can be formally given by
l,m(κl,m)=Rl+mκl,m(s1,··· ,sl,t1,··· ,tm)
a∗
s1···a∗
slat1···atmds1···dsldt1···dtm.
In this way l, m(κl, m) can be considered as the operator
polynomials of degree l+massociated to the distribution
κl,m∈(S⊗(l+m)
C)
sym(l,m)as coefficient; and therefore every
white noise operator is a “function” of the annihilation op-
erators and the creation operators.
2.2. QWN-White noise operators
From Ref. [8], (see also Ref. [1]), we summarize the
novel formalism of QWN-derivatives. For ζ∈SC,then a(ζ)

H. Rguigui / Chaos, Solitons and Fractals 84 (2016) 41–48 43
extends to a continuous linear operator from F∗
θinto it-
self (denoted by the same symbol) and a∗(ζ)(restricted
to Fθ)is a continuous linear operator from Fθinto itself.
Hence, for any white noise operator ∈L(Fθ,F∗
θ),the
commutators
[a(ζ),]=a(ζ) −a(ζ),
[a∗(ζ),]=a∗(ζ) −a∗(ζ),
are well defined white noise operators in L(Fθ,F∗
θ).The
QWN-derivatives are defined by
D+
ζ=[a(ζ),],D−
ζ=−[a∗(ζ),].(8)
These are called the creation derivative and annihilation
derivative of , respectively. Note that, for z∈SC,the QWN-
derivatives D±
zand (D±
z)∗are continuous linear operators
from L(F∗
θ,Fθ)into itself and from L(Fθ,F∗
θ)into itself,
i.e.,
D±
z,(D±
z)∗∈L(L(F∗
θ,Fθ)) ∩L(L(Fθ,F∗
θ)).
Moreover, for z∈SCand ∈L(Fθ,F∗
θ),we have
(D+
z)∗=a∗(z),(D−
z)∗=a(z).
For more details, see [15].
The operator j, k, l, m(κ) is defined through two canon-
ical bilinear forms as follows:
 j,k,l,m(κ)S,T =κ, (D+
s1)∗···(D+
sj)∗(D−
t1)∗···
(D−
tk)∗D+
u1···D+
ulD−
v1···D−
vmS,T 
where S,T∈L(F∗
θ,Fθ)and for S=l,ml,m(sl,m)in
L(F∗
θ,Fθ)and T=l,ml,m(tl,m)in L(Fθ,F∗
θ),such that
tl,m∈(S
C)
⊗(l+m)and sl,m∈S
⊗(l+m)
C,the duality between
the two spaces L(F∗
θ,Fθ)and L(Fθ,F∗
θ),denoted by
., .,isdefinedasfollows
 T,S :=
∞

l,m=0
l!m!tl,m,sl,m.(9)
The operator j, k, l, m(κ) can be expressed as the following
integral:
j,k,l,m(κ)
=Rj+k+l+m
κ(s1,··· ,sj,t1,··· ,tk;u1,··· ,ul,v1,··· ,vm)
(D+
s1)∗···(D+
sj)∗(D−
t1)∗···(D−
tk)∗D+
u1···D+
ulD−
v1···D−
vm
ds1···dsjdt1···dtkdu1···duldv1···dvm.
We call j, k, l, m aQWN-integral operator with kernel distri-
bution κ. (See [5]). For any Q∈L(L(F∗
θ,Fθ),L(Fθ,F∗
θ))
there exists a unique family of distributions κj,k,l,m∈
(S⊗(j+k+l+m)
C)
sym(j,k,l,m),such that
QS=
∞

j,k,l,m=0
j,k,l,m(κj,k,l,m)S,S∈L(F∗
θ,Fθ),
where the right hand side converges in L(Fθ,F∗
θ).
3. Wick derivation
A quantum white noise operator D∈L(L(F∗
θ,Fθ),
L(Fθ,F∗
θ)) is called a Wick derivation (see [9])if
D(ST)=D(S)T+SD(T),S,T∈L(F∗
θ,Fθ).
Theorem 3.1. Let D∈L(L(F∗
θ,Fθ),L(Fθ,F∗
θ)).ThenDis a
Wick derivation if and only if it admits the following decom-
position
D=
∞

j,k=0
j,k,0,1(κj,k,0,1)+
∞

j,k=0
j,k,1,0(κj,k,1,0),(10)
where, κj,k,0,1,κj,k,1,0∈(S
C)
⊗j+k+1.
Proof. Dis a wick derivation if and only if
D(ST)=D(S)T+SD(T),S,T∈L(F∗
θ,Fθ).(11)
Let the operator Sa, b (for a,b∈SC)begivenby
Sa,b≡
∞

l,m=0
l,m(κl,m(a,b)) ∈L(Fθ,F∗
θ),
where κl,m(a,b)=1
l!m!a⊗l⊗b⊗m. Then, by the density of
{Sa,b,a,b∈SC}in L(F∗
θ,Fθ),Eq. (11) is equivalent to
 D(Sa,bSa1,b1),Sc,d = D(Sa+a1,b+b1),Sc,d
= D(Sa,b)Sa1,b1,Sc,d
+ Sa,bD(Sa1,b1),Sc,d .(12)
Therefore, for D=∞
j,k,l,m=0j,k,l,m(κj,k,l,m),Eq. (12) is
equivalent to
∞

j,k,l,m=0
κj,k,l,m,c⊗j⊗d⊗k⊗(a+a1)⊗l⊗(b+b1)⊗m
=
∞

j,k,l,m=0
κj,k,l,m,c⊗j⊗d⊗k⊗(a⊗l⊗b⊗m+a⊗l
1⊗b⊗m
1).
(13)
If κj, k, l, m = 0forallj, k, l, m ≥0. Then to obtain (13)
it must be m+l=1,i.e., (m=0andl=1) or (m=1and
l=0). Hence we show the necessity. By reversing the rea-
soning we get the sufficiency. 
As examples, the QWN-derivatives D±
zand the QWN-
conservation operator NQare Wick derivation, (see Ref.
[15]).
4. The characterization of the QWN-conservation
operator
Let O(S(R),L2(R)) given by (see [13])
O(S(R),L2(R)) ={B∈GL(S(R));|Bξ|0=|ξ|0∀ξ∈S(R)},
which is called infinite dimensional rotation group. We say
that a continuous operator from L(F∗
θ,Fθ)into L(Fθ,F∗
θ)
is rotation-invariant if
(Q(B))∗QQ(B)=Q,∀B∈O(S(R),L2(R)),(14)
where Q(B)isgivenby
Q(B) =
l,m
l,m(B⊗(l+m)κl,m),
for =l,ml,m(κl,m)in L(F∗
θ,Fθ).Formoredetailssee
[5]. By definition, if Qis rotation-invariant, so is (Q)∗.
We recall that, (see [13]), F∈(S⊗n
C)is rotation-invariant
if (B⊗n)∗F=Ffor all B∈O(S(R),L2(R)).LetQ∈

44 H. Rguigui / Chaos, Solitons and Fractals 84 (2016) 41–48
L(L(F∗
θ,Fθ),L(Fθ,F∗
θ)) and Q=∞
j,k,l,m=0j,k,l,m(κ),
where κ∈(S⊗(j+k+l+m)
C)
sym(j,k,l,m).ThenQis rotation-
invariant if and only if κis rotation invariant, see [5].
Theorem 4.1. Let P ∈L(L(F∗
θ,Fθ),L(Fθ,F∗
θ)).ThenPis
equal to NQup to a constant factor if and only if
•P is Wick derivation
•P=P∗
•P is rotation-invariant.
Proof. The necessity is obvious (see also [15]). We
need only to show the sufficiency. To this end let P=
∞
j,k,l,m=0j,k,l,m(κj,k,l,m)in L(L(F∗
θ,Fθ),L(Fθ,F∗
θ)) and
put Pj,k,l,m=j,k,l,m(κj,k,l,m). Using Theorem 3.1,wede-
duce that for all (l, m)= (0, 1) and (l, m)= (1, 0), Pj,k,l,m=
0. From the condition P=P∗,we get
Pj,k,l,m=P∗
l,m,j,k,∀j,k,l,m≥0.
Then, for (j, k)= (0, 1) and (j, k)= (1, 0), we get
Pj,k,l,m=P∗
l,m,j,k=0.
Therefore, from (10),wededucethat
P=P0,1,0,1+P1,0,1,0
=0,1,0,1(κ0,1,0,1)+1,0,1,0(κ1,0,1,0).
But from the above discussions, we know that j, k, l, m(κ)
is rotation invariant (for κ∈(S⊗(j+k+l+m)
C)) if and only if κ
is rotation invariant. Moreover, if j+k+l+m=2p,then
κis a linear combination of (τ⊗p)σfor σ∈Sj+k+l+m,(see
[5] for more details). From which we deduce that κ1, 0, 1, 0
and κ0, 1, 0, 1 are linear combinations of τ. This gives the
desired statement. 
Theorem 4.2. P∈L(L(F∗
θ,Fθ),L(Fθ,F∗
θ)) is equal to NQif
and only if
•P is Wick derivation
•P(N)=2N.
Proof. We know that NQis a Wick derivation. Moreover,
we have
NQ(N)=R4
τ(s,t)(D+
s)∗D+
t(τ(u,v)a∗
uav)dsdtdudv
+R4
τ(s,t)(D−
s)∗D−
t(τ(u,v)a∗
uav)dsdtdudv
=R4
τ(s,t)a∗
sD+
t(a∗
uav)τ(u,v)dsdtdudv
+R4
τ(s,t)asD−
t(a∗
uav)τ(u,v)dsdtdudv.
But, we know that
a∗
uav=a∗
uav∀u,v∈R
then, we get
D+
t(a∗
uav)=D+
t(a∗
u)av+a∗
uD+
t(av)
=δt(u)av(15)
and similarly
D−
t(a∗
uav)=D−
t(a∗
u)av+a∗
uD−
t(av)
=δt(v)a∗
u(16)
where δt(v)isequalto1ift=vand equal to zero if t= v.
Then, we obtain
NQ(N)=R3
τ(s,t)τ(t,v)a∗
savdsdtdv
+R3
τ(s,t)τ(u,t)asa∗
udsdtdu
=R3
τ(s,t)τ(t,v)a∗
savdsdtdv
+R3
τ(s,t)τ(t,v)a∗
uasdudsdt
=21,1(τ
⊗1τ).
We know that the trace τbelongs to (S
C⊗SC)and to
(SC⊗S
C),moreover we have
τ
⊗1τ=
∞

i,j=1
(ei⊗ei)
⊗1(ej⊗ej)
=
∞

i,j=1
ej,eiei⊗ej
=
∞

j=1
ej⊗ej
=τ
where {en;n≥1} form an orthonormal basis for L2(R)and
each enis an element of S(R). Hence, we obtain
NQ(N)=21,1(τ)=2N
which shows the necessity. To show the sufficiency,
let P, a continuous linear operator from L(F∗
θ,Fθ)into
L(Fθ,F∗
θ)),be a Wick derivation and P(N)=2N.From
(10),Pis given by
P=
∞

j,k=0
j,k,1,0(κj,k,1,0)+
∞

j,k=0
j,k,0,1(κj,k,0,1).
Using the definition of the number operator, we get
j,k,0,1(κj,k,0,1)(N)
=Rj+k+3
κj,k,0,1(s1,...,sj,t1,...,tk,t)τ(u,v)
(D+
s1)∗···(D+
sj)∗(D−
t1)∗···(D−
tk)∗D−
t(a∗
uav)
ds1···dsjdt1dtkdt dudv
=Rj+k+3
κj,k,0,1(s1,...,sj,t1,...,tk,t)τ(u,v)
(D+
s1)∗···(D+
sj)∗(D−
t1)∗···(D−
tk)∗D−
t(a∗
uav)
a∗
s1··· a∗
sjat1···atka∗
uδt(v)
ds1···dsjdt1···dtkdt dudv
where we have used (16),then
j,k,0,1(κj,k,0,1)(N)
=Rj+k+2
κj,k,0,1(s1,...,sj,t1,...,tk,t)τ(u,v)
a∗
s1···a∗
sja∗
uat1···atkds1···dsjdt1···dtkdtdu
=j+1,k(κj,k,0,1
⊗1τ).
Similarly, we get
j,k,1,0(κj,k,1,0)(N)=j,k+1(κj,k,1,0
⊗1τ).

H. Rguigui / Chaos, Solitons and Fractals 84 (2016) 41–48 45
On the other hand, we have
κj,k,1,0
⊗1τ=
∞

i1,...,ij+k+1=1
∞

n=1
κj,k,1,0,ei1⊗···⊗eij+k+1
(ei1⊗··· ⊗eij+k+1)
⊗1(en⊗en)
=
∞

i1,...,ij+k+1=1
κj,k,1,0,ei1⊗···⊗eij+k+1
(ei1⊗··· ⊗eij+k)⊗∞

n=1
en,eij+k+1en
=
i1,...,ij+k+1
κj,k,1,0,ei1⊗···⊗eij+k+1ei1
⊗···⊗ eij+k+1
=κj,k,1,0.
Hence, we obtain
P(N)=
∞

j,k=0
j,k+1(κj,k,1,0)+
∞

j,k=0
j+1,k(κj,k,0,1).
Finally, by uniqueness of the integral kernel decomposition
of the white noise operator P(N)andbytheequality
P(N)=21,1(τ)
we get the following
κj,k,1,0=τif (j,k)=(1,0)
0 otherwise
κj,k,0,1=τif (j,k)=(0,1)
0 otherwise
Which implies that
P=1,0,1,0(τ)+0,1,0,1(τ).
This completes the proof. 
Theorem 4.3. P∈L(L(F∗
θ,Fθ),L(Fθ,F∗
θ)) is equal to the
QWN-conservation operator NQif and only if
•P is Wick derivation
•[P,(D+
x)∗]=(D+
x)∗,∀x∈SC
•[P,(D−
y)∗]=(D−
y)∗,∀y∈SC.
Proof. NQis a Wick derivation (see [15]). Using the inte-
gral representation (3) of NQ,weget
[NQ,(D+
x)∗]=R
a∗
sD+
s((D+
x)∗)ds
−R
(D+
x)∗(a∗
sD+
s())ds
=R
a∗
sD+
s(a∗
x)ds
−R
a∗
xa∗
sD+
s()ds.
Using the fact D+
sis Wick derivation, we obtain
[NQ,(D+
x)∗]=R
a∗
sD+
s(a∗
x)ds
=R
x(s)a∗
sds 
=a∗
x,∀∈L(F∗
θ,Fθ),∀x∈SC.
Similarly, we can verify that
[NQ,(D−
y)∗]=ay,∀∈L(F∗
θ,Fθ),∀y∈SC.
Conversely, let P∈L(L(F∗
θ,Fθ),L(Fθ,F∗
θ)) given
by j, k, l, mj, k , l, m(κj, k , l, m), where κj,k,l,m∈
(S⊗j+k+l+m
C)
sym(j,k,l,m)verifying: P is a Wick deriva-
tion, [P,(D+
x)∗]=(D+
x)∗and [P,(D−
y)∗]=(D−
y)∗.Weneed
to show that P=NQ. Using (10), then the following
equation
[P,(D+
x)∗]=a∗
x
is equivalent to
a∗
x=
∞

j,k=0
j,k,1,0(κj,k,1,0)(a∗
x)
+
∞

j,k=0
j,k,0,1(κj,k,0,1)(a∗
x)
−
∞

j,k=0
a∗
xj,k,1,0(κj,k,1,0)()
−
∞

j,k=0
a∗
xj,k,0,1(κj,k,0,1)().(17)
Using the fact that D±
sare Wick derivation, we get
j,k,1,0(κj,k,1,0)(a∗
x)
=Rj+k+1
κj,k,1,0(s1,...,sj,t1,...,tk,s)
(D+
s1)∗···(D+
sj)∗(D−
t1)∗···(D−
tk)∗D+
s(a∗
x)
ds1···dsjdt1···dtkds
=Rj+k+1
κj,k,1,0(s1,...,sj,t1,...,tk,s)a∗
s1··· a∗
sj
at1···atk(a∗
xD+
s() +D+
s(a∗
x))
ds1···dsjdt1···dtkds
=a∗
xj,k,1,0(κj,k,1,0)()
+Rj+k+1
κj,k,1,0(s1,...,sj,t1,...,tk,s)x(s)
(D+
s1)∗···(D+
sj)∗(D−
t1)∗···(D−
tk)∗()
ds1···dsjdt1···dtkds
=a∗
xj,k,1,0(κj,k,1,0)() +j,k,0,0(κj,k,1,0
⊗1x)()
and similarly, from the fact that D−
s(a∗
x)=0,we obtain
j,k,0,1(κj,k,0,1)(a∗
x) =a∗
xj,k,0,1(κj,k,0,1)().
Then, from (17), we obtain
a∗
x=
∞

j,k=0
j,k,0,0(κj,k,1,0
⊗1x)().
Which is equivalent to
ω(a∗
x)(ξ,η)ω()(ξ,η)
=
∞

j,k=0
ω(j,k,0,0(κj,k,1,0
⊗1x)())(ξ,η)

46 H. Rguigui / Chaos, Solitons and Fractals 84 (2016) 41–48
=
∞

j,k=0Rj+k+1
κj,k,1,0(s1,...,sj,t1,...,tk,s)x(s)
ω(a∗
s1)(ξ,η)···ω(a∗
sj)(ξ,η)ω(a∗
t1)(ξ,η)···ω(a∗
tk)(ξ,η)
ω()(ξ,η)ds1···dsjdt1···dtk.
This gives
x,ηω()(ξ,η)
=
∞

j,k=0
κj,k,1,0
⊗1x,η⊗j⊗ξ⊗kω()(ξ,η),
for all ξ,η∈SCand for all ∈L(F∗
θ,Fθ). Which yields
that κj,k,1,0=0∀j= 1, ∀k= 0andκ1,0,1,0
⊗1x=x. Then,
κ1,0,1,0
⊗1x,η=x,η∀x,η∈SC.
From which we get
κ1,0,1,0,η⊗x=τ,η⊗x∀x,η∈SC.
This gives κ1,0,1,0=τ. Then, we obtain
P=1,0,1,0(τ)+
∞

j,k=0
j,k,0,1(κj,k,0,1).
Similarly to the previous discussions, we get
P=
∞

j,k=0
j,k,1,0(κj,k,1,0)+0,1,0,1(τ).
By identification we obtain
P=1,0,1,0(τ)+0,1,0,1(τ)
=R2
τ(s,t)(D+
s)∗D+
tdsdt +R2
τ(s,t)(D−
s)∗D−
tdsdt
=NQ.
This gives the desired statement. 
5. Application to Wick differential equation
Let βbe a Young function satisfying the condition
lim sup
x→∞
β(x)
x2<+∞.
and put θ=(eβ∗−1)∗.Forϒ∈L(Fβ,F∗
β)the exponential
Wick denoted by wexp(ϒ)isdefinedby
wexp(ϒ ) =
∞

n=0
1
n!ϒn,
belongs to L(Fθ,F∗
θ),see [10]. In the following we study
some Wick differential equations for white noise operators.
Theorem 5.1. The unique solution of the following Wick dif-
ferential equation:
∂
∂tut+NQ(ut)=Nut
u0=N
(18)
is given by
ut=e−2tNwexp1
2(1−e−2t)N(19)
Proof. Applying the operator ∂
∂tto Eq. (19) we get
∂
∂tut=−2Nwex p1
2(1−e−2t)N
+e−2tNNwexp1
2(1−e−2t)N
=−2ut+e−2tNut.(20)
On the other hand, using the fact that NQis a Wick deriva-
tion (see Theorem 4.2), we get
NQ(ut)=e−2tNQ(N)wexp1
2(1−e−2t)N
+e−2tNNQwexp1
2(1−e−2t)N
But one can prove easily that
NQ(Nn)=nNQ(N)N(n−1).
Then, we obtain
NQ(ut)=e−2tNQ(N)wexp1
2(1−e−2t)N
+e−2t1
2(1−e−2t)NNQ(N)
wexp1
2(1−e−2t)N
Now, using Theorem 4.2,weget
NQ(ut)=2e−2tNwexp1
2(1−e−2t)N
+(1−e−2t)e−2tNNwexp1
2(1−e−2t)N
=2e−2tut+(1−e−2t)Nut.(21)
From (20) and (21), we obtain
∂
∂tut+NQ(ut)=Nut
which shows that utis solution of (18).Now,letutbe an
arbitrary solution of (18) and put
Ft=utwexp1
2(e−2t−1)N
then, we get
∂
∂tFt=∂
∂tutwexp1
2(1−e−2t)N
+ut(−e−2t)Nwexp1
2(1−e−2t)N
=∂
∂tutwexp1
2(1−e−2t)N
−NQwexp1
2(1−e−2t)Nut−NFt
=∂
∂tut−Nutwexp1
2(1−e−2t)N
−NQwexp1
2(1−e−2t)Nut.
Using (18) and Theorem 4.2, we obtain

H. Rguigui / Chaos, Solitons and Fractals 84 (2016) 41–48 47
∂
∂tFt=−NQ(ut)wex p1
2(1−e−2t)N
−NQwexp1
2(1−e−2t)Nut
=−NQ(Ft).
From which we deduce that
∂
∂tFt+NQ(Ft)=0.(22)
A simple calculus similar to the same used in [3] and [15],
one can show that the unique solution of (22) is given by
Ft=e−2tN
Then, we deduce that
ut=Ftwexp1
2(1−e−2t)N
=e−2tNwexp1
2(1−e−2t)N.
This completes the proof. 
Theorem 5.2. Let x ∈SC,then the unique solution of the fol-
lowing Wick differential equation
⎧
⎨
⎩
∂
∂tut+NQ(D+
x)∗(ut)=(D+
x)∗NQ(ut)+tut
u0=∈L(Fθ,F∗
θ)
is given by
ut=wexp{−ta∗(x)}wexpt
0
sds.(23)
Proof. Applying the operator ∂
∂tto (23) we get
∂
∂tut=−a∗(x)ut+tut
using the fact that (see [15]),
(D+
x)∗(T)=a∗(x)T
for all T∈L(Fθ,F∗
θ).Thenweget
∂
∂tut+(D+
x)∗(ut)=tut.
Hence, using Theorem 4.3, we obtain
∂
∂tut+NQ(D+
x)∗(ut)−(D+
x)∗NQ(ut)=tut.
This shows that utis solution of (5.2). To show the unique-
ness, let utbe an arbitrary solution of (5.2) and put
Ft=utwexp−t
0
sds.
Then, we get
∂
∂tFt=∂
∂tutwexp−t
0
sds−tut
wexp−t
0
sds.
Using (5.2) and Theorem 4.3, we obtain
∂
∂tFt=(−a∗(x)ut+tut)
wexp−t
0
sds−tFt
=−a∗(x)ut+tFt−tFt
=−a∗(x)ut
which has a unique solution given by
Ft=wexp{−ta∗(x)}.
Then,wededucethat
ut=Ftwexpt
0
sds
=wexp{−ta∗(x)}wexpt
0
sds.
Which completes the proof. 
Let G(x)=a∗(x)+a(x),where x∈S(R)such that
|x|0=1. It is well known that the operator G(x)hasa
standard Gaussian distribution in the vacuum state. G(x)is
called a quantum Gaussian random variable.
Theorem 5.3. Let P a wick derivation. Then wexp{G(x)} is
the solution (up to constant) of
P() =G(x)(24)
if and only if
[P,(D+
x)∗+(D−
x)∗]=(D+
x)∗+(D−
+)∗(25)
Proof. Let P a Wick derivation and suppose that
wexp{G(x)} is the solution of (24). Applying P to the
operator wexp{G(x)}, we get
P(wexp{G(x)})=P(G(x)) wexp{G(x)}
which can be shown using the fact that P is Wick deriva-
tion. By Identification, we get
P(G(x)) =G(x).
Then, using Theorem 3.1 and similar discussions to those
used in the proof of Theorem 4.3, one can obtain
P=NQ.
Then, using Theorem 4.3,weget
[P,(D+
x)∗]=(D+
x)∗
and
[P,(D−
x)∗]=(D−
x)∗.
From which we obtain
[P,(D+
x)∗+(D−
x)∗]=(D+
x)∗+(D−
x)∗.
Conversely, let P a wick derivation satisfying (25). Then, we
get
P(a∗(x)) −a∗(x)P() +P(a(x)) −a(x)P()
=G(x).
This implies that

48 H. Rguigui / Chaos, Solitons and Fractals 84 (2016) 41–48
a∗(x)P() +P(a∗(x)) −a∗(x)P()
+a(x)P() +P(a(x)) −a(x)P()
=G(x)
which gives
P(G(x)) =G(x)
from which we get
P=NQ.
Then, it is obvious that wexp{G(x)} is solution of (24) by
replacing P by NQ. Suppose that is an arbitrary solution
of (24) and let F given by
F=wexp{−G(x)}.
Then, we get
NQ(F)=NQ() wex p{−G(x)}
−NQ(G(x)) wexp{−G(x)}
=G(x)F−G(x)F
=0.
Then, F is equal to some constant c multiplied by I. From
which we obtain
cI =wexp{−G(x)}.
Hence, we get
=cwexp{G(x)}.
This completes the proof. 
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