ArticlePDF Available

Abstract

In this paper we introduce a new notion of λ −order homogeneous operators on the nuclear algebra of white noise operators. Then, we give their Fock expansion in terms of quantum white noise (QWN) fields {at,at∗;t∈ℝ}{at,at;  tR}\{a_{t},\: a^{*}_{t}\, ; \; t\in \mathbb {R}\}. The quantum extension of the scaling transform enables us to prove Euler’s theorem in quantum white noise setting.
Mathematical Physics, Analysis and Geometry manuscript No.
(will be inserted by the editor)
Euler’s Theorem For Homogeneous White Noise
Operators
Abdessatar Barhoumi ·Hafedh Rguigui
Received: date / Accepted: date
Abstract In this paper we introduce a new notion of λorder homogeneous op-
erators on the nuclear algebra of white noise operators. Then, we give their Fock
expansion in terms of quantum white noise (QWN) fields {at, a
t;tR}. The
quantum extension of the scaling transform enables us to prove Euler’s theorem
in quantum white noise setting.
Keywords QWN-Euler operator, Euler’s Theorem, QWN-scaling operator, Homoge-
neous operator, QWN-derivatives.
Mathematics Subject Classification (2000) 60H40, 46A32, 46F25, 46G20.
1 Introduction and Preliminaries
Let Hbe the real Hilbert space of square integrable functions on Rwith norm
| · |0and E≡ S(R) be the Schwartz space consisting of rapidly decreasing C-
functions. Then, the nuclear Gel’fand triple
S(R)L2(R, dx)⊂ S0(R) (1)
can be reconstructed in a standard way (see Ref. [18]) by the harmonic oscillator
A= 1 + t2d2/dt2and H. The eigenvalues of Aare 2n, n = 1,2,· · · , the
corresponding eigenfunctions {en;n1}form an orthonormal basis for L2(R).
In fact (en) are the Hermite functions and therefore each enis an element of E.
The space Eis a nuclear space equipped with the Hilbertian norms
|ξ|p=|Apξ|0, ξ E, p R
A. Barhoumi
Carthage University, Tunisia, Nabeul Preparatory Engineering Institute, Department of Math-
ematics, Campus Universitaire - Mrezgua - 8000 Nabeul
E-mail: abdessatar.barhoumi@ipein.rnu.tn
H. Rguigui
Department of Mathematics, Faculty of Sciences of Tunis, University of Tunis El-Manar, 1060
Tunis, Tunisia
E-mail: hafedh.rguigui@yahoo.fr
2 Abdessatar Barhoumi, Hafedh Rguigui
and we have
E= proj lim
p→∞ Ep, E0= ind lim
p→∞ Ep,
where, for p0, Epis the completion of Ewith respect to the norm | · |pand Ep
is the topological dual space of Ep. We denote by N=E+iE and Np=Ep+iEp,
pZ, the complexifications of Eand Ep, respectively. Throughout, we fix a Young
function θsatisfying the condition
lim sup
x→∞
θ(x)
x2<+∞· (2)
Its polar function θis the Young function defined by
θ(x) = sup
t0
(tx θ(t)), x 0.
For more details , see Refs. [8].
For a complex Banach space (B, k · k), H(B) denotes the space of all entire
functions on Band for m > 0, Exp(B, θ, m) is the Banach space
Exp(B, θ, m) = nf∈ H(B); kfkθ,m := sup
zB
|f(z)|eθ(mkzk)<o.
The projective system {Exp(Np, θ, m); pN, m > 0}and the inductive system
{Exp(Np, θ, m); pN, m > 0}give the two nuclear spaces
Fθ(N0) = proj lim
p→∞;m0Exp(Np, θ, m),Gθ(N) = ind lim
p→∞;m0Exp(Np, θ, m).
(3)
It is noteworthy that, for each ξN, the exponential function
eξ(z) := ehz,ξi, z N0,
belongs to Fθ(N0) and the set of such test functions spans a dense subspace of
Fθ(N0). In the remainder of this paper we use simply Fθto denote the space
Fθ(N0). The space of continuous linear operators from Fθinto its topological dual
space F
θis denoted by L(Fθ,F
θ) and assumed to carry the bounded convergence
topology. For zN0and ϕ∈ Fθwith Taylor expansions P
n=0hxn, fni, the
holomorphic derivative of ϕat xN0in the direction zis defined by
(a(z)ϕ)(x) := lim
λ0
ϕ(x+λz)ϕ(x)
λ.(4)
We can check that the limit always exists and a(z)∈ L(Fθ,Fθ). Let a(z)
L(F
θ,F
θ) be the dual adjoint of a(z), i.e., for Φ∈ F
θand φ∈ Fθ,hha(z)Φ, φii =
hhΦ, a(z)φii, where h,·ii denotes the standard bilinear form on F
θ× Fθ. Similarly,
for ψ∈ Gθ(N) with Taylor expansion ψ(ξ) = P
n=0hψn, ξ niwe use the common
notation a(z)ψfor the derivative (4) with zN.
The Wick symbol of Ξ∈ L(Fθ,F
θ) is by definition [18] a C-valued function
on N×Ndefined by
σ(Ξ)(ξ, η) = hhΞeξ, eηiie−hξ,η i, ξ, η N. (5)
By a density argument, every operator in L(Fθ,F
θ) is uniquely determined by its
Wick symbol. In fact, if Gθ(NN) denotes the nuclear space obtained as in (3)
by replacing Npby Np×Np, we have the following characterization theorem for
operator Wick symbols.
Euler’s Theorem For Homogeneous White Noise Operators 3
Theorem 1 (See Ref. [13]) The Wick symbol map σyields a topological isomor-
phism between L(Fθ,F
θ)and Gθ(NN).
It is a fundamental fact in quantum white noise theory [18] (see, also Ref. [13])
that every white noise operator Ξ∈ L(Fθ,F
θ) admits a unique Fock expansion
Ξ=
X
l,m=0
Ξl,m(κl,m ),(6)
where, for each pairing l, m 0, κl,m (N(l+m))0
sym(l,m)and Ξl,m(κl,m ) is the
integral kernel operator uniquely specified via the Wick symbol transform by
σ(Ξl,m(κl,m ))(ξ, η) = hκl,m, ηlξmi, ξ, η N. (7)
For any S1, S2∈ L(Fθ,F
θ), there exists a unique Ξ∈ L(Fθ,F
θ), denoted S1S2,
such that
σ(S1S2) = σ(S1)σ(S2).(8)
The operator S1S2will be referred to as the Wick product of S1and S2.
Let θnbe given by θn= infr>0eθ(r)/rn, n N. Then, for pNand γ1, γ2>0,
we define the Hilbert space
Fθ,γ12(NpNp) =
n
ϕ= (ϕl,m)
l,m=0;ϕl,m (Nl
pNm
p)sym(l,m),
X
l,m=0
(θlθm)2γl
1γm
2|ϕl,m|2
p<o
Put
Fθ(NN) = \
pN1>02>0
Fθ,γ12(NpNp).
Theorem 2 ([4]) An operator Ξ∈ L(F
θ,Fθ)if and only if there exists a unique
(κl,m)l,m Fθ(NN)such that
Ξ=Ξτ
X
l,m=0
Ξl,m(κl,m ),(9)
where τis the usual trace on NN, i.e., hτ, ξ ηi=hξ, η iand
Ξ±τ=
X
k=0
(±1)k
k!Ξk,k(τk).
Let Uθbe the space of white noise operators given by
Uθ=nΞ=
X
l,m=0
Ξl,m(κl,m ); (κl,m)l,m Fθ(NN)o·
For x, y N, we put κl,m (x, y) = xl
l!ym
m!and Ξx,y := P
l,m=0 Ξl,m(κl,m (x, y)).
Then, the set {Ξx,y ;x, y N}spans a dense subspace of Uθ.
4 Abdessatar Barhoumi, Hafedh Rguigui
Theorem 3 ([4]) The map fτdefined by
fτ:L(F
θ,Fθ)→ Uθ, Ξ 7−ΞτΞ ,
is a topological isomorphism.
We recall from Ref. [4] the dual pairing: for T=P
l,m=0 Ξl,m(Φl,m )∈ Uθand
Ξ=P
l,m=0 Ξl,m(κl,m )∈ L(Fθ,F
θ), we define
Ξ, T :=
X
l,m=0
l!m!hκl,m, Φl,m i.
For more details see [4], [5], [6],[22], [23] and [24].
In mathematics, a homogeneous function is a function with multiplicative scal-
ing behavior: if the argument is multiplied by a factor, then the result is multiplied
by some power of this factor. More precisely, for fL2(Rd) and tR\{0}, put
Stf(x) = f(tx), xRd. For a given λR, an element fL2(Rd) is said to be
λorder homogeneous if Stf(x) = tλf(x) for each tR\{0}and xRd. It is
well known that fis λorder homogeneous if and only if it satisfies the so-called
Euler equation
d
X
i=1
xi
∂xi
f=λf·(10)
In infinite dimension analysis, an analogue of the Euler operator
d
X
i=1
xi
∂xi
was
introduced in [17] as follows
E=
X
i=1
(a(ei) + a(ei))a(ei) =
X
i=1
h · , eiia(ei).
Moreover, the scaling transformation Stis defined at ϕ(x) = P
n=0hxn, ϕni ∈ Fθ
by
Stϕ(x) =
X
n=0 Dxn,tn
n!
X
l=0
(t21)l(n+ 2l)!
l!2lτlb
lϕn+2lE, x N0·
For λR,ϕis said to be λorder homogeneous if Stϕ=tλϕfor any tR\{0}.
It is proved in [20] that ϕis λorder homogeneous if and only if it satisfies the
Euler equation
Eϕ=λϕ·(11)
The main purpose of this paper is the study of the QWN-analogue of (11). We start
by introducing a QWN-Scaling transformation and a QWN-second quantization. These
transformations will be used to introduce the notion of λorder homogeneous
operators. Then, as a first main result we give their Fock expansions (see Theorem
5). Our second main result is stated in Theorem 7, where we show that a white
noise operator Ξis λorder homogeneous if and only if it satisfies the following
QWN-Euler equation
Q
EΞ=λΞ.
Here Q
Eis the QWN-Euler operator defined in [6].
Euler’s Theorem For Homogeneous White Noise Operators 5
2 Fundamental QWN-Operators
2.1 QWN-Laplacians
From [6], the QWN-Gross Laplacian and QWN-conservation operator can be defined
through Theorem 3 on Uθ, respectively, by
Q
G=
X
j=1
D+
ejD+
ej+
X
j=1
D
ejD
ej,
NQ=
X
j=1
(D+
ej)(Dej)++
X
j=1
(D
ej)D
ej,
where, for ζN,
D+
ζΞ= [a(ζ), Ξ], D
ζΞ=[a(ζ), Ξ] (12)
are the creation derivative and annihilation derivative of Ξ, (see [12]).
Lemma 1 For any Ξ=P
l,m=0 Ξl,m(κl,m )∈ Uθ, we have
NQΞ=
X
l,m=0
(l+m)Ξl,m(κl,m ).(13)
Proof From [6], we have, for x, y N
σ(NQΞx,y)(ξ , η) = (hx, ηi+hy, ξ i)σ(Ξx,y)(ξ, η).
On the other hand, denoting the right hand side of (13) by AQ, we get
σ(AQΞx,y)(ξ , η)
=
X
l=1
X
m=0
hx, ηihx, ηil1
(l1)!
hy, ξim
m!+
X
l=0
X
m=1
hx, ηil
l!hy, ξihy , ξim1
(m1)!
= (hx, ηi+hy, ξ i)σ(Ξx,y)(ξ, η ).
Then, by a density argument we complete the proof.
It is noteworthy that the identity (13) holds true for Ξ∈ L(Fθ,F
θ).
Proposition 1 Let T∈ L(Fθ,F
θ). Then, we have
(Q
G)T={Ξ2,0(τ) + Ξ0,2(τ)}  T(14)
(NQ)T=NQT. (15)
6 Abdessatar Barhoumi, Hafedh Rguigui
Proof From [1], for Ξ=P
l,m=0 Ξl,m(κl,m )∈ Uθ, we have
Q
GΞ=
X
l,m=0
(l+2)(l+1)Ξl,m (τ2κl+2,m)+
X
l,m=0
(m+2)(m+1)Ξl,m (κl,m+22τ),
(16)
where, for zp(Np)0, and ξl+mpN(l+mp),pl+m, the contractions
zppκl,m and κl,m pzpare defined by
hzppκl,m, ξlp+mi=hκl,m , zpξlp+mi,
hκl,m pzp, ξl+mpi=hκl,m, ξl+mpzpi.
Then, for T=P
l,m=0 Ξl,m(Φl,m )∈ L(Fθ,F
θ), we obtain
T, ∆Q
GΞ
=
X
l,m=0
l!m!(l+ 2)(l+ 1)hΦl,m, τ 2κl+2,mi
+
X
l,m=0
l!m!(m+ 2)(m+ 1)hΦl,m, κl,m+2 2τi
=
X
l=2
X
m=0
l!m!hτΦl2,m, κl,m i+
X
l=0
X
m=2
l!m!hΦl,m2τ, κl,m i.
Therefore, we get
(Q
G)T=
X
l=2
X
m=0
Ξl,m(τΦl2,m ) +
X
l=0
X
m=2
Ξl,m(Φl,m2τ),
which yields
σ((Q
G)T)(ξ, η) =
X
l=2
X
m=0
hτΦl2,m, ηlξmi
+
X
l=0
X
m=2
hΦl,m2τ, η lξmi
={hη, ηi+hξ , ξi}σ(T)(ξ, η)
=σ(Ξ2,0(τ) + Ξ0,2(τ))(ξ, η)σ(T)(ξ, η).
This gives
(Q
G)T={Ξ2,0(τ) + Ξ0,2(τ)}  T
as desired. (15) follows from (13).
Euler’s Theorem For Homogeneous White Noise Operators 7
2.2 QWN-Second Quantization
We start by clarifying the topology of the nuclear algebra L(Fθ,F
θ). From The-
orem 1, we have the topological isomorphism:
L(Fθ,F
θ)' Gθ(NN) = [
p0,γ>0
Exp(NpNp, θ, γ).
For p0 and γ > 0, let Lθ ,p,γ (Fθ,F
θ) denotes the subspace of all Ξ
L(Fθ,F
θ) which correspond to elements in Exp(NpNp, θ, γ). The topology of
Lθ,p,γ (Fθ,F
θ) is naturally induced from the norm of the Banach space Exp(Np
Np, θ, γ) which will be denoted by ||| · |||θ,p,γ , i.e., for Ξ∈ Lθ,p,γ(Fθ,F
θ),
|||Ξ|||θ,p,γ =kσΞ kθ,p,γ = sup
ξ,ηNp
|σ(Ξ)(ξ, η)|eθ(γ|ξ|p)θ(γ|η|p).
For Ξ=P
l,m=0 Ξl,m(Φl,m )∈ L(Fθ,F
θ) and tR, we define the operator ΓQ(t)
by
ΓQ(t)Ξ=
X
l,m=0
Ξl,m(tl+mΦl,m ).(17)
We denote by GL(L(Fθ,F
θ)) the group of all linear homeomorphisms from L(Fθ,F
θ)
onto itself.
Proposition 2 {ΓQ(et)}tRis a regular one-parameter subgroup of GL(L(Fθ,F
θ))
with infinitesimal generator NQ.
Proof The proof of the fact that {ΓQ(et)}tRis a one-parameter subgroup of
GL(L(Fθ,F
θ)) is straightforward. Since we have
et(l+m)= 1 + t(l+m) + t2
X
k=0
tk
(k+ 2)!(l+m)(k+2)
then, for Ξ=P
l,m=0 Ξl,m(Φl,m )∈ L(Fθ,F
θ), one can write
ΓQ(et)Ξ=Ξ+tNQ(Ξ) + t2Λ(t)(Ξ) (18)
where
Λ(t)(Ξ) =
X
l,m=0
Ξl,m(Λl,m (t)Φl,m)
with
Λl,m(t) =
X
k=0
tk
(k+ 2)!(l+m)(k+2) .
Now, for |t| ≤ 1, using a similar computation as in [7], one can show that, there
exist c, r, r0>0 and p, q 0 such that
σ(ΓQ(et)Ξ)σ(Ξ)
tσ(NQΞ)
θ,p,r0
c|t|kσ(Ξ)kθ,q,r ·
It then follows
lim
t0sup
kσ(Ξ)kθ,q,r 1
σ(ΓQ(et)Ξ)σ(Ξ)
tσ(NQT)
θ,p,γ
= 0.
This proves the desired statement.
8 Abdessatar Barhoumi, Hafedh Rguigui
2.3 QWN-Scaling Transformation
Motivated by the classical case studied in [17] and [20], we define the QWN-scaling
transformation acting on Ξ=P
l,m=0 Ξl,m(κl,m )∈ Uθby
SQ
t(Ξ) :=
X
j,k,l,m=0
tl+m(t21)j+k(l+ 2j)!(m+ 2k)!
2j+kj!k!l!m!Ξl,m(τj2jκl+2j,m+2k2kτk).(19)
We recall from Ref. [6] and Theorem 3 that the QWN-Fourier-Gauss transform
GQ
K1,K2;B1,B2is a continuous linear operator from Uθinto itself defined by
GQ
K1,K2;B1,B2Ξ=
X
l,m
Ξl,m(gl,m ) (20)
where Ki, Bi∈ L(N0, N 0)∩ L(N, N ), i= 1,2 and gl,m is given by
gl,m =
X
j,k=0
(l+ 2j)!(m+ 2k)!
l!m!j!k!Bl
1Bm
2τj
K12jκl+2j,m+2k2kτk
K2.
In our setting, we observe that SQ
t=GQ
1
2(t21)I, 1
2(t21)I;tI,tI .
Theorem 4 Let Ξ=P
l,m=0 Ξl,m(κl,m )∈ L(Fθ,F
θ). Then, (SQ
t)Ξis given by
(SQ
t)Ξ=F(t)ΓQ(t)(Ξ),
where F(t)is given by
F(t) =
X
j,k=0
(t21)j+k
2j+kj!k!Ξ2j,2k(τjτk).(21)
Proof For Ξ=Pl,m=0 Ξl,m(κl,m)∈ L(Fθ,F
θ) and T=Pl,m=0 Ξl,m(Φl,m )
Uθ, we have
Ξ, S Q
tT
=
X
j,k,l,m=0
l!m!Dκl,m,(t21)j+ktl+m(l+ 2j)!(m+ 2k)!
2j+kj!l!τj2jΦl+2j,m+2k2kτkE
=
X
p,q=0
p!q!D[p/2]
X
j=0
[q/2]
X
k=0
(t21)j+k
2j+kj!k!tp+q2j2kτjκp2j,q2kτk, Φp,q E.
This yields
(SQ
t)Ξ=
X
p,q=0
[p/2]
X
j=0
[q/2]
X
k=0
tp+q2j2k(t21)j+k
2j+kj!k!Ξp,q(τjκp2j,q 2kτk).
(22)
Euler’s Theorem For Homogeneous White Noise Operators 9
On the other hand, we have
σ(F(t)ΓQ(t)(Ξ))(ξ, η)
=σ(F(t))(ξ, η)σ(ΓQ(t)(Ξ))(ξ , η)
=
X
j,k,l,m=0
tl+m(t21)j+k
2j+kj!k!Dτjκl,m τk, ηl+2jξm+2kE
=
X
p,q=0
[p/2]
X
j=0
[q/2]
X
k=0
tp+q2j2k(t21)j+k
2j+kj!k!Dτjκp2j,q2kτk, ηpξqE
or equivalently
F(t)ΓQ(t)(Ξ)
=
X
p,q=0
[p/2]
X
j=0
[q/2]
X
k=0
tp+q2j2k(t21)j+k
2j+kj!k!Ξp,q(τjκp2j,q 2kτk).
Comparing with (22), the statement follows.
Remark 1 Using (19), for x, y N, we have
SQ
t(Ξx,y) = exp 1
2(t21)hx, xi+1
2(t21)hy, yiΞtx,ty ·(23)
Then, for all s, t R\{0}, by a density argument, one can verify that
SQ
sSQ
t=SQ
st·(24)
In particular, for all sRand tR\{0}, we get
SQ
1+ s
t
SQ
t=SQ
s+t·(25)
3 Euler’s Theorem For Homogeneous Operator
3.1 Homogeneous Operator
Definition 1 Let Ξ∈ Uθand λR. We say that Ξis λ-order homogeneous if
for each tR\{0}we have
SQ
t(Ξ) = tλΞ. (26)
This definition is motivated by the classical case studied in [17].
Lemma 2 Let l,m 0and κl,m (NlNm)sym(l,m). Then, for Ki,Bi
L(N0, N 0)∩ L(N, N ),i= 1,2, we have
GQ
K1,K2;B1,B2Ξl,m(κl,m ) =
[l/2]
X
p=0
[m/2]
X
q=0
Ξl2p,m2q(gl2p,m2q),(27)
where gl2p,m2qis given by
gl2p,m2q=
l!m!
(l2p)!(m2q)!p!q!B(l2p)
1B(m2q)
2τp
K12pκl,m 2qτq
K2(28)
and τKiis the Ki-trace defined by hτKi, z wi=hKiz , wi.
10 Abdessatar Barhoumi, Hafedh Rguigui
Proof The operator Ξl,m (κl,m) can be rewritten as
Ξl,m(κl,m ) =
X
α,β=0
Ξα,β (fα,β ),
where fα,β is defined by
fα,β =κl,m if (l, m) = (α, β)
0if (l, m)6= (α, β).(29)
Then, by using (20), we get
GQ
K1,K2;B1,B2Ξl,m(κl,m ) =
X
α,β=0
Ξα,β (gα,β ),
with
gα,β =
X
j,k=0
(α+ 2j)!(β+ 2k)!
α!β!j!k!Bα
1Bβ
2τj
K12jfα+2j,β+2k2kτk
K2.
From (29), we observe that gα,β = 0 for α>lor β > m. Thus, we obtain
GQ
K1,K2;B1,B2Ξ=X
0αlX
0βm
Ξα,β (gα,β ),
with
gα,β =
X
j,k=0 X
2j=lαX
2k=mβ
l!m!
α!β!j!k!Bα
1Bβ
2τj
K12jκl,m 2kτk
K2.
Moreover, when lα= 2p+ 1 or mβ= 2q+ 1, we have gα,β = 0. The case
lα= 2pand mβ= 2qgives
gα,β =l!m!
α!β!p!q!Bα
1Bβ
2τp
K12pκl,m 2qτq
K2.
Replacing αby l2pand βby m2q, we get the desired statement.
The following theorem gives the Fock expansion of the λorder homogeneous
operator in Uθ.
Theorem 5 Let λNand Ξ=P
l,m=0 Ξl,m(κl,m )∈ Uθ. Then, Ξis λ-order
homogeneous if and only if
Ξ=
λ
X
l=0
[l/2]
X
p=0
[λl
2]
X
q=0 ZRλ2p2q
Υl2p,λl2q(s1,· · · , sl2p, t1,· · · , tλl2q)
a
s1· · · a
sl2pat1· · · atλl2qds1· · · dsl2pdt1· · · dtλl2q,
where Υl2p,λl2qis given by
Υl2p,λl2q=
X
j,k=0
(l+ 2j)!(λl+ 2k)!(1)j+k
j!k!p!q!(l2p)!(λl2q)!2j+k2p+qτ(j+p)2(j+p)κl+2j,λl+2k2(k+q)τ(k+q).
Euler’s Theorem For Homogeneous White Noise Operators 11
Proof In the following we set
GQ:= GQ
1
2I,1
2I;iI,iI .
Motivated by the classical case (see [16]), we can show that GQis a topological
isomorphism from Uθinto itself. Moreover,
(GQ)1Ξ=GQ
1
2I,1
2I;iI,iI Ξ . (30)
For any Ξ=P
l,m=0 Ξl,m(κl,m )∈ Uθand tR, the technical identity
SQ
t(Ξ) = (GQ)1ΓQ(t)GQ(Ξ) (31)
holds true. Indeed, by direct computation, we have
GQΞx,y = exp 1
2hx, xi − 1
2hy, yiΞix,iy , x, y N.
Therefore,
ΓQ(t)GQ(Ξx,y) = exp 1
2hx, xi − 1
2hy, yiΞitx,ity .
Then, we obtain
(GQ)1ΓQ(t)GQ(Ξx,y) = exp 1
2(t21)(hx, xi+hy, yi)Ξtx,ty .
Hence, by (23) we deduce that
SQ
t(Ξx,y) = (GQ)1ΓQ(t)GQ(Ξx,y ),
which proves (31) by density argument.
In view of (31), equation (26) can be rewritten as follows
ΓQ(t)GQ(Ξ) = tλGQ(Ξ).(32)
Let T=P
l,m=0 Ξl,m(Φl,m )∈ Uθbe the unique Fock expansion of the operator
T=GQ(Ξ), where Φl,m is given by
Φl,m =
X
j,k=0
(l+ 2j)!(m+ 2k)!(i)l+m(1)j+k
l!m!j!k!2j+kτj2jΦl+2j,m+2k2kτk.
Then (32) can be rewritten as
ΓQ(t)T=tλT, (33)
or equivalently
X
l,m=0
tl+mΞl,m(Φl,m ) =
X
l,m=0
tλΞl,m(Φl,m ).
12 Abdessatar Barhoumi, Hafedh Rguigui
From the uniqueness of the Fock expansion, this last equation is satisfied if and
only if λ=l+m. Then, Tsatisfies (33) if and only if
T=
λ
X
l=0
Ξl,λl(Φl,λl).
Therefore, by Eq. (27), we obtain
Ξ= (GQ)1T
=
λ
X
l=0
GQ
1
2I,1
2I;iI,iI (Ξl,λl(Φl,λl))
=
λ
X
l=0
[l/2]
X
p=0
[λl
2]
X
q=0
Ξl2p,λl2q(Υl2p,λl2q)
where Υl2p,λl2qis given by
Υl2p,λl2q=l!(λl)!iλ
(l2p)!(λl2p)!p!q!2p+qτp2pΦl,λl2qτq
and
Φl,λl=
X
j,k=0
(l+ 2j)!(λl+ 2k)!(1)λ(1)j+k
l!(λl)!j!k!2j+kτj2jκl+2j,λl+2k2kτk.
Then, we get
Υl2p,λl2q=
X
j,k=0
(l+ 2j)!(λl+ 2k)!(1)j+k
j!k!p!q!(l2p)!(λl2q)!2j+k2p+q
×τ(j+p)2(j+p)κl+2j,λl+2k2(k+q)τ(k+p)
as desired.
Example 1 (The 1-order homogeneous operators). For z, w N, the operator
Ξ=a(z) + a(w)
is a 1-order homogeneous operator. In particular, for z=w, the multiplication
operator Ξ=M,ziis 1-order homogeneous.
Example 2 (The 2-order homogeneous operators). For κ0,2,κ2,0,κ1,1NN,
the operators
Ξ2,0(κ2,0) + Ξ0,0(hτ, κ2,0i),
Ξ0,2(κ0,2) + Ξ0,0(hτ, κ0,2i),
Ξ1,1(κ1,1)
are 2-order homogeneous. Note that if we take κ0,2=κ2,0=τKfor K∈ L(N0, N)
such that hτ, τKi= 0, then the KGross Laplacian G(K) = Ξ2,0(τK) and its
dual
G(K) = Ξ0,2(τK) are 2-order homogeneous operators. Moreover, for B
L(N0, N ), the conservation operator N(B) = Ξ1,1(τB) is 2-order homogeneous
operator.
Euler’s Theorem For Homogeneous White Noise Operators 13
Remark 2 Let λN. Then, using Theorem 3, Ξ∈ L(F
θ,Fθ) is λ-order homoge-
neous if and only if
Ξ=ΞτΞh
where Ξhis λ-order homogeneous in Uθ.
3.2 Euler’s Theorem For Homogeneous Operator
From Ref. [6], the QWN-Euler operator can be represented, via Theorem 3, as a
continuous linear operator on Uθby
Q
E:= Q
G+NQ=
X
j=1
MQ+
,ejiD+
ej+
X
j=1
MQ
,ejiD
ej,
where for zN0,
MQ
,zi=σ1(M,ziI)σ, M Q+
,zi=σ1(IM,zi)σ,
and M,ziis the multiplication operator by , zi, see [18].
Theorem 6 Let Ξ∈ Uθand tR\{0}. Then for each T∈ L(Fθ,F
θ)
lim
s0T, SQ
t+sΞSQ
tΞ
s=1
tT, ∆Q
ESQ
tΞ.
Proof By Theorem 4 and Eq. (25) we have
lim
s0T, SQ
t+sΞSQ
tΞ
s= lim
s01
snF(1 + s
t)ΓQ(1 + s
t)(T)To, SQ
tΞ
= lim
s0F(1 + s
t)1
s(ΓQ(1 + s
t)(T)T)
+1
s(F(1 + s
t)I)T, SQ
tΞ.
Since, we have
lim
s0n(1 + s
t)l+m1
so=d
ds 1 + s
tl+ms=0 =1
t(l+m),
for T=P
l,m=0 Ξl,m(Φl,m ), we get
lim
s0
ΓQ(1 + s
t)(T)T
s=
X
l,m=0
1
t(l+m)Ξl,m(Φl,m ) = 1
tNQT. (34)
On the other hand using (21), we have
lim
s0σ(1
s{F(1 + s
t)I})(ξ, η)
14 Abdessatar Barhoumi, Hafedh Rguigui
= lim
s0
1
s(exp "(1 + s
t)21
2(hξ, ξi+hη, ηi)#1)
=1
thξ, ξi+hη, ηi
=1
tσΞ0,2(τ) + Ξ2,0(τ)(ξ, η).
Then, from Eqs. (14) and (15) we get
lim
s0T, SQ
t+sΞSQ
tΞ
s=1
tNQT+1
t(Q
G)T, SQ
tΞ
=1
tT, ∆Q
ESQ
tΞ.
Which gives the desired statement.
Remark 3 Using Theorem 6, for Ξ∈ Uθ, we have
lim
s0
SQ
esΞΞ
s= lim
s0es1
sSQ
(es1+1)ΞΞ
es1=Q
EΞ.
This shows that {SQ
et}is a semigroup on Uθwith infinitesimal generator Q
E.
Hence, we deduce that SQ
etU0is the unique solution of the Cauchy problem
∂t Ut=Q
EUt, U0∈ Uθ.
Theorem 7 (Euler’s theorem). Let Ξ∈ Uθ. Then Ξis λorder homogeneous if
and only if it satisfies the following QWN-Euler equation
Q
EΞ=λΞ·(35)
Proof If Ξis λ-order homogeneous, then by (26) we have
Q
EΞ= lim
t1
SQ
t(Ξ)SQ
1(Ξ)
t1= lim
t1
tλ1
t1Ξ=λΞ.
Conversely, suppose that (35) is satisfied. Let tR\{0}. Put G(t) = tλSQ
t(Ξ).
Then, by (35) and Theorem 6 we get
lim
s0
G(t+s)G(t)
s= lim
s0
1
sn(t+s)λSQ
t+s(Ξ)tλSQ
t(Ξ)o
= lim
s0
1
sn(t+s)λ(SQ
t+s(Ξ)SQ
t(Ξ))o
+ lim
s0
1
sn(t+s)λtλoSQ
t(Ξ)
=t(λ+1)Q
E(SQ
t(Ξ)) λt(λ+1)SQ
t(Ξ).(36)
Now, let T∈ L(Fθ,F
θ). Then, by Theorem 4 and Theorem 6, we have
T, SQ
t(Q
EΞ)=F(t)ΓQ(t)(T), ∆Q
EΞ
= lim
s0F(t)ΓQ(t)(T),SQ
1+sΞΞ
s
= lim
s0T, SQ
tSQ
1+sΞSQ
tΞ
s.
Euler’s Theorem For Homogeneous White Noise Operators 15
But, by applying the QWN-Scaling transformation on Ξx,y, for x, y N, we can
show that SQ
uSQ
v=SQ
vSQ
ufor all u, v R. Then we get
T, SQ
t(Q
EΞ)= lim
s0T, SQ
1+sSQ
tΞSQ
tΞ
s.
Hence, using Theorem 6 , we obtain
T, SQ
t(Q
EΞ)=T, ∆Q
E(SQ
tΞ),
from which we deduce that
SQ
t(Q
EΞ) = Q
E(SQ
tΞ).
Therefore, using (35) we get
Q
E(SQ
t(Ξ)) = λSQ
t(Ξ).
Thus, from (36) we deduce that G0(t) = 0 for all tR\{0}. In particular, G(t) =
G(1), i.e.,
tλSQ
t(Ξ) = SQ
1(Ξ) = Ξ·
From which we deduce the desired statement.
Remark 4 Euler’s theorem remains valid in L(F
θ,Fθ) where Q
Eis replaced by
e
Q
Eacting on L(F
θ,Fθ) as follows
e
Q
E(Ξ) = ΞτQ
E(ΞτΞ).
Corollary 1 Let λNand Ξ=P
l,m=0 Ξl,m(κl,m )∈ Uθsuch that Q
G(Ξ)=0.
Then Ξis λorder homogeneous if and only if Ξ=Pλ
l=0 Ξl,λl(κl,λl).
Proposition 3 Let Ξ∈ Uθbe a λorder homogeneous operator such that Q
G(Ξ) =
0. Then for each ξN,D±
ξ(Ξ)are (λ1)order homogeneous and (D±
ξ)(Ξ)
are (λ+ 1)order homogeneous.
Proof We recall from [6] that, for any ξN, the following identities hold true
D+
ξΞl,m(κl,m ) = l1,m(ξ1κl,m )
D
ξΞl,m(κl,m ) = l,m1(κl,m 1ξ)
(D+
ξ)Ξl,m(κl,m ) = Ξl+1,m(ξκl,m )
(D
ξ)Ξl,m(κl,m ) = Ξl,m+1(κl,m ξ).
Then, if Ξ=Pλ
l=0 Ξl,λl(κl,λl), we have
D+
ξΞ=
λ1
X
l=0
(l+ 1)Ξl,λl1(ξ1κl+11l).(37)
Thus, the fact Q
G(D+
ξΞ) = D+
ξ(Q
GΞ) = 0 and identity (37) proves the state-
ment for D+
ξ(Ξ) via Corollary 1. The others statements can be verified by slight
modification.
16 Abdessatar Barhoumi, Hafedh Rguigui
Theorem 8 Let Ξl,m(κl,m)∈ Uθ. Then Ξl,m (κl,m)is (l+m)order homogeneous
if and only if Q
G(Ξl,m(κl,m )) = 0.
Proof From (16) we have
Q
GΞl,m(κl,m ) = l(l1)Ξl2,m(τ2κl,m ) + m(m1)Ξl,m2(κl,m 2τ).(38)
Then by iterating (38) we get
(Q
G)kΞl,m(κl,m ) =
[l/2]
X
p=0
[m/2]
X
p+q=k
l!m!k!
(l2p)!(m2q)!p!q!Ξl2p,m2q(τp2pκl,m 2qτq).(39)
On the other hand from (27), we obtain
SQ
tΞl,m(κl,m ) =
[l/2]
X
p=0
[m/2]
X
q=0
l!m!
(l2p)!(m2q)!p!q!tl+m2p2q1
2(t21)p+q
×Ξl2p,m2q(τp2pκl,m 2qτq).(40)
Then in view of (39), (40) becomes
SQ
tΞl,m(κl,m )
=
[(l+m)/2]
X
k=0
1
k!tl+m2k1
2(t21)k(Q
G)kΞl,m(κl,m )
=tl+mΞl,m(κl,m ) +
[(l+m)/2]
X
k=1
1
k!tl+m2k1
2(t21)k(Q
G)kΞl,m(κl,m ).
It then follows that Ξl,m(κl,m) is (l+m)homogeneous if and only if
[(l+m)/2]
X
k=1
1
k!tl+m2k1
2(t21)k(Q
G)kΞl,m(κl,m ) = 0.(41)
Hence, using the fact that {Pk(X) = Xl+m2k(X21)k;k= 1,2,· · · ,[(l+m)/2]}
is a linearly independent family of polynomials, one can show that (41) holds if
and only if
(Q
G)kΞl,m(κl,m ) = 0,k= 1,2,· · · ,[(l+m)/2].
This implies in particular that Q
GΞl,m(κl,m ) = 0. The converse is straightforward
by Euler’s theorem.
Euler’s Theorem For Homogeneous White Noise Operators 17
References
1. Accardi, L., Barhoumi, A., Ji, U.C.: Quantum Laplacians on Generalized Operators on
Boson Fock space. Probability and Mathematical Statistics, Vol. 31, 1-24 (2011).
2. Accardi, L., Smolyanov, O.G.: On Laplacians and Traces. Conferenze del Seminario di
Matematica dell’Universit`a di Bari, Vol. 250, 1-28 (1993).
3. Accardi, L., Smolyanov, O.G.: Transformations of Gaussian measures generated by the
evy Laplacian and generalized traces. Dokl. Akad. Nauk SSSR, Vol. 350, 5-8 (1996).
4. Barhoumi, A., Lanconelli, A., Rguigui, H.: QWN-Convolution operators with application to
differential equations. Random Operators and Stochastic Equations, Vol. 22 (4), 195-211
(2014).
5. Barhoumi, A., Ouerdiane, H., Rguigui, H.: Stochastic Heat Equation on Algebra of Gener-
alized Functions. Infinite Dimensional Analysis, Quantum Probability and Related Topics,
Vol. 15, No. 4, 1250026 (18 pages) (2012).
6. Barhoumi, A., Ouerdiane, H., Rguigui, H.: QWN-Euler Operator And Associated Cauchy
problem. Infinite Dimensional Analysis Quantum Probability and Related Topics, Vol. 15,
No. 1, 1250004 (20 pages) (2012).
7. Barhoumi, A., Ouerdiane, H., Rguigui, H.: Generalized Euler heat equation. Quantum
Probability and White Noise Analysis, Vol. 25, 99–116 (2010).
8. Gannoun, R., Hachaichi, R., Ouerdiane, H., Rezgi, A.: Un th´eor`eme de dualit´e entre espace
de fonction holomorphes `a croissance exponentielle. J. Funct. Anal., Vol. 171, 1-14 (2000).
9. Huang, Z.Y. , Hu, X.S., Wang, X.J.: Explicit forms of Wick tensor powers in general white
noise spaces. IJMMS, Vol. 31, 413–420 (2002).
10. Gel’fand, I.M., Shilov, G.E.: Generalized Functions, Vol. I. Academic Press, Inc., New
York (1968).
11. Grothaus, M., Streit, L.: Construction of relativistic quantum fields in the framework of
white noise analysis. J. Math. Phys. Vol. 40, 5387–5405 (1999).
12. Ji, U.C., Obata, N.: Annihilation-derivative, creation-derivative and representation of
quantum martingales. Commun. Math. Phys., Vol. 286, 751-775 (2009).
13. Ji, U.C., Obata, N., Ouerdiane, H.: Analytic characterization of generalized Fock space op-
erators as two-variable entire function with growth condition. Infinite Dimensional Analysis
Quantum Probability and Related Topics, Vol. 5, No 3, 395-407 (2002).
14. Kuo, H.H.: On Fourier transform of generalized Brownian functionals. J. Multivariate
Anal. Vol. 12, 415-431 (1982).
15. Kuo, H.H.: The Fourier transform in white noise calculus. J. Multivariate Analysis, Vol.
31, 311-327 (1989).
16. Kuo, H.H.: White noise distribution theory. CRC press, Boca Raton (1996).
17. Liu, K., Yan, J.A.: Euler operator and Homogenoeus Hida distributions. Acta Mathematica
Sinica, Vol. 10, No 4, (1994), 439-445.
18. Obata, N.: White noise calculus and Fock spaces. Lecture notes in Mathematics 1577,
Spriger-Verlag (1994).
19. Ouerdiane, H., Rguigui, H.: QWN-Conservation Operator And Associated Differential Equa-
tion. Communication on stochastic analysis, Vol. 6, No. 3, 437-450 (2012).
20. Yan, J.A.: Products and Transforms of white noise Functionals. Appl. Math. Optim. Vol.
31, 137-153 (1995).
21. Potthoff, J., Yan, A.: Some results about test and generalized functionals of white noise.
Proc. Singapore Prob. Conf. L.Y. Chen et al.(eds.) , 121-145 (1989).
22. Rguigui, H.: Quantum Ornstein-Uhlenbeck semigroups. Quantum Studies: Math. and
Foundations, Vol. 2, 159-175 (2015).
23. Rguigui, H.: Quantum λ-potentials associated to quantum Ornstein - Uhlenbeck semi-
groups. Chaos, Solitons & Fractals, Vol. 73, 80-89, (2015).
24. Rguigui, H.: Characterization of the QWN-conservation operator. Chaos, Solitons & Fractals,
Vol. 84, 41-48 (2016).
... The operator n j=1 x j ∂ ∂ x j is called the Euler operator (see [2; 9; 11; 19]). The quantum white noise [17; 18] counterpart of this theorem, as well as the associated Cauchy problem and potential, were studied in [2]. ...
Article
Full-text available
We present a new characterization of real homogeneous functions of a negative degree by a new counterpart of Euler’s homogeneous function theorem using quantum calculus and replacing the classical derivative operator by a (p, q)-derivative operator. As an application we study the solution of the Cauchy problem associated to the (p, q)-analogue of the Euler operator. Using this solution, a probabilistic interpretation is given in some details; more specifically, we prove that this solution is a stochastically continuous markovian transition operator. Finally, we study it’s associated subordinated stochastically markovian transition operator.
Article
Full-text available
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.
Article
Full-text available
In this paper we introduce a quantum white noise (QWN) convolution calculus over a nuclear algebra of operators. We use this calculus to discuss new solutions of some linear and non-linear differential equations.
Article
Full-text available
Based on nuclear infinite-dimensional algebra of entire functions with a certain exponential growth condition with two variables, we define a class of operators which gives in particular three semigroups acting on continuous linear operators, called the quantum Ornstein-Uhlenbeck (O-U) semigroup, the left quantum O-U semigroup and the right quantum O-U semigroup. Then, we prove that the solution of the Cauchy problem associated with the quantum number operator, the left quantum number operator and the right quantum number operator, respectively, can be expressed in terms of such semigroups. Moreover, probabilistic representations of these solutions are given. Eventually, using a new notion of positive white noise operators, we show that the aforementioned semigroups are Markovian.
Article
Full-text available
Using the quantum Ornstein–Uhlenbeck (O–U) semigroups (introduced in Rguigui [21]) and based on nuclear infinite dimensional algebra of entire functions with a certain exponential growth condition with two variables, the quantum -potential and the generalised quantum -potential appear naturally for . We prove that the solution of Poisson equations associated with the suitable quantum number operators can be expressed in terms of these potentials. Using a useful criterion for the positivity of generalised operators, we demonstrate that the solutions of the Cauchy problems associated to the quantum number operators are positive operators if the initial condition is also positive. In this case, we show that these solutions, the quantum -potential and the generalised quantum -potential have integral representations given by positive Borel measures. Based on a new notion of positivity of white noise operators, the aforementioned potentials are shown to be Markovian operators whenever and .
Article
Full-text available
The quantum white noise (QWN)-Euler operator Δ E Q is defined as the sum Δ G Q +N Q , where Δ G Q and N Q stand for appropriate QWN counterparts of the Gross Laplacian and the conservation operator, respectively. It is shown that Δ E Q has an integral representation in terms of the QWN-derivatives {D t - ,D t + ;t∈ℝ} as a kind of functional integral acting on nuclear algebra of white noise operators. The solution of the Cauchy problem associated to the QWN-Euler operator is worked out in the basis of the QWN coordinate system.
Article
Full-text available
By adapting the white noise theory, the quantum analogues of the (classical) Gross Laplacian and Lévy Laplacian, the so-called quantum Gross Laplacian and quantum Lévy Laplacian, respectively, are introduced as the Laplacians acting on the spaces of generalized operators. Then the integral representations of the quantum Laplacians are studied in terms of quantum white noise derivatives. Correspondences of the classical Laplacians and quantum Laplacians are studied. The solutions of heat equations associated with the quantum Laplacians are obtained from a normal-ordered white noise differential equation.
Article
Full-text available
In [AGV] it has been proved that the Yang Mills equations are equivalent to the Laplace equation with respect to a generalized Levy laplacian. In that paper an infinite hierarchy of exotic laplacians was introduced, of which the Levy laplacian is the simplest example. In the present paper a one-to-one correspondence between laplacians and traces on *-algebras of operators is established. The Volterra laplacian corresponds to the trace class operators; the Levy laplacian to the algebra of multiplication by functions of the position operator; the exotic laplacians, to *-algebras of unbounded operators with exotic traces. The analogue, for exotic laplacians, of the characterization of the Levy laplacian as ergodic average of the second derivatives, is established. An existence and uniqueness theorem for the heat equation associated to the Levy laplacian is proved by means of an explicit formula for the fundamental solution.