Complex Anal. Oper. Theory https://doi.org/10.1007/s11785-018-0773-x

Complex Analysis and Operator Theory

Characterization Theorems for the Quantum White Noise Gross Laplacian and Applications Hafedh Rguigui1,2

Received: 14 January 2017 / Accepted: 19 January 2018 © Springer International Publishing AG, part of Springer Nature 2018

Abstract This paper reports on the characterization of the quantum white noise (QWN) Gross Laplacian based on nuclear algebra of white noise operators acting on spaces of entire functions with θ -exponential growth of minimal type. First, we use extended techniques of rotation invariance operators, the commutation relations with respect to the QWN-derivatives and the QWN-conservation operator. Second, we employ the new concept of QWN-convolution operators. As application, we study and characterize the powers of the QWN-Gross Laplacian. As for their associated Cauchy problem it is solved using a QWN-convolution and Wick calculus. Keywords QWN-Gross Laplacian · QWN-convolution operators · Rotation invariance operators · Wick product · Cauchy problem Mathematics Subject Classification Primary 60H40; Secondary 46A32 · 46F25 · 46G20

Contents 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Communicated by ILWOO CHO.

B

Hafedh Rguigui [email protected]

1

Department of Mathematics, High School of Sciences and Technology of Hammam Sousse, University of Sousse, Rue Lamine Abassi, 4011 Hammam Sousse, Tunisia

2

Department of Mathematics, AL-Qunfudhah University college, Umm Al-Qura University, Mecca, Saudi Arabia

H. Rguigui 2.1 White Noise Operators . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 QWN-White Noise Operators . . . . . . . . . . . . . . . . . . . . . . . 3 Characterization of the QWN-Gross Laplacian . . . . . . . . . . . . . . . . 4 Differential Equation Associated with the Power of the QWN-Gross Laplacian 4.1 The Power of the QWN-Gross Laplacian . . . . . . . . . . . . . . . . . 4.2 Cauchy Problem Associated with the Power of the QWN-Gross Laplacian References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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1 Introduction By the end of 18th century, Pierre Laplace was the first scientist to have used an operator acting on the functions of finitely many variables called later the Laplacian operator. We denote this finite dimensional Laplacian by  which admits on Rd the expression: d  ∂2 = ∂ x 2j j=1 when  acts on the Shwartz space S(Rd ). Around the mid of the 20th century, Gross [10] proposed a new operator, later on known as Gross Laplacian and denoted by G . This operator is an infinite dimensional analogues of the classical Laplacian  introduced by Laplace. Moreover, Gross has studied the infinite dimensional analogue of the heat equation in an abstract Wiener space. Many authors investigated G by focusing on various directions ([2,11,12,15,20]). Many current equations associated with G have resulted from modern science problems. It appears in superconductivity theory, the theory of control systems, Gauss random field theory and the theory of gauge fields (the Yang-Mills equation). H-H Kuo, in his book [16] published in 1996, has used G in white setting as continuous linear operator on functions belonging to test space. In the general theory [10], the Gross Laplacian G is expressed as follows:  G = τ (s, t)as at dsdt (1.1) R2

where at is the annihilation operator. Moreover, if we employ a discrete coordinate, it has the following expression: ∞  G = ∂e2k (1.2) k=1

with {en ; n ≥ 1} being an arbitrary orthonormal basis for L 2 (R), ∂ek the derivative in the direction ek acting on a test space (Fθ (N  ) as example). For details see [17]. Recently Barhoumi et al. ([6]) gave an extension of the Gross Laplacian acting on generalized space Fθ∗ (N  ). In the same period, Barhoumi, Ouerdiane and Rguigui [5], based on the quantum white noise derivatives developed by Ji and Obata [13], presented Q ) of the Gross Laplacian. As a result, a quantum white noise analogues (denoted by G it has been shown in [5] that had representation with respect to the quantum white noise derivatives {Dt− , Dt+ ; t ∈ R} as follows (by taking K 1 = K 2 = I ):

Characterization Theorems for the QWN-Gross Laplacian…



Q

G = =

R2 ∞ 

τ (s, t)Ds− Dt− dsdt + De−j De−j +

j=1

∞ 

 R2

τ (s, t)Ds+ Dt+ dsdt

(1.3)

De+j De+j

(1.4)

j=1 Q

In this paper, our goal is to characterize the QWN-Gross Laplacian G by using an extended idea of rotation invariant operator, the commutation relations with respect to the QWN-derivatives, the QWN-conservation operator and the concept of the QWNconvolution operators. The paper is organized as follows. In Sect. 2, we briefly recall well-known results on nuclear algebra of entire holomorphic functions and the Fock expansion of white noise operators. In Sect. 3, we characterize the QWN-Gross Laplacian. In Sect. 4, we characterize the power of the QWN-Gross Laplacian and we use these techniques to solve the associated Cauchy problem.

2 Preliminaries Let L 2 (R) be the real Hilbert space of square integrable functions on R with norm |·|0 . It is well known (see Refs. [16,17]) that the Schwarz space S(R) and its topological dual space S  (R) can be reconstructed using L 2 (R) and the harmonic oscillator A = 1 + t 2 − d 2 /dt 2 . More precisely, they are given by S  (R) = ind lim S− p (R),

S(R) = proj lim S p (R) , p→∞

p→∞

where, for p ≥ 0, S p (R) is the completion of S(R) with respect to the norm |.| p = |A p .|0 and S− p (R) is the topological dual space of S p (R). Denoting by SC , S p,C , S− p,C and SC the complexifications of S(R), S p (R), S− p (R) and S  (R), respectively, where the complexification of any real locally convex space X is given by XC = X + iX. Throughout the paper, we fix a Young function θ and θ ∗ its polar function given by θ ∗ (x) = sup(t x − θ (t)), x ≥ 0, t≥0

see Refs. [9] and [18]. The test space Fθ (SC ) and the space Gθ ∗ (SC ⊕ SC ) are defined as follows: Fθ (SC ) := proj lim



p→∞;m↓0

Gθ ∗ (SC ⊕ SC )

:= ind lim

p→∞;m,m  →0



f ∈ H(S− p,C );

 sup | f (z)|e−θ(m|z|− p ) < ∞ ,

z∈S− p,C

f ∈ H(S p,C ⊕ S p,C );

sup

ξ,η∈S p,C

| f (ξ, η)|e−θ

∗ (m|ξ |

p)

e−θ

∗ (m  |η|

p)

 <∞

H. Rguigui

where H(B) denotes the space of all entire functions on a complex Banach space B, see Refs. [4–6,8,9,18,19,22,23] and [24]. In all the remainder of this paper we denote by Fθ the test space Fθ (SC ) and its topological dual space is denoted by Fθ∗ . 2.1 White Noise Operators The space of continuous linear operators from a nuclear space X to another nuclear space Y is denoted by L(X, Y) and assumed to carry the bounded convergence topology. For z ∈ SC and ϕ in Fθ , the holomorphic derivative of ϕ at x ∈ SC in the direction z is defined by ϕ(x + λz) − ϕ(x) . (2.1) (a(z)ϕ)(x) := lim λ→0 λ We can check that a(z) ∈ L(Fθ , Fθ ) and a ∗ (z) ∈ L(Fθ∗ , Fθ∗ ), where a ∗ (z) is the adjoint of a(z) with respect to the duality between Fθ∗ and Fθ . Now, if z = δt ∈ S  (R) we simply write at instead of a(δt ) and the pair at and at∗ are 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) := ez,ξ , z ∈ SC , belongs to Fθ and the set of such test functions spans a dense subspace of Fθ . The Wick symbol of a white noise operator  ∈ L(Fθ , Fθ∗ ) is by definition [17] a C-valued function on SC × SC defined by ω()(ξ, η) = eξ , eη e−ξ,η , ξ, η ∈ SC .

(2.2)

By a density argument, every operator in L(Fθ , Fθ∗ ) is uniquely determined by its Wick symbol. Moreover, we have the following characterization theorem. Theorem 2.1 (See Ref. [14]) The Wick symbol map yields a topological isomorphism between L(Fθ , Fθ∗ ) and Gθ ∗ (SC ⊕ SC ). In Ref. [14], it is shown that Gθ ∗ (SC ⊕SC ) is closed under pointwise multiplication. Then, for any 1 , 2 ∈ L(Fθ , Fθ∗ ), there exists a unique  ∈ L(Fθ , Fθ∗ ) such that ω() = ω(1 )ω(2 ). The operator  will be denoted 1 2 and it will be referred to as the Wick product of 1 and 2 . It is noteworthy that, endowed with the Wick product , L(Fθ , Fθ∗ ) becomes a commutative algebra. It is a fundamental fact in QWN theory (see [17] and [14]) that every white noise operator  ∈ L(Fθ , Fθ∗ ) admits a unique Fock expansion ∞  = l,m (κl,m ), (2.3) l,m=0

where, for each pairing l, m ≥ 0, κl,m ∈ (SC⊗(l+m) )sym(l,m) and l,m (κl,m ) is the integral kernel operator characterized via the Wick symbol transform by ω(l,m (κl,m ))(ξ, η) = κl,m , η⊗l ⊗ ξ ⊗m , ξ, η ∈ SC .

(2.4)

Characterization Theorems for the QWN-Gross Laplacian…

Which can be formally given by  l,m (κl,m ) = Rl+m κl,m (s1 , . . . , sl , t1 , . . . , tm ) as∗1 · · · as∗l at1 · · · atm ds1 · · · dsl dt1 · · · dtm . In this way l,m (κl,m ) can be considered as the operator polynomials of degree l + m ⊗(l+m)  )sym(l,m) as coefficient; and therefore associated to the distribution κl,m ∈ (SC every white noise operator is a “function” of the annihilation operators and the creation operators. 2.2 QWN-White Noise Operators From Ref. [13], we summarize the novel formalism of QWN-derivatives. For ζ ∈ SC , then a(ζ ) extends to a continuous linear operator from Fθ∗ into itself (denoted by the same symbol) and a ∗ (ζ ) (restricted to Fθ ) is a continuous linear operator from Fθ into itself. Hence, for any white noise operator  ∈ L(Fθ , Fθ∗ ), the commutators [a(ζ ), ] = a(ζ ) −  a(ζ ), [a ∗ (ζ ), ] = a ∗ (ζ ) −  a ∗ (ζ ), are well defined white noise operators in L(Fθ , Fθ∗ ). The QWN-derivatives are defined by (2.5) Dζ+  = [a(ζ ), ], Dζ−  = −[a ∗ (ζ ), ]. These are called the creation derivative and annihilation derivative of , respectively. Note that, for z ∈ SC , the QWN-derivatives Dz± and (Dz± )∗ are continuous linear operators from L(Fθ∗ , Fθ ) into itself and from L(Fθ , Fθ∗ ) into itself, i.e., Dz± , (Dz± )∗ ∈ L(L(Fθ∗ , Fθ )) ∩ L(L(Fθ , Fθ∗ )). Moreover, for z ∈ SC and  ∈ L(Fθ , Fθ∗ ), we have (Dz+ )∗  = a ∗ (z) , (Dz− )∗  = a(z) . For more details, see [19]. The operator  j,k,l,m (κ) is defined through two canonical bilinear forms as follows:  j,k,l,m (κ)S, T = κ, (Ds+1 )∗ · · · (Ds+j )∗ (Dt−1 )∗ · · · (Dt−k )∗ Du+1 · · · Du+l Dv−1 · · · Dv−m S, T  ∗ where S, T ∈ L(Fθ∗ , Fθ ) and for S = l,m l,m (sl,m ) in L(Fθ , Fθ ) and T =  (l+m) ⊗  ∗  ⊗(l+m) and s , l,m ∈ SC l,m l,m (tl,m ) in L(Fθ , Fθ ), such that tl,m ∈ (SC ) the duality between the two spaces L(Fθ∗ , Fθ ) and L(Fθ , Fθ∗ ), denoted by ., . , is defined as follows ∞  T, S := l!m!tl,m , sl,m . (2.6) l,m=0

H. Rguigui

The operator  j,k,l,m (κ) can be expressed as the following integral:   j,k,l,m (κ) =

κ(s1 , . . . , s j , t1 , . . . , tk ; u 1 , . . . , u l , v1 , . . . , vm ) R j+k+l+m (Ds+1 )∗ · · · (Ds+j )∗ (Dt−1 )∗ · · · (Dt−k )∗ Du+1 · · · Du+l Dv−1 · · · Dv−m ds1 · · · ds j dt1 · · · dtk du 1 · · · du l dv1 · · · dvm .

We call  j,k,l,m a QWN-integral operator with kernel distribution κ. (See [3]). For any  Q ∈ L(L(Fθ∗ , Fθ ), L(Fθ , Fθ∗ )) there exists a unique family of distributions ⊗( j+k+l+m)  )sym( j,k,l,m) , such that κ j,k,l,m ∈ (SC Q S =

∞ 

 j,k,l,m (κ j,k,l,m )S, S ∈ L(Fθ∗ , Fθ ),

j,k,l,m=0

where the right hand side converges in L(Fθ , Fθ∗ ).

3 Characterization of the QWN-Gross Laplacian Recall that from Ref. [3], the infinite-dimensional rotation group O(S(R), L 2 ) is given by O(S(R), L 2 ) = {B ∈ G L(S(R)), | B ξ |0 = | ξ |, ∀ξ ∈ S(R)}. We say that a continuous linear operator  Q from L(Fθ∗ , Fθ ) into L(Fθ , Fθ∗ ) is rotation invariant if ( Q (B))∗  Q  Q (B) =  Q , ∀B ∈ O(S(R), L 2 ) where,  Q (B) is defined by  Q (B) Q =

∞ 

l,m (B ⊗(l+m) κl,m )

l,m=0

∞ for all  ∈ L(Fθ∗ , Fθ ) given by  = l,m=0 l,m (κl,m ). Let us denoting by N Q+ and N Q− the following operators: N Q+ = N Q− =

∞  j=1 ∞  j=1

De+j De−j

∗

∗

De+j

(3.1)

De−j .

(3.2)

Characterization Theorems for the QWN-Gross Laplacian…

Lemma 3.1 Let n 1 , n 2 , k1 , k2 ∈ {0, 1, 2, . . .}. Then, we have [N Q+ , n 1 ,n 2 ,k1 ,k2 (κ)] = (n 1 − k1 )n 1 ,n 2 ,k1 ,k2 (κ). [N Q− , n 1 ,n 2 ,k1 ,k2 (κ)] = (n 2 − k2 )n 1 ,n 2 ,k1 ,k2 (κ). Proof We know that, N Q+ =

∞ 

De+j

∗

De+j ,

N Q− =

∞ 

j=1

De−j

∗

De−j .

j=1

Then, we need to compute [(De±j )∗ De±j , n 1 ,n 2 ,k1 ,k2 (κ)]. For this, we start by  ∗ [De+j , Dx+ ] = x, ej I, x ∈ SC ∗ [De+j , D − ] = 0, y ∈ SC . y Therefore, by using Lemma 4.2 in [7] (for n 1 = 1, n 2 = 0, k1 = 1 and k2 = 0), we get

∗

n 

n 

k 

k  1 −∗ 2 + 1 − 2 D D D De+j , D +∗ ] [ De+j     

ξ





= n 1 ξ , e j De+j

∗

η

D +∗ 

n  −1 1

ξ

α

D −∗ 

n  2

η

β

D +

k 

α

1

D −

k 

2

β

n 

n 

k  −1



k    1 2 1 2 D −∗ D + De+j D − − k1 α , e j D +∗   ξ

η

α

β

But we know that, ∞  ± x, ej Dej = Dx± , x ∈ SC j=1 ∞ 

± ∗ x, ej (Dej ) = (Dx± )∗ , x ∈ SC .

j=1

Then, using (3.1) and (3.3), we get

n 

n 

k 

k  1 −∗ 2 + 1 − 2 [N Q+ , D +∗ D D D ]     



ξ

= n 1 D +∗  ξ

n  1

η

D −∗  η

n  2

α

D +

k 

α

1

β

D −

k 

β

2



n 

n 

k 

k   1 2 1 2 D −∗ D + D − − k1 D +∗   ξ

η

α

β

(3.3)

H. Rguigui



n 

n 

k 

k    +∗ 1 −∗ 2 + 1 − 2 D  D  D  = n 1 − k1 D  . ξ

η

α

β

Hence, we deduce that 







[N Q+ , n 1 ,n 2 ,k1 ,k2 (κ)] = (n 1 − k1 )n 1 ,n 2 ,k1 ,k2 (κ). Similarly, we obtain [N Q− , n 1 ,n 2 ,k1 ,k2 (κ)] = (n 2 − k2 )n 1 ,n 2 ,k1 ,k2 (κ).  

This completes the proof.

By uniqueness of the Fock expansion of T Q ∈ L(L(Fθ∗ , Fθ ), L(Fθ , Fθ∗ )) (see [3]) and using Lemma 3.2, we get Lemma 3.2 Let T Q ∈ L(L(Fθ∗ , Fθ ), L(Fθ , Fθ∗ )) and r ≥ 0 be an integer. Then • [T Q , N Q− ] = r T Q if and only if T Q is given by T

Q

∞ 

=

 j,k, j+r,m (κ j,k, j+r,m ),

j,k,m=0

• [T Q , N Q+ ] = r T Q if and only if T Q is given by TQ =

∞ 

 j,k,l,k+r (κ j,k,l,k+r ).

j,k,l=0

Lemma 3.3 Let T Q ∈ L(L(Fθ∗ , Fθ ), L(Fθ , Fθ∗ )). Then, ∀a ∈ SC , we have [T Q , Da+ ] = 0 ⇔ T Q =

∞ 

0,n 2 ,k1 ,k2 (κ0,n 2 ,k1 ,k2 )

n 2 ,k1 ,k2 =0

and [T Q , Da− ] = 0 ⇔ T Q =

∞ 

n 1 ,0,k1 ,k2 (κn 1 ,0,k1 ,k2 ) .

n 1 ,k1 ,k2 =0 

Proof Let ξ, η, α, β ∈ SC and a ∈ SC . Using Lemma 4.2 in [7], we get [(Dξ+∗ )n 1 (Dη−∗ )n 2 (Dα+ )k1 (Dβ− )k2 , Da+ ] = −n 1 ξ, a (Dξ+∗ )n 1 −1 (Dη−∗ )n 2 (Dα+ )k1 (Dβ− )k2 .

Characterization Theorems for the QWN-Gross Laplacian…

From which we obtain ˆ 1 κn 1 −1,n 2 ,k1 ,k2 ) [n 1 ,n 2 ,k1 ,k2 (κn 1 ,n 2 ,k1 ,k2 ), Da+ ] = −n 1 n 1 −1,n 2 ,k1 ,k2 (a ⊗ ˆ 1 denote the contraction. Then, using Proposition 4 in [3], [T Q , Da+ ] = 0 where ⊗ gives n 1 = 0. Conversely, using Lemma 4.2 in [7], one can get [0,n 2 ,k1 ,k2 (κ0,n 2 ,k1 ,k2 ), Da+ ] = 0. This proves the first assertion. The second assertion is similar to the first.

 

The QWN-conservation operator N Q introduced in [19] is given by 

 τ (s, t)(Ds+ )∗ Dt+ dsdt + τ (s, t)(Ds− )∗ Dt− dsdt 2 2 R R  + ∗ + − ∗ − = (Ds ) Ds ds + (Ds ) Ds ds.

NQ =

R

R

(3.4) (3.5)

Obviously, using (3.1) and (3.2), we have N Q = N Q+ + N Q− . For more details see Ref. [19]. Q Theorem 3.1 Let T Q ∈ L(L(Fθ∗ , Fθ ), L(Fθ , Fθ∗ )). Then, T Q is equal to G up to a constant factor if and only if the following conditions are satisfied:

1. [N Q , T Q ] = −2T Q 2. ∀a ∈ SC , [T Q , Da+ ] = [T Q , Da− ] = 0 3. T Q is rotation invariant. Proof Using Eq. (1.3), the quantum white noise Gross Laplacian is given by Q G = 0,0,2,0 (τ )+ 0,0,0,2 (τ )

Then, using Lemma 3.2, we get Q

Q

Q

[N Q , G ] = [N Q+ , G ] + [N Q− , G ] = [N Q+ , 0,0,2,0 (τ )] + [N Q+ , 0,0,0,2 (τ )] +[N Q− , 0,0,2,0 (τ )] + [N Q− , 0,0,0,2 (τ )] = −20,0,2,0 (τ ) + 0 + 0 + (−2)0,0,0,2 (τ ) Q = −2G .

This proves the condition 1.

H. Rguigui

From the fact (see [7]), − − + − − + [Dx+ , D + y ] = [D x , D y ] = [D x , D y ] = [D x , D y ] = 0

we get  

Q Q G , Da+ = G , Da− = 0 This shows the condition 2. To prove condition 3., we use Theorem 14 in [3]. Conversely, let T Q ∈ L(L(Fθ∗ , Fθ ), L(Fθ , Fθ∗ )) verifying conditions 1., 2. and 3. and given by ∞ 

TQ =

n 1 ,n 2 ,k1 ,k2 (κ).

n 1 ,n 2 ,k1 ,k2 =0

Using condition 2. and Lemma 3.3, we get TQ =

∞ 

0,0,k1 ,k2 (κ).

k1 ,k2 =0

Then, by condition 1. and Lemma 3.2, we obtain k1 + k2 = 2 and then, T Q is given by TQ =



0,0,k1 ,k2 (κ).

k1 +k2 =2

Hence, using condition 3. and in view of Theorem 14 in [3] (for λ = 1 and α = β = 0) we get the result.   Recall that from [4], a QWN-convolution operator on L(Fθ∗ , Fθ ) is a continuous linear mapping C Q which commutes with all QWN-translation operators. More precisely, for any  ∈ L(Fθ∗ , Fθ ) and c,d ∈ SC we have Q Q ) = T−c,−d (C Q ) C Q (T−c,−d Q , for c, d ∈ SC , is the QWN-translation operator given by (see [4]) where T−c,−d Q T−c,−d

=

∞  l,m=0

1 l!m!

 Rl+m

d(s1 ) · · · d(sl )c(t1 ) · · · c(tm )Ds+1 · · · Ds+l Dt−1

· · · Dt−m ds1 · · · dsl dt1 · · · dtm

Characterization Theorems for the QWN-Gross Laplacian… ∞ 

=

l,m=0

1 0,0,l,m (d ⊗l ⊗ c⊗m )· l!m!

The set of all QWN-convolution operators on L(Fθ∗ , Fθ ) is denoted by C Q .  Q Let S = i,∞j=0 i, j (si, j ) ∈ L(Fθ , Fθ∗ ). Then, the QWN-convolution operator C S , defined in [4], coincides with C SQ =

∞   i, j=0

Ri+ j

si, j (u 1 , . . . , u i , v1 , . . . , v j )Du+1 · · · Du+i Dv−1 · · · Dv−j du 1

· · · du i dv1 · · · dv j ∞  = 0,0,i, j (si, j )·

(3.6)

i, j=0 Q Theorem 3.2 Let T Q ∈ L(L(Fθ∗ , Fθ ), L(Fθ , Fθ∗ )). Then, T Q is equal to G up to a constant factor if and only if the following conditions are satisfied:

1. T Q is a QWN-convolution operator 2. T Q (G ) = (G + ∗G )  G Q

Proof We recall that the QWN-(K 1 , K 2 )-Gross Laplacian G (K 1 , K 2 ) defined as in Ref. [5], via the symbol map, by:  (K 1 , K 2 )σ (). G (K 1 , K 2 ) = σ −1  G Q

Q

Q

G (K 1 , K 2 ) is a QWN-convolution operator and we have Q

Q

G (K 1 , K 2 ) = CG (K 2 )+∗ (K 1 ) G

For more details see [4]. For K 1 = K 2 = I , we get Q

Q

G = CG +∗

(3.7)

G

Which means that the QWN-Gross Laplacian is a convolution operator and we prove condition 1. Moreover, Eq. (3.7) gives that G (G ) = CG +∗ (G ) = (G + ∗G )  G Q

Q

G

Which proves condition 2. Conversely, using Theorem 4.5 in [4], for any T Q ∈ C Q there exists a unique Q S ∈ L(Fθ , Fθ∗ ) such that T Q = C S . Then, using condition 1., we get Q

T Q (G ) = C S (G ) = S  G

H. Rguigui

Therefore, from condition 2., we obtain S  G = (G + ∗G )  G Then, by the unicity of S, we get S = (G + ∗G ) Hence, we obtain Q T Q = CQG +∗ = G G

 

This completes the proof.

Q Theorem 3.3 Let T Q ∈ L(L(Fθ∗ , Fθ ), L(Fθ , Fθ∗ )). Then, T Q is equal to G up to a constant factor if and only if the following conditions are satisfied:

1. T Q is a QWN-convolution operator 2. [N Q , T Q ] = −2T Q 3. T Q is rotation invariant. Proof The fact that the QWN-Gross Laplacian is a convolution operator is shown in Theorem 3.2 which proves condition 1. Now, we know that the quantum white noise Gross Laplacian is given by Q

G = 0,0,2,0 (τ )+ 0,0,0,2 (τ ). As was shown previously, using Lemma 3.2, we get

 Q Q N Q , G = −2G .

This proves the condition 2. To prove condition 3., we use obviously Theorem 14 in [3]. Conversely, let T Q ∈ L(L(Fθ∗ , Fθ ), L(Fθ , Fθ∗ )) verifying conditions 1., 2. and 3. and given by ∞ 

TQ =

n 1 ,n 2 ,k1 ,k2 (κ).

n 1 ,n 2 ,k1 ,k2 =0

Then, using (3.6), we get TQ =

∞  k1 ,k2 =0

0,0,k1 ,k2 (κ).

Characterization Theorems for the QWN-Gross Laplacian…

Then, by condition 2. and Lemma 3.2, we obtain k1 + k2 = 2 and then, T Q is given by 

TQ =

0,0,k1 ,k2 (κ).

k1 +k2 =2

Hence, using condition 3. and in view of Theorem 14 in [3] (for λ = 1 and α = β = 0) we get the result.  

4 Differential Equation Associated with the Power of the QWN-Gross Laplacian 4.1 The Power of the QWN-Gross Laplacian Q

Q

Let p ∈ N and (G ) p is the power of order p of G . Lemma 4.1 Let p ∈ N. Then p Q Q G = C(G +∗ ) p

(4.1)

G

Proof Let  ∈ L(Fθ∗ , Fθ ). Then, by induction we should prove (4.1): (1) Case p=1 is given in (3.7). We prove case p = 2: 2  Q G () = (G + ∗G )  (G + ∗G )   Q

Q

= CG +∗ ◦ CG +∗ () G

G

It is noteworthy that, endowed with the Wick product , L(Fθ , Fθ∗ ) becomes a commutative algebra and from Ref. [4], we know that for S1 , S2 ∈ L(Fθ , Fθ∗ ), we have Q

Q

Q

C S1 ◦ C S2 = C S1 S2 .

(4.2)

Then, we get 2 Q Q G () = C(

∗ 2 G +G )

().

Q p Q ) = C( (2) Suppose that (G ∗ p . Then G + ) G

p+1 p

Q Q Q G G () = G

H. Rguigui Q

Q

= C(G +∗ ) p ◦ CG +∗ () G

G

Then, using (4.2), we obtain ( p+1) Q Q G () = C(

∗ ( p+1) G +G )

()  

which completes the proof. Lemma 4.2 Let p ∈ N. Then, we have

p p Q Q N Q , G = −2 p G .

(4.3)

Proof We should prove (4.3) by induction. (1) Case p = 1: see Theorem 3.1 or Theorem 3.3. (2) Suppose that the identity (4.3) is verified for p ≥ 1. Then 

p+1 p+1 p+1  Q Q Q = N Q G N Q , G − G NQ p p Q Q Q Q = N Q G G − G G N Q p p

Q Q Q Q G = N Q G G − G

p Q Q Q + N Q G − G N Q G p p

Q Q Q Q N  . +G −  N Q G G

Then, using (4.3) and Theorem 3.3, we get p+1 p p Q Q Q Q Q [N Q , G ] = [N Q , G ] G + G [N Q , G ] p p Q Q Q Q = −2G G + G (−2 p G ) p+1 p+1 Q Q = −2 G − 2 p G p+1 Q = −2( p + 1) G . This completes the proof.

 

 Theorem 4.1 Let p ∈ N and T Q ∈ L L(Fθ∗ , Fθ ), L(Fθ∗ , Fθ ) . Then T Q is equal to Q p ) up to a constant factor if and only if the following conditions are satisfied: (G 1) [N Q , T Q ] = −2 pT Q 2) T Q is a QWN-convolution operator 3) T Q is rotation invariant.

Characterization Theorems for the QWN-Gross Laplacian… Q

Proof Let T Q be equal to (G ) p up to a constant factor. Then, using (4.3), we show the condition 1). Condition 2) is shown obviously by (4.1). Now, using Theorem 14 in [3] (by taking α = β = 0 and λ = p), we show condition 3). Conversely, let T Q ∈ L L(Fθ∗ , Fθ ), L(Fθ , Fθ∗ ) verifying conditions 1), 2), 3) and given by ∞ 

TQ =

n 1 ,n 2 ,k1 ,k2 (κ).

n 1 ,n 2 ,k1 ,k2 =0

Using condition 2), we get ∞ 

TQ =

0,0,k1 ,k2 (κ)

k1 ,k2 =0

(see [4] for more details). Then, using Lemma 3.2 and condition 1), we obtain k1 + k2 = 2 p and then 

TQ =

0,0,k1 ,k2 (κ).

k1 +k2 =2 p

Hence, using condition 3) and the same argument as in Theorem 14 in [3] (by taking λ = 2 p j, k = 0, l = 2 p, m = 2 p), we obtain the desired statement.   4.2 Cauchy Problem Associated with the Power of the QWN-Gross Laplacian From [14], we have the following Lemma 4.3 Let S ∈ L(Fθ , Fθ∗ ), then the operator wex p{S} given by wex p{S} :=

∞  S n n! n=0

∗ belongs to L(F(eθ ∗ −1)∗ , F(e θ ∗ −1)∗ ).

Let θ1 and θ2 be two fixed young functions. In the following we assume that there ∗ exists a constant α > 0 such that eθ1 (r ) − 1 ≤ αθ2∗ (r ) for r large enough. Let p ∈ N and consider the following Cauchy problem: 

p Q = G (Ut ) + Vt U0 =  ∈ L(Fθ∗2 , Fθ2 ) ∂ ∂t Ut

where {Vt }t is a continuous L(Fθ2 , Fθ∗2 ) quantum process.

(4.4)

H. Rguigui

Theorem 4.2 The Cauchy problem (4.4) has a unique solution in L(Fθ2 , Fθ∗2 ) given by   Ut = wex p t (G + ∗G ) p   +

 0

t

  wex p (t − s)(G + ∗G ) p  Vs ds (4.5)

Proof By Picard’s iteration procedure, we shall prove the solution of the Cauchy problem (4.4) is the one given in the identity (4.5). Uniqueness follows from the general theory. Denoting Ut = U (t), we should apply the iteration to the differential equation ∂ U (t) = f (t, U (t)), ∂t where f (t, U (t)) = A  U (t) + Vt with A = (G + ∗G ) p and the initial condition U (0) =  ∈ L(Fθ2 , Fθ∗2 ). Then  t f (s, U0 )ds U1 (t) = U0 + 0  t = + (A   + Vs )ds 0  t = +tA+ Vs ds. 0

Next, we iterate once more to get the second guess:  t U2 (t) = U0 + f (s, U1 (s))ds 0  t  = + A  U1 (s) + V (s) ds 0    t A   + sA   + = + 0

s



Vu du + V s ds

0

t2 =  + t A   + A  (A  ) 2    t  t s + V (s)ds + (A  Vu )du ds. 0

0

0

Then, using (4.2), we get Q

Q

A  (A  ) = C A ◦ C A () Q

= C A A ()



Characterization Theorems for the QWN-Gross Laplacian…

= (A A)   = A   2

Then, we obtain t2 2 U2 (t) =  + t A   + A   2   t  t  s + Vs ds + (A  Vu )du ds. 0

0

0

Hence, by induction we have, for any k ≥ 1 Un (t) =  +

n  tk

k!

k=1

A

k

+

n 



t

G k (t) +

Vs ds 0

k=1

where G k (t) is given by G k (t) =

 t  0



uk

u1

...

0

(A

k





 Vu )du du 1 . . . du k

(4.6)

0

Then, we prove the following equality by induction on k:  t

G k (t) =

 (t − s)k k A  Vs ds k!

0

(4.7)

for any k ≥ 1. (1) The case k = 1, we have G 1 (t) =

 t  0

u1

 (A  Vu )du du 1 .

0

Denoting the right hand side of (4.7) by γk (t), i.e., γk (t) =

 t 0

 (t − s)k k A  Vs ds. k!

Then  γ1 (t) =

t

((t − s)A  Vs )ds.

0 

Let γ1 (t) =

d dt γ1 (t).

Then, using the symbol map, we get 

γ1 (t) =



t 0

(A  Vs )ds

(4.8)

H. Rguigui

 = A

t

Vs ds 0

which gives γ1 (t) =

 t  0

u

 (A  Vs )ds du

0

= G 1 (t). (2) Suppose the equality holds for the order k ≥ 1. Then, using (4.8), we get 

γk+1 (t) = = = =

 t

 (k + 1)(t − s)k (k+1) A  Vs ds (k + 1)! 0   t (t − s)k k A A  Vs ds k! 0 A γk (t) A G k (t).

Then  γk+1 (t) =

t

(A G k (s))ds   s  u k  u 1    t = A ... (A k  Vu )du du 1 . . . du k ds 0 0 0 0   u 1    t  u k+1  u k = ... (A (k+1)  Vu )du du 1 . . . du k du k+1 0

0

0

0

0

= G k+1 (t) as desired. It then follows that for any n ≥ 1, Un (t) =

n  tk

A k   k! k=0   t  n (t − s)k k A  Vs ds + k! 0 k=0

and hence the exact solution of our Cauchy problem (4.4) is given by U (t) = lim Un (t) n→∞

= wex p{t A}   +

 t

wex p{(t − s)A}  Vs ds 0

= wex p{t (G + ∗G ) p }  

Characterization Theorems for the QWN-Gross Laplacian…

+

 t 0

wex p{(t − s)(G + ∗G ) p }  Vs ds.

 Now, the operator wex p {t (G + ∗G ) p } belongs to L F

∗ (eθ1 −1)∗

, F ∗ θ∗

(e 1 −1)

 , by ∗

Lemma 4.3. Then, using the condition ∗

eθ1 − 1 ≤ αθ2∗ , we obtain  L F

∗ (eθ1 −1)∗

, F ∗ θ∗ (e

1 −1)∗



 ⊂ L Fθ2 , Fθ∗2 .

Hence, the operator wex p{t (G +∗G ) p } belongs to L(Fθ2 , Fθ∗2 ) and then Ut is well   defined operator in L(Fθ2 , Fθ∗2 ).

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