Accepted Manuscript
Quantum Generalized Euler Heat Equation Salha Alshaikey, Narjess T. Khalifa, Hakeem A. Othman, Hafedh Rguigui PII: DOI: Reference:
S2405-7223(16)30024-X 10.1016/j.spjpm.2017.10.006 SPJPM 163
To appear in:
St. Petersburg Polytechnical University Journal: Physics and Mathematics
Received date: Accepted date:
25 March 2016 9 October 2017
Please cite this article as: Salha Alshaikey, Narjess T. Khalifa, Hakeem A. Othman, Hafedh Rguigui, Quantum Generalized Euler Heat Equation, St. Petersburg Polytechnical University Journal: Physics and Mathematics (2017), doi: 10.1016/j.spjpm.2017.10.006
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Quantum Generalized Euler Heat Equation Salha Alshaikey Umm Al-Qura University, University College Al-Qunfudha, Saudi Arabia
Narjess T. Khalifa
Hakeem A. Othman
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Umm Al-Qura University, University College Al-Qunfudha, Saudi Arabia
Umm Al-Qura University, University College Al-Qunfudha, Saudi Arabia
Hafedh Rguigui∗
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Department of Mathematics, AL-Qunfudhah University college, Umm Al-Qura University, KSA. Second address: High School of Sciences and Technology of Hammam Sousse, University of Sousse, Tunisia
Abstract
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Based on nuclear algebra of operators acting on spaces of entire functions with θ-exponential growth of minimal type, we introduce the quantum generalized Fourier-Gauss transform, the quantum second quantization as well as the quantum generalized Euler operator of which the quantum differential second quantization and the quantum generalized Gross Laplacian are particular examples. Important relation between the quantum generalized Fourier-Gauss transform, the quantum second quantization and the quantum convolution operator is given. Then, using this relation and under some conditions, we investigate the solution of a initial-value problem associated to the quantum generalized Euler operator. More precisely, we show that the aforementioned solution is the composition of a quantum second quantization and a quantum convolution operator.
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1. Introduction
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Keywords: Quantum Generalized Euler Heat Equation, quantum convolution operator, quantum generalized Fourier-Gauss transform, quantum generalized Euler operator. 2010 MSC: 60H40, 46A32, 46F25, 46G20.
AC
As infinite dimensional analogue of the so-called Euler operator which is defined as the first order differential P operator di=1 xi ∂x∂ i on Rd (see Ref. [11]), the operator ∆E := ∆G + N was studied ([8, 9, 26]) based on the white noise analysis, where ∆G is the Gross Laplacian introduced by Gross [12] and N is the number operator studied in [25] by Piech on infinite dimensional abstract Wiener space as infinite dimensional analogue of the finite dimensional Laplacian. Later on, for K ∈ L(XC , XC0 ) and B ∈ L(XC , XC ) where XC is the complexification of some real nuclear space X, the operator 1 ∆E (K, B) = ∆G ( K) + N(B) 2 was introduced (see [5]) which is called the (infinite dimensional) generalized Euler operator. Moreover it was shown that, under some conditions, ∆E (K, B) is the generator of a one-parameter group transformation using the GK,B transform studied in [16, 8] as well as the existence of a solution of the following generalized Euler heat equation was ∗ Corresponding
author Email address:
[email protected] (Hafedh Rguigui )
Preprint submitted to St. Petersburg Polytechnical University Journal: Physics and Mathematics
November 10, 2017
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studied (see [5]):
∂u(t) = ∆E (K, B)u(t), u(0) = ϕ ∈ Fθ (XC0 ) ∂t where Fθ (XC0 ) is a test space of entire functions with θ−exponential growth (θ is a Young function). Using a technic of calculus based on holomorphic spaces and the Wick symbol map, a QWN-Euler operator ∆Q E was defined (see [6]) as the sum ∆GQ + N Q , where ∆GQ and N Q stand for appropriate QWN counterparts of the Gross Laplacian and the conservation operator, respectively, i.e, for K1 , K2 ∈ L(XC0 , XC0 ), the QWN-Euler operator given by (1)
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Q Q ∆Q E (K1 , K2 ) := ∆G (K1 , K2 ) + NK1 ,K2 .
For a particular case of K1 and K2 , the solution of the Cauchy problem associated to the QWN-Euler operator was 0 worked out in the basis of the QWN coordinate system. In this paper, based on the kernel theorem, for K1 ∈ L(XC,1 , XC,1 ), 0 K2 ∈ L(XC,2 , XC,2 ), B1 ∈ L(XC,1 , XC,1 ) and B2 ∈ L(XC,2 , XC,2 ), where XC,i (i = 1, 2) is the complexification of some real nuclear space Xi , a quantum generalized Euler operator is defined as follows: 1 Q 1 Q ∆Q E (K1 , K2 ; B1 , B2 ) := ∆G ( K1 , K2 ) + N (B1 , B2 ). 2 2
(2)
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The quantum generalized Euler operator in (2) generalizes (1) and most importantly it can give the QWN-Gross Laplacian and the QWN-conservation operator studied in [6], [15] and [23] contrary to the definition in (1). Moreover, under some conditions, we give the solution of the following quantum generalized Euler heat equation ∂Ξt = ∆Q E (K1 , K2 ; B1 , B2 )Ξt , ∂t
0 0 ), Fθ2 (XC,2 )). Ξ0 ∈ L(Fθ∗1 (XC,1
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The paper is organized as follows. In Section 2 we briefly recall well-known results on nuclear algebras of entire holomorphic functions. In Section 3, the quantum generalized Euler operator and the quantum second quantization are studied. In Section 4, we introduce the quantum generalized Fourier-Gauss transform G Q K1 ,K2 ;B1 ,B2 acting on nuclear algebra of white noise operators and some basic properties are shown. In Section 5, we investigate the solution of a initial-value problem associated to the quantum generalized Euler operator ∆Q E (K1 , K2 ; B1 , B2 ).
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2. preliminaries
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First we review basic concepts, notations, and some results which will be needed in the present paper. Development of these and similar results can be found in Refs. [3, 5, 10, 18, 21, 22, 23, 27, 28]. In mathematics, a nuclear space is a locally convex topological vector space such that for any seminorm p we can find a larger seminorm q so that the natural map from Vq to V p is nuclear. Such spaces preserve many of the good properties of finite-dimensional vector spaces. A typical examples of a nuclear spaces are the Schwartz space of smooth functions for which the derivatives of all orders are rapidly decreasing and the space of entire holomorphic functions on the complex plane with θ−exponential growth. Using a separable Hilbert space and a positive self-adjoint operator with Hilbert-Schmidt inverse we can construct a real nuclear space. For i = 1, 2, let Hi be a real separable (infinite dimensional) Hilbert space with inner product h·, ·i and norm | · |0 . Let Ai ≥ 1 be a positive self-adjoint operator in Hi with Hilbert-Schmidt inverse. Then there exist a sequence of positive numbers 1 < λi,1 ≤ λi,2 ≤ · · · and a complete orthonormal basis of Hi , ei,n ∞ n=1 ⊆ Dom(Ai ) such that
For every p ∈ R we define:
Ai ei,n = λi,n ei,n ,
∞ X n=1
|ξ|2p :=
∞ X n=1
−1
2 λ−2 i,n = Ai HS < ∞.
p 2 hξ, ei,n i2 λ2p i,n = Ai ξ 0 ,
ξ ∈ Hi .
For p ≥ 0, the space (Xi ) p , of all ξ ∈ Hi with |ξ| p < ∞, is a Hilbert space with norm | · | p and, if p ≤ q, then (Xi )q ⊆ (Xi ) p . Let (Xi )−p denoting the | · |−p -completion of Hi (p ≥ 0), if 0 ≤ p ≤ q, then (Xi )−p ⊆ (Xi )−q . Hence, we 2
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n o obtain a decreasing chain of Hilbert spaces (Xi ) p with natural continuous inclusions iq,p : (Xi )q ,→ (Xi ) p (p ≤ q). p∈R Defining the following countably Hilbert nuclear space: \ Xi := projlim p→∞ (Xi ) p (Xi ) p p≥0
Its strong dual space Xi0 is given by:
This leads to the triple
p≥0
(Xi )−p .
Xi ⊂ Hi ≡ Hi0 ⊂ Xi0
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[
Xi0 := indlim p→∞ (Xi )−p
(3)
is called a real standard triple [21]. For i = 1, 2, let XC,i be the complexification of the real nuclear space Xi . For 0 p ∈ N, we denote by (XC,i ) p the complexification of (Xi ) p and by (XC,i )−p respectively XC,i the strong dual space of (XC,i ) p and XC,i . Then, we obtain XC,i = pro j lim (XC,i ) p
and
0 = ind lim (XC,i )−p . XC,i p→∞
(4)
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p→∞
0 The spaces XC,i and XC,i are respectively equipped with the projective and inductive limit topology. For all p ∈ N, we 0 denote by |.|−p the norm on (XC,i )−p and by h., .i the C−bilinear form on XC,i × XC,i . In the following, H denote by
b ⊗n the direct Hilbertian sum of (XC,1 )0 and (XC,2 )0 , i.e., H = (XC,1 )0 ⊕ (XC,2 )0 . For n ∈ N, we denote by XC,i the n-fold
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symmetric tensor product on XC,i equipped with the π−topology and by (XC,i )b⊗p n the n-fold symmetric Hilbertian tensor product on (XC,i ) p . We will preserve the notation |.| p and |.|−p for the norms on (XC,i )b⊗p n and (XC,i )b⊗−pn , respectively. For a later use, let θ be a Young function, i.e., it is a continuous, convex and increasing function defined on R+ θ(r) = ∞. Obviously, the conjugate function θ∗ of θ defined by and satisfies the two conditions: θ(0) = 0 and lim r→∞ r
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∀x ≥ 0,
is also a Young function. For every n ∈ N, let
θ∗ (x) := sup(tx − θ(t)), t≥0
eθ(r) . (5) r>0 r n Throughout the paper, we fix a pair of Young function (θ1 , θ2 ). From now on we assume that the Young functions θi (i=1,2) satisfy θi (r) lim < ∞. (6) r→∞ r 2 Note that, if a Young function θ satisfies condition (6), there exist constant numbers α and γ such that
AC
CE
PT
(θ)n = inf
and, for r > 0 such that rγ < 1, we have
(θ)n ≤ α( ∞ X n=0
2eγ n/2 ) n
rn n!(θ)2n < ∞.
(7)
(8)
For a complex Banach space (C, k · k), let H(C) denotes the space of all entire functions on C, i.e. of all continuous C-valued functions on C whose restrictions to all affine lines of C are entire on C. For each m > 0 we denote by Exp(C, θ, m) the space of all entire functions on C with θ−exponential growth of finite type m, i.e. Exp(C, θ, m) = f ∈ H(C); k f kθ,m := sup | f (z)|e−θ(mkzk) < ∞ . z∈C
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The projective system {Exp((XC,i )−p , θ, m); p ∈ N, m > 0} gives the space 0 ) := projlim Fθ (XC,i
p→∞;m↓0
Exp((XC,i )−p , θ, m).
(9)
It is noteworthy that, for each ξ ∈ XC,i , the exponential function eξ (z) := ehz,ξi ,
0 , z ∈ XC,i
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0 0 belongs to Fθ (XC,i ) and the set of such test functions spans a dense subspace of Fθ (XC,i ). For all positive numbers m1 , m2 > 0 and all integers (p1 , p2 ) ∈ N × N, we define the space of all entire functions on (XC,1 )−p1 ⊕ (XC,2 )−p2 with (θ1 , θ2 )−exponential growth by
Exp((XC,1 )−p1 ⊕ (XC,2 )−p2 , (θ1 , θ2 ), (m1 , m2 )) = { f ∈ H((XC,1 )−p1 ⊕ (XC,2 )−p2 ); k f k(θ1 ,θ2 );(p1 ,p2 );(m1 ,m2 ) < ∞} where H((XC,1 )−p1 ⊕ (XC,2 )−p2 ) is the space of all entire functions on (XC,1 )−p1 ⊕ (XC,2 )−p2 and
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k f k(θ1 ,θ2 );(p1 ,p2 );(m1 ,m2 ) = sup{| f (z1 , z2 )|e−θ1 (m1 |z1 |−p1 )−θ2 (m2 |z2 |−p2 ) }
for (z1 , z2 ) ∈ (XC,1 )−p1 ⊕(XC,2 )−p2 . So, the space of all entire functions on (XC,1 )−p1 ⊕(XC,2 )−p2 with (θ1 , θ2 )−exponential growth of minimal type is naturally defined by 0 0 F(θ1 ,θ2 ) (XC,1 ⊕ XC,2 ) = prolim j Exp((XC,1 )−p1 ⊕ (XC,2 )−p2 , (θ1 , θ2 ), (m1 , m2 )). p1 ,p2 →∞,m1 ,m2 ↓0
(10)
0 0 By definition, ϕ ∈ F(θ1 ,θ2 ) (XC,1 ⊕ XC,2 ) admits the Taylor expansions: ∞ X
n,m=0
hx⊗n ⊗ y⊗m , ϕn,m i,
M
ϕ(x, y) =
0 0 (x, y) ∈ XC,1 × XC,2
(11)
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b b ⊗n ⊗m where for all n, m ∈ N we have ϕn,m ∈ XC,1 ⊗ XC,2 and we used the common symbol h., .i for the canonical C−bilinear ⊗n ⊗m 0 ⊗n ⊗m 0 0 form on (XC,1 × XC,2 ) × XC,1 × XC,2 . So, we identify in the next all test function ϕ ∈ F(θ1 ,θ2 ) (XC,1 ⊕ XC,2 ) by their coefficients of its Taylors series expansion at the origin (ϕn,m )n,m∈N and we write ϕ ∼ (ϕn,m )n≥0 . As important example 0 0 of elements in F(θ1 ,θ2 ) (XC,1 ⊕ XC,2 ), we define the exponential function as follows. For a fixed (ξ, η) ∈ XC,1 × XC,2 ,
PT
e(ξ,η) (a, b) = (eξ ⊗ eη )(a, b) = exp{ha, ξi + hb, ηi},
0 0 (a, b) ∈ XC,1 × XC,2 .
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0 0 Let ϕ ∼ (ϕn,m )n≥0 in F(θ1 ,θ2 ) (XC,1 ⊕ XC,2 ). Then, from [18] for any p1 , p2 ≥ 0 and m1 , m2 > 0, there exist q1 > p1 and q2 > p2 such that
|ϕn,m | p1 ,p2
≤
n m en+m (θ1 )n (θ2 )m mn1 mm 2 kiq1 ,p1 kHS kiq2 ,p2 kHS kϕk(θ1 ,θ2 );(q1 ,q2 );(m1 ,m2 ) .
(12)
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0 0 0 0 0 0 Denoted by F(θ∗1 ,θ2 ) (XC,1 ⊕XC,2 ) the topological dual of F(θ1 ,θ2 ) (XC,1 ⊕XC,2 ) called the space of distribution on XC,1 ⊕XC,2 .
b b ⊗n ⊗m 0 0 0 The element Φ ∈ F(θ∗1 ,θ2 ) (XC,1 ⊕ XC,2 ) is given, via its Taylor series, by Φ ∼ (Φn,m )n≥0 where Φn,m ∈ (XC,1 ⊗ XC,2 ) . The 0 0 0 space F(θ1 ,θ2 ) (XC,1 ⊕ XC,2 ) can be considered as a generalization of the space Fθ1 (XC,1 ), i.e., we take XC,2 = {0}, we obtain the following identification 0 0 F(θ1 ,θ2 ) (XC,1 ⊕ {0}) = Fθ1 (XC,1 ) 0 0 0 and F(θ∗1 ,θ2 ) (XC,1 ⊕ XC,2 ) can be considered as a generalization of the space Fθ∗1 (XC,1 ), i.e., 0 0 ). F(θ∗1 ,θ2 ) (XC,1 ⊕ {0}) = Fθ∗1 (XC,1
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3. Quantum generalized Euler operator and quantum second quantization For locally convex spaces X and Y we denote by L(X, Y) the set of all continuous linear operators from X into Y. 0 Let be given Ai ∈ L(XC,i , XC,i ) and Ki ∈ L(XC,i , XC,i ). Motivated by the classical case studied in [5, 7, 8, 19, 21], for ∞ X 0 0 ϕ in the test space F(θ1 ,θ2 ) (XC,1 ⊕ XC,2 ) given by ϕ(x, y) = hx⊗nb ⊗y⊗m , ϕn,m i, we define the operators ∆G (K1 , K2 ), ∆G (K1 , K2 )ϕ(x, y) =
∞ X
n,m=0
n,m=0
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0 0 Γ(A1 , A2 ) and N(A1 , A2 ) on F(θ1 ,θ2 ) (XC,1 ⊕ XC,2 ) as follows:
hx⊗n ⊗ y⊗m , {(n + 2)(n + 1)τ1 (K1 )b ⊗2,0 ϕn+2,m + (m + 2)(m + 1)τ2 (K2 )b ⊗0,2 ϕn,m+2 }i, ∞ X
⊗m hx⊗n ⊗ y⊗m , {A⊗n 1 ⊗ A2 }ϕn,m i,
(14)
hx⊗n ⊗ y⊗m , {γn (A1 ) ⊗ I ⊗m + I ⊗n ⊗ γm (A2 )}ϕn,m i
(15)
Γ(A1 , A2 )ϕ(x, y) = ∞ X
n,m=0
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n,m=0
N(A1 , A2 )ϕ(x, y) =
(13)
where γ j (Ai ), j = n, m and i = 1, 2, is given by j−1 X γ j (Ai ) = I ⊗k ⊗ Ai ⊗ I ⊗( j−1−k) , j ≥ 1 k=0 γ0 (Ai ) = 0.
For completeness the following result is given with proof ultimately connected to our setting.
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0 Proposition 3.1. For any Ai ∈ L(XC,i , XC,i ) and Ki ∈ L(XC,i , XC,i ), the operators ∆G (K1 , K2 ), Γ(A1 , A2 ) and N(A1 , A2 ) 0 0 are linear continuous from F(θ1 ,θ2 ) (XC,1 ⊕ XC,2 ) into itself.
ED
0 0 Proof. Consider ϕ in F(θ1 ,θ2 ) (XC,1 ⊕ XC,2 ) given by
ϕ(x, y) =
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Then, from (13), for p ≥ 0,
AC
CE
|∆G (K1 , K2 )ϕ(x, y)| ≤ +
∞ X
n,m=0 ∞ X
n,m=0
∞ X
n,m=0
hx⊗n ⊗ y⊗m , ϕn,m i.
(n + 2)(n + 1)|τ1 (K1 )|−p |ϕn+2,m | p1 ,p2 |x|n−p1 |y|m −p2 (m + 2)(m + 1)|τ2 (K2 )|−p2 |ϕn,m+2 | p1 ,p2 |x|n−p1 |y|m −p2 .
From (12), it follows that |∆G (K1 , K2 )ϕ(x, y)| ≤
|τ1 (K1 )|−p1 kϕk(θ1 ,θ2 );(q1 ,q2 );(m1 ,m2 ) ∞ X n+2 m × (n + 2)(n + 1)|x|n−p1 en+2 (θ1 )n+2 mn+2 1 kiq1 ,p1 kHS |y|−p2 ×
×
n,m=0 m em (θ2 )m mm 2 kiq2 ,p2 kHS ∞ X n,m=0
+ |τ2 (K2 )|−p2 kϕk(θ1 ,θ2 );(q2 ,q2 );(m1 ,m2 )
m+2 m n n n n (m + 2)(m + 1)|y|m+2 (θ2 )m+2 mm+2 −p2 e 2 kiq2 ,p2 kHS |x|−p1 e (θ1 )n m1 kiq1 ,p1 kHS
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Thus, for any m01 > 0 and m02 > 0, we have 0
0
|∆G (K1 K2 )ϕ(x, y)|e−θ1 (m1 |x|−p1 )−θ2 (m2 |y|−p2 ) ≤ {c1 |τ1 (K1 )|−p1 c2 |τ2 (K2 )|−p2 }kϕk(θ1 ,θ2 );(q1 ,q2 );(m1 ,m2 )
where c1 and c2 are given by ∞ X
n,m=0
c2 =
∞ X
n,m=0
n+2 m m m m (n + 2)(n + 1)|x|n−p1 en+2 (θ1 )n+2 mn+2 1 kiq1 ,p1 kHS |y|−p2 e (θ2 )m m2 kiq2 ,p2 kHS
m+2 n n m+2 n n (m + 2)(m + 1)|y|m+2 (θ2 )m+2 mm+2 −p2 e 2 kiq2 ,p2 kHS |x|−p1 e (θ1 )n m1 kiq1 ,p1 kHS
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c1 =
It remains to show that the constant c1 and c2 are finite number. Obviously we have the inequalities n2 ≤ 22n and θn+m ≤ 2n+m θn θm . Therefore, we have
(n + 2)(n + 1)θn+2 ≤ 23n+6 θn θ2 .
m1 8e m 0 kiq1 ,p1 k HS 1
(16) (17)
2 em m02 kiq2 ,p2 kHS
|Γ(A1 , A2 )ϕ(x, y)| ≤
∞ X
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< 1 and < 1, we conclude Thus using (5), q1 > p1 , q2 > p2 , m1 > 0 and m2 > 0 such that that c1 is finite. Similarly we prove that c2 is finite. This completes the proof of the statement for ∆G (K1 , K2 ). Let be given p1 , p2 ≥ 0. We have
n,m=0
M
Then, using (12), for m01 , m02 > 0, we get
n m |x|n−p1 |y|m −p2 {kA1 k kA2 k }|ϕn,m | p1 ,p2 .
|Γ(A1 , A2 )ϕ(x, y)| ≤
ED
kϕk(θ1 ,θ2 );(q1 ,q2 );(m1 ,m2 ) )n ( )m ( ∞ X 0 n m1 0 m m2 (m1 (θ1 )n |x|−p1 ) × ekiq1 ,p1 kHS kA1 k (θ2 )m (m2 |y|−p2 ) ekiq2 ,p2 kHS kA2 k . m01 m02 n,m=0 m1 m01 ekiq1 ,p1 kHS kA1 k
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Therefore, using (5), for m01 > 0 and m02 > 0 such that 0
0
AC
CE
|Γ(A1 , A2 )ϕ(x, y)| e−θ1 (m1 |x|−p1 )−θ2 (m2 |y|−p2 )
≤
< 1 and
m2 m02 ekiq2 ,p2 kHS kA2 k
< 1, we deduce
kϕk(θ1 ,θ2 );(q1 ,q2 );(m1 ,m2 ) )n ∞ ( X m1 eki k kA k × q1 ,p1 HS 1 m01 n=0 )m ∞ ( X m2 × eki k kA k q2 ,p2 HS 2 m02 m=0
which follows the proof of the statement for Γ(A1 , A2 ). For p1 , p2 ≥ 0 and m1 , m2 > 0, we have |N(A1 , A2 )ϕ(x, y)| ≤
∞ X {γ (A ) + γ (A )}ϕ n 1 m 2 n,m p
n,m=0
1 ,p2
Then, by using (5), (12) and the obvious inequalities γn (A1 )ϕn,m p ,p ≤ 2n kA1 k|ϕn,m | p1 ,p2 , 1 2 γm (A2 )ϕn,m p ,p ≤ 2m kA2 k|ϕn,m | p1 ,p2 . 1
2
6
|x|n−p1 |y|m −p2 .
(18)
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we get ≤
kϕk(θ1 ,θ2 );(q1 ,q2 );(m1 ,m2 ) {kA1 k + kA2 k} )n X )m ∞ ( ∞ ( X m1 m2 × 2e 0 ekiq1 ,p1 kHS eki k q ,p HS 2 2 m1 m02 n=0 m=0
1 Thus, with the assumption 2e m m01 ekiq1 ,p1 kHS k < 1 and N(A1 , A2 ).
m2 m02 ekiq2 ,p2 kHS
< 1, we complete the proof of the statement for
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kN(A1 , A2 )ϕk(θ1 ,θ2 );(p1 ,p2 );(m01 ,m02 )
0 From the nuclearity of the spaces Fθi (XC,i ), we have by Kernel Theorem the following isomorphisms 0 0 0 0 0 0 ) ⊗ Fθ2 (XC,2 )) ' Fθ1 (XC,1 ), Fθ2 (XC,2 L(Fθ∗1 (XC,1 ). ⊕ XC,2 ) ' F(θ1 ,θ2 ) (XC,1
(19)
0 0 0 0 So, for every Ξ ∈ L(Fθ∗1 (XC,1 ), Fθ2 (XC,2 )), the associated kernel ΦΞ ∈ F(θ1 ,θ2 ) (XC,1 ⊕ XC,2 ) is defined by
hhΞϕ, ψii = hhΦΞ , ϕ ⊗ ψii,
0 0 ). ), ∀ψ ∈ Fθ∗2 (XC,2 ∀ϕ ∈ Fθ∗1 (XC,1
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Using the topological isomorphism:
0 0 0 0 ), Fθ2 (XC,2 )) 3 Ξ 7−→ KΞ = ΦΞ ∈ F(θ1 ,θ2 ) (XC,1 ⊕ XC,2 ), L(Fθ∗1 (XC,1
(20)
(21)
0 for Ki ∈ L(XC,i , XC,i ) and Bi ∈ L(XC,i , XC,i ), the quantum generalized Euler operator is defined by
1 Q 1 Q ∆Q E (K1 , K2 ; B1 , B2 ) := ∆G ( K1 , K2 ) + N (B1 , B2 ) 2 2
M
where ∆GQ (K1 , K2 ) and N Q (B1 , B2 ) are given by
∆GQ (K1 , K2 ) = K −1 ∆G (K1 , K2 )K,
ED
N Q (B1 , B2 ) = K −1 N(B1 , B2 )K.
4. Quantum generalized Fourier-Gauss
CE
PT
For any Bi , Ci ∈ L(XC,i , XC,i ), where i = 1, 2, let A1 = C1∗C1 and A2 = C2∗C2 . Then define the generalized Fourier-Gauss transform by Z G A1 ,A2 ;B1 ,B2 ϕ(y1 , y2 ) = ϕ(C1∗ x1 + B∗1 y1 , C2∗ x2 + B∗2 y2 )dµ(x1 )dµ(x2 ) X10 ×X20
AC
0 0 where ϕ ∈ F(θ1 ,θ2 ) (XC,1 ⊕ XC,2 ) given by
ϕ(x1 , x2 ) =
∞ X
n,m=0
hx1⊗n ⊗ x2⊗m , ϕn,m i.
Let τi (Ai ) given by hτi (Ai ), ξ ⊗ ηi = hAi ξ, ηi, ξ, η ∈ XC,i .
Therefore, by a technic similar to the same used in Refs. [5, 27], we can show that G A1 ,A2 ;B1 ,B2 is continuous linear 0 0 operator from F(θ1 ,θ2 ) (XC,1 ⊕ XC,2 ) into itself and G A1 ,A2 ;B1 ,B2 ϕ(y1 , y2 ) =
∞ X
n,m=0
7
⊗m hy⊗n 1 ⊗ y2 , gn,m i
(22)
ACCEPTED MANUSCRIPT
where gn,m is given by gn,m =
∞ ⊗m X (B⊗n (n + 2k)!(m + 2l)! 1 ⊗ B2 ) {(τ1 (A1 ))⊗kb ⊗(τ2 (A2 ))⊗l }b ⊗2k,2l ϕn+2k,m+2l l+k k!l! n!m! 2 k,l=0
⊗q 0 ⊗(m−q) ⊗p 0 ⊗(l−p) ⊗l ⊗m with, for u p ∈ (XC,1 ) , zq ∈ (XC,2 ) ξl−p+m−q ∈ XC,1 ⊗ XC,2 , p ≤ l, q ≤ m, κl,m ∈ XC,1 ⊗ XC,2 , the contraction (u p ⊗ zq ) ⊗ p,q κl,m is defined by
Using the topological isomorphism:
CR IP T
h(u p ⊗ zq ) ⊗ p,q κl,m , ξl−p+m−q i = hκl,m , u p ⊗ ξl−p+m−q ⊗ zq i.
0 0 0 0 ), ), Fθ2 (XC,2 )) 3 Ξ 7−→ KΞ = ΦΞ ∈ Fθ1 ,θ2 (XC,1 ⊕ XC,2 L(Fθ∗1 (XC,1
we can define the quantum generalized Fourier-Gauss transform as follows
(23)
−1 ∗ 0 0 GQ A1 ,A2 ;B1 ,B2 := K G A1 ,A2 ;B1 ,B2 K ∈ L(L(Fθ1 (XC,1 ), Fθ2 (XC,2 ))).
AN US
0 0 0 0 Let Ξξ,η = K −1 (e(ξ,η) ), for (ξ, η) ∈ (XC,1 ×XC,2 ). The set of such operators spans a dense subspace of L(Fθ∗1 (XC,1 ), Fθ2 (XC,2 )). 0 Theorem 4.1. Let Ai ∈ L(XC,i , XC,i ) and Bi ∈ L(XC,i , XC,i ). Then, the quantum generalized Fourier-Gauss transform Q 0 0 G A1 ,A2 ;B1 ,B2 realizes a topological isomorphism on L(Fθ∗1 (XC,1 ), Fθ2 (XC,2 )) if and only if B01 B1 = I, B02 B2 = I, A1 + ∗ 0 ∗ 0 B1 A1 B1 = 0 and A2 + B2 A2 B2 = 0 (zero operator). 0 Proof. Let Ai ∈ L(XC,i , XC,i ) and Bi ∈ L(XC,i , XC,i ), then we get
) hA1 ξ1 , ξ1 i hA2 ξ2 , ξ2 i + e(B1 ξ1 ,B2 ξ2 ) , G A1 ,A2 ;B1 ,B2 (e(ξ1 ,ξ2 ) ) = exp 2 2
ED
for any ξi ∈ XC,i . By Eq. (24), we have (G Q GQ A1 ,A2 ;B1 ,B2 Ξξ1 ,ξ2 ) = A0 ,A0 ;B0 ,B0 1
2
1
M
(
2
PT
=
AC
CE
=
= × = × =
K −1G A01 ,A02 ;B01 ,B02 G A1 ,A2 ;B1 ,B2 (e(ξ1 ,ξ2 ) ) ( ) ! hA1 ξ1 , ξ1 i hA2 ξ2 , ξ2 i K −1G A01 ,A02 ;B01 ,B02 exp + e(B1 ξ1 ,B2 ξ2 ) 2 2 ) ( hA1 ξ1 , ξ1 i hA2 ξ2 , ξ2 i exp + K −1G A01 ,A02 ;B01 ,B02 (e(B1 ξ1 ,B2 ξ2 ) ) 2 2 ( ) hA1 ξ1 , ξ1 i hA2 ξ2 , ξ2 i exp + 2 2 ( 0 ) hA1 B1 ξ1 , B1 ξ1 i hA02 B2 ξ2 , B2 ξ2 i exp + K −1 (e(B01 B1 ξ1 ,B02 B2 ξ2 ) ) 2 2 ) ( h(A1 + B∗1 A01 B1 )ξ1 , ξ1 i h(A2 + B∗2 A02 B2 )ξ2 , ξ2 i + exp 2 2
K −1 (e(B01 B1 ξ1 ,B02 B2 ξ2 ) ) ( ) h(A1 + B∗1 A01 B1 )ξ1 , ξ1 i h(A2 + B∗2 A02 B2 )ξ2 , ξ2 i exp + ΞB01 B1 ξ1 ,B02 B2 ξ2 . 2 2
Using Eq. (24), we get GQ A1 ,A2 ;B1 ,B2 Ξξ,η = exp{
hA1 ξ, ξ A2 η, ηi + }ΞB1 ξ,B2 η . 2 2
From which we obtain 8
(24)
ACCEPTED MANUSCRIPT
Q GQ (G Q A1 ,A2 ;B1 ,B2 Ξξ1 ,ξ2 ) = G A1 +B∗ A0 B1 ,A2 +B∗ A0 B2 ;B0 B1 ,B0 B2 Ξξ1 ,ξ2 . A0 ,A0 ;B0 ,B0 1
2
1
2
1 1
2 2
1
2
Hence, by a density argument, in order to get GQ GQ Ξ=Ξ A0 ,A0 ;B0 ,B0 A1 ,A2 ;B1 ,B2 1
2
1
2
0 0 for any Ξ ∈ L(Fθ∗1 (XC,1 ), Fθ2 (XC,2 )), it is necessary and sufficient that B01 B1 = I, B02 B2 = I, A1 + B∗1 A01 B1 = 0 and ∗ 0 A2 + B2 A2 B2 = 0 (zero operator).
Xi0
for more details see [5, 19]. 0 0 Proposition 4.2. For each Ξ ∈ L(Fθ∗1 (XC,1 ), Fθ2 (XC,2 )), we have
AN US
GQ A1 ,A2 ;B1 B2 (Ξ) = G A1 ,B1 ΞFA2 ,B2 ,
CR IP T
Recall that, for B, C ∈ L(XC,i , XC,i ) and A = C ∗C, the classical G A,B −transform is defined by Z 0 0 ) , ϕ ∈ Fθ (XC,i G A,B ϕ(y) = ϕ(C ∗ x + B∗ y)dµ(x) , y ∈ XC,i
where FA2 ,B2 is the adjoint of G A2 ,B2 . Proof. From equation (24), one can obtain
(25)
(26)
G A1 ,A2 ;B1 B2 = G A1 ,B1 ⊗ G A2 ,B2 .
M
0 0 0 0 Let Ξ ∈ L(Fθ∗1 (XC,1 ), Fθ2 (XC,2 )), ψ ∈ Fθ1 (XC,1 ) and ϕ ∈ Fθ2 (XC,2 ). Then, we get
ED
hhG Q A1 ,A2 ;B1 ,B2 (Ξ)ψ, ϕii
which implies (26).
=
= =
hhG A1 ,A2 ;B1 ,B2 (KΞ), ϕ ⊗ ψii
hhKΞ, (FA1 ,B1 ϕ) ⊗ (FA2 ,B2 ψ)ii hhG A1 ,B1 ΞFA2 ,B2 ψ, ϕii
PT
0 0 0 0 Let Φ ∈ F(θ∗1 ,θ2 ) (EC,1 ⊕ EC,2 ) and ϕ ∈ F(θ1 ,θ2 ) (EC,1 ⊕ EC,2 ) be given. Then, the convolution Φ ∗ ϕ is defined by
(Φ ∗ ϕ)(x, y) := Φ, t(x,y) ϕ ,
CE
with t(x,y) ϕ ∈ Fθ1 ,θ2 (N10 ⊕ N20 ) being the translation operator of ϕ, i.e., (t(x,y) ϕ)(a, b) := ϕ(x + a, y + b),
AC
0 0 0 0 0 0 where x, a ∈ EC,1 and y, b ∈ EC,2 . Let Ξ1 ∈ L(Fθ1 (EC,1 ), Fθ∗2 (EC,2 )) and Ξ2 ∈ L(Fθ∗1 (EC,1 ), Fθ2 (EC,2 )). Then, we define Q the convolution of Ξ1 and Ξ2 denoted by Ξ1 ∗ Ξ2 as follows
Ξ1 ∗Q Ξ2 = K −1 (K(Ξ1 ) ∗ K(Ξ2 )).
The last quantum convolution product Ξ1 ∗Q Ξ2 can be written as follows Ξ1 ∗Q Ξ2 = CΞQ1 (Ξ2 ) where CΞQ1 denote the quantum convolution operator.
9
ACCEPTED MANUSCRIPT
0 0 Theorem 4.3. Let Ξ ∈ L(Fθ∗1 (EC,1 ), Fθ2 (EC,2 )). Then, we have
Q Q Q GQ A1 ,A2 ;B1 ,B2 (Ξ) = Γ (B1 , B2 ) µA1 ,A2 ∗ Ξ
or equivalently
Q Q GQ A1 ,A2 ;B1 ,B2 = Γ (B1 , B2 )oC Q µA
1 ,A2
µQ A1 ,A2
where
µn,m =
(
−1
= K µA1 ,A2 K, µA1 ,A2 ∈
0
if
(τ1 (A1 ))⊗k b ⊗(τ2 (A2 ))⊗l if 2k+l k!l! 0 0 EC,2 ) and Φ ∈ F(θ∗1 ,θ2 ) (EC,1
0 Proof. Let ϕ ∈ F(θ1 ,θ2 ) (EC,1 ⊕ Φ ∼ (Φn,m )n,m∈N respectively, then
Φ∗ϕ= see [15]. Using (27) and (22), we get ⊕
0 EC,2 )
µn,m = Then, using (22), we get
(
n = 2k + 1 or m = 2l + 1 n = 2k and m = 2l
0 ⊕ EC,2 ) given by ϕ(x1 , x2 ) =
∞ X (n + k)!(m + l)! Φk,lb ⊗k,l ϕn+k,m+l n!m! n,m∈N k,l=0
G A1 ,A2 ,I,I ϕ = µA1 ,A2 ∗ ϕ
P∞
⊗n n,m=0 hx1
⊗ x2⊗m , ϕn,m i and (27)
given by µA1 ,A2 ∼ (µn,m )n,m and µn,m is defined by 0
if
(τ1 (A1 ))⊗k b ⊗(τ2 (A2 ))⊗l 2k+l k!l!
G A1 ,A2 ;B1 ,B2 ϕ(y1 , y2 )
if
n = 2k + 1 or m = 2l + 1 n = 2k and m = 2l
= G A1 ,A2 ;I,I ϕ(B∗1 y1 , B∗2 y2 )
M
= (µA1 ,A2 ∗ ϕ)(B∗1 y1 , B∗2 y2 )
On the other hand,
ED
Γ(B1 , B2 )ϕ(y1 , y2 ) =
CE
PT
=
Then, we obtain
0 ⊕ EC,2 ) given by µA1 ,A2 ∼ (µn,m )n,m and
AN US
where µA1 ,A2 ∈
0 F(θ∗1 ,θ2 ) (EC,1
0 F(θ∗1 ,θ2 ) (EC,1
CR IP T
0 0 on L(Fθ∗1 (EC,1 ), Fθ2 (EC,2 )), µn,m is defined by
From which we get
n,m=0 ∞ X
⊗m ⊗n ⊗m hy⊗n 1 ⊗ y2 , (B1 ⊗ B2 )ϕn,m i
h(B∗1 , y1 )⊗n ⊗ (B∗2 y2 )⊗m , ϕn,m i
n,m=0 ϕ(B∗1 y1 , B∗2 y2 )
G A1 ,A2 ;B1 ,B2 ϕ(y1 , y2 ) = ΓQ (B1 , B2 )(µA1 ,A2 ∗ ϕ)(y1 , y2 ). Q Q Q GQ A1 ,A2 ;B1 ,B2 (Ξ) = Γ (B1 , B2 )((µA1 ,A2 ∗ Ξ))
AC
This completes the proof.
=
∞ X
5. Quantum generalized Euler heat equation
In this section we show that, under some conditions, ∆Q E (K1 , K2 ; B1 , B2 ) is the generator of a one-parameter group transformation and we use the quantum generalized Fourier-Gauss transform G Q K1 ,K2 ;B1 ,B2 to investigate the existence of a solution of the following quantum generalized Euler heat equation ∂Ξt 0 0 = ∆Q Ξ0 ∈ L(Fθ∗1 (XC,1 ), Fθ2 (XC,2 )). (28) E (K1 , K2 ; B1 , B2 )Ξt , ∂t 0 0 0 0 We denote by GL(F(θ1 ,θ2 ) (XC,1 ⊕ XC,2 )) the group of all linear homeomorphisms from F(θ1 ,θ2 ) (XC,1 ⊕ XC,2 ) onto itself. 10
ACCEPTED MANUSCRIPT
0 Lemma 5.1. Let Ai ∈L(XC,i , XC,i ), then {Γ(etA1 , etA2 )}t∈R is a regular one- parameter subgroup of GL(F(θ1 ,θ2 ) (XC,1 ⊕ 0 XC,2 )) with infinitesimal generator the operator N(A1 , A2 ).
Proof. We have i = 1, 2 and j = n, m, (e )
⊗j
=
I
+
j−1 X k=0
where C kj =
j! k!( j−k)! .
+ t jI
j−2 X ⊗Ai + C kj I ⊗kb ⊗(tAi )⊗( j−k)
⊗( j−1)b
k=0
C kj (I + tAi )⊗kb ⊗{t2 A2i
∞ X (tAi )2 ⊗( j−k) } (l + 2)! l=0
0 0 Then, for ϕ in F(θ1 ,θ2 ) (XC,1 ⊕ XC,2 ) given by ϕ(x, y) = tA1
Γ(e
tA2
,e
CR IP T
tAi ⊗ j
∞ X
n,m=0
hx⊗n ⊗ y⊗m , ϕn,m i, one can write
)ϕ(x, y) = ϕ(x, y) + tN(A1 , A2 )ϕ(x, y) + t2 ε(t, A1 , A2 )ϕ(x, y)
where ε(t, A1 , A2 ) is given by +
+ +
{[I ⊗n ⊗ (ε1m (A2 ) + ε2m (A2 )) + γn (A1 ) ⊗ γm (A2 ) γn (A1 ) ⊗ (ε1m (A2 ) + ε2m (A2 ))
AN US
ε(t, A1 , A2 )ϕ ∼
(29)
(ε1n (A1 ) + ε2n (A1 )) ⊗ (I ⊗m + γm (A2 ))
(ε1n (A1 ) + ε2n (A1 )) ⊗ (ε1m (A2 ) + ε2m (A2 ))]ϕn,m }
k=0
ED
Now, for |t| ≤ 1, we can show easily that (
M
where ε1j (Ai ) and ε2j (Ai ), j = n, m and i = 1, 2, are given by j−1 ∞ X X (tAi )l ⊗( j−k) k 2 j−2k−2 ⊗k b 2 1 ε (A ) = C t (I + tA ) ) ⊗ (A i i i j j (l + 2)! k=0 l=0 j−2 X j−k) 2 b t j−k−2C kj A⊗( ⊗I ⊗k . (A ) = ε i i j |ε1j (Ai )| pi ≤ (1 + kAi k + kAi k2 ekAi k ) j |ε2j (Ai )| pi ≤ (1 + kAi k) j .
We get
⊗n ⊗ (ε1m (A2 ) + ε2m (A2 ))}ϕn,m p ,p |x|n−p1 |y|m −p2 {I
AC
∞ X
CE
PT
Step 1 : Let p1 , p2 ≥ 0 and m1 , m2 > 0. From (12) and (5), for q1 > p1 , q2 > p2 and m01 , m02 > 0 such that ( ) m1 m2 2 kA2 k max ekiq1 ,p1 kHS , 0 ekiq2 ,p2 kHS (2 + 2kA2 k + kA2 k e ) < 1. m01 m2
n,m=0
1
2
∞ X
(m01 |x|−p1 )n (θ1 )n {
m1 ekiq1 ,p1 kHS }n m01
≤
kϕk(θ1 ,θ2 );(q1 ,q2 );(m1 ,m2 )
×
∞ X m2 (m02 |y|−p2 )m (θ2 )m { 0 ekiq2 ,p2 kHS (2 + 2kA2 k + kA2 k2 ekA2 k )}m m 2 m=0
≤ ×
n=0
0
0
kϕk(θ1 ,θ2 );(q1 ,q2 );(m1 ,m2 ) eθ1 (m1 |x|−p1 )+θ2 (m2 |y|−p2 ) ∞ X m1 m2 { 0 ekiq1 ,p1 kHS }n { 0 ekiq2 ,p2 kHS (2 + 2kA2 k + kA2 k2 ekA2 k )}m m m2 1 n,m=0 11
(30)
ACCEPTED MANUSCRIPT
the last summation is a finite number s1 . Then we have ∞ X
n,m=0
⊗n 0 0 |x|n−p1 |y|m ⊗ (ε1m (A2 ) + ε2m (A2 ))}ϕn,m p ,p ≤ s1 kϕk(θ1 ,θ2 );(q1 ,q2 );(m1 ,m2 ) eθ1 (m1 |x|−p1 )+θ2 (m2 |y|−p2 ) . −p2 {I 1
2
Similarly, we get
where
1
s2 =
∞ X m2 m1 { 0 ekiq1 ,p1 kHS (2 + 2kA1 k + kA1 k2 ekA1 k )}n { 0 ekiq2 ,p2 kHS }m < ∞. m1 m2 n,m=0
Step 2 : Using (30) and (18), we get {γn (A1 )b ⊗(ε1m (A2 ) + ε2m (A2 ))}ϕn,m p Therefore, in view of (12) and (5) we obtain
n,m=0
1 ,p2
≤ 2n kA1 k(2 + 2kA2 k + kA2 k2 ekA2 k )m |ϕn,m | p1 ,p2 .
2 b 1 |x|n−p1 |y|m −p2 {γn (A1 )}⊗(εm (A2 ) + εm (A2 ))}ϕn,m p
≤
1 ,p2
×
M
∞ X
2
CR IP T
n,m=0
⊗n 0 0 |x|n−p1 |y|m ⊗ (ε1n (A1 ) + ε2n (A1 ))}ϕn,m p ,p ≤ s2 kϕk(θ1 ,θ2 );(q1 ,q2 );(m1 ,m2 ) eθ1 (m1 |x|−p1 )+θ2 (m2 |y|−p2 ) −p2 {I
AN US
∞ X
(
∞ X
0
2m1 ekiq1 ,p1 kHS }n m01 n,m=0 m2 {(2 + 2kA2 k + kA2 k2 ekA2 k ) 0 ekiq2 ,p2 kHS }m . m2 kA1 k{
ED
Under the assumption
×
0
kϕk(θ1 ,θ2 );(q1 ,q2 );(m1 ,m2 ) eθ1 (m1 |x|−p1 )+θ2 (m2 |y|−p2 )
m1 m2 max ekiq1 ,p1 kHS , (2 + 2kA2 k + kA2 k2 ekA2 k ) 0 ekiq2 ,p2 kHS 2m01 m2
)
< 1,
∞ X
where
1 2 b |x|n−p1 |y|m −p2 {(εn (A1 ) + εn (A1 ))⊗γm }ϕn,m p
CE
n,m=0
PT
the last summation converges to a finite number s3 . Similarly, we get
AC
s4 =
which is finite.
Step 3 : Using (18), we get
∞ X
n,m=0
0
1 ,p2
kA2 k{(2 + 2kA1 k + kA1 k2 ekA1 k )
{γn (A1 )b ⊗γm (A2 )}ϕn,m p
1 ,p2
m1 n 2m2 ekiq2 ,p2 kHS }m 0 ekiq1 ,p1 kHS } { m1 m02
≤ 2n+m kA1 kkA2 k|ϕn,m | p1 ,p2 .
Therefore, in view of (12) and (5) we obtain
12
0
≤ s4 kϕk(θ1 ,θ2 );(q1 ,q2 );(m1 ,m2 ) eθ1 (m1 |x|−p1 )+θ2 (m2 |y|−p2 )
ACCEPTED MANUSCRIPT
∞ X
n,m=0
b |x|n−p1 |y|m −p2 {γn (A1 )⊗γm (A2 )}ϕn,m p
1 ,p2
0
0
kϕk(θ1 ,θ2 );(q1 ,q2 );(m1 ,m2 ) eθ1 (m1 |x|−p1 )+θ2 (m2 |y|−p2 ) ∞ ∞ X X m2 m1 { 0 2ekiq2 ,p2 kHS }m kA1 kkA2 k. { 0 2ekiq1 ,p1 kHS }n m m 1 2 n,m=0 n,m=0
≤
×
CR IP T
The last summation converges to a finite positive number s5 with the assumption ) ( m2 m1 2eki k , 2eki k max q1 ,p1 HS q2 ,p2 HS < 1. m01 m02 Step 4 : Using (30), (12) and (5) we get
n,m=0
1 2 2 b 1 |x|n−p1 |y|m −p2 {(εn (A1 ) + εn (A1 ))⊗(εm (A2 ) + εm (A2 ))}ϕn,m p ≤
× ×
1 ,p2
AN US
∞ X
0
0
kϕk(θ1 ,θ2 );(q1 ,q2 );(m1 ,m2 ) eθ1 (m1 |x|−p1 )+θ2 (m2 |y|−p2 ) ∞ X m1 {(2 + 2kA1 k + kA1 k2 ekA1 k ) 0 2ekiq1 ,p1 kHS }n m1 n=0 ∞ X m=0
{(2 + 2kA2 k + kA2 k2 ekA2 k )
M
for j = 1, 2 with the assumptions
(2 + 2kA j k + kA j k2 ekA j k )
m2 2ekiq2 ,p2 kHS }m m02
mj ekiq j ,p j kHS < 1, m0j
ED
the last summation is some constant number s6 . Finally, we get kε(t, A1 , A2 )ϕk
≤ kϕk(θ1 ,θ2 );(q1 ,q2 );(m1 ,m2 )
PT
(θ1 ,θ2 );(p1 ,p2 );(m01 ,m02 )
6 X
si .
i=1
From (29) it follows that
CE
tA tA
Γ(e 1 , e 2 )ϕ − ϕ
− N(A , A )ϕ
1 2
(θ ,θ t 1
0 0 2 );(p1 ,p2 );(m1 ,m2 )
≤ |t|kϕk(θ1 ,θ2 );(q1 ,q2 );(m1 ,m2 )
which completes the proof.
6 X
si
i=1
AC
0 For i = 1, 2, let Ki ∈ L(XC,i , XC,i ), for T i ∈ L(XC,i ⊗ XC,i , XC,i ⊗ XC,i ), we define τi (Ki ) ◦ T i ∈ (XC,i ⊗ XC,i ) such that for every f, g ∈ XC,i hτi (Ki ) ◦ T i , f ⊗ gi := hτi (Ki ), T i ( f ⊗ g)i. 0 By the kernel theorem, there exist a unique operator ΞTKii ∈ L(XC,i , XC,i ) with kernel τi (Ki ) ◦ T i . In the particular case Bi ⊗2 where T Bi := (e ) − I ⊗ I, for Bi ∈ L(XC,i , XC,i ), is the kernel of the operator TB
∗
ΞKi i = e(Bi ) Ki eBi − Ki . Now, for α ∈ C, we define
2 ΥKB11,B ,K2 ;t
GtK1 ,tK2 ;etB1 ,etB2 , := G 1 ΞTKtB1 , 1 ΞTKtB2 ;etB1 ,etB2 , 2α
1
2α
13
2
if α = 0 i f α , 0.
ACCEPTED MANUSCRIPT
0 0 0 0 2 It is easy to show that {ΥKB11,B ,K2 ;t }t∈R ⊂ L(F(θ1 ,θ2 ) (XC,1 ⊕ XC,2 ), F(θ1 ,θ2 ) (XC,1 ⊕ XC,2 )) is a one-parameter transformation group. We put −1 B1 ,B2 2 QKB11,B ,K2 := K ΥK1 ,K2 ;t K.
Then, using this transformation group, we give in the following theorem the explicit solution of the Euler equation (28).
2α ΞK1
1;
2α ΞK2
CR IP T
0 Theorem 5.1. Let Ki ∈ L(XC,i , XC,i ) and Bi ∈ L(XC,i , XC,i ) satisfying τi (Ki ) ◦ (Bi ⊗ I) = ατi (Ki ) for i = 1, 2 and α ∈ C. Then Q tB tB Q Q 1 2 Γ (e , e ) µ ∗ Ξ ifα = 0 0 tK ,tK Q1 2 2 Ξt = QKB11,B Q Q tB1 tB2 ,K2 ;t Ξ0 = ∗ Ξ i f α,0 Γ (e , e ) µ 0 T tB T tB 1 1 2
is the unique solution of the generalized Euler heat equation (28).
2 Proof. To prove the statement, we shall prove that {ΥKB11,B ,K2 ;t }t∈R is a differentiable one-parameter transformation group 0 with infinitesimal generator the operator ∆E (K1 , K2 ; B1 , B2 ), i.e, for any p1 , p2 ≥ 0, m01 , m02 > 0 and ϕ ∈ F(θ1 ,θ2 ) (XC,1 ⊕ 0 XC,2 ), we have
B ,B
ΥK11 ,K22 ;t ϕ − ϕ 1 = 0. − ( ∆G (K1 , K2 ) + N(B1 , B2 ))ϕ
lim
t→0 t 2
∞ X
Let ϕ(x, y) =
0 0 hx⊗n y⊗m , ϕn,m i ∈ F(θ1 ,θ2 ) (XC,1 ⊕ XC,2 ) be given. Then we have
n,m=0
B ,B
ΥK11 ,K22 ;t ϕ−ϕ
1 − ( ∆ (K , K ) + N(B , B ))ϕ
1 2 1 2 t 2 G (θ ,θ 1
+
where
=
1 2α τi (Ki )
tB tB
Γ(e 1 , e 2 )ϕ − ϕ
− N(B1 , B2 ))ϕ
(θ ,θ );(p ,p );(m0 ,m0 ) t 1 2 1 2 1 2
(t)
(t)
F1 (θ ,θ );(p ,p );(m0 ,m0 ) + F2 (θ ,θ );(p ,p );(m0 ,m0 ) 1 2 1 2 1 2 1 2 1 2 1 2
(t)
(t)
F3 (θ ,θ );(p ,p );(m0 ,m0 ) + F4 (θ ,θ );(p ,p );(m0 ,m0 )
ED
+
0 0 2 );(p1 ,p2 );(m1 ,m2 )
M
≤
τ(t) i (Ki )
AN US
(θ1 ,θ2 );(p1 ,p2 );(m01 ,m02 )
1
2
1
2
1
1
2
2
1
2
1
2
◦ T tBi for i = 1, 2 and
PT
F1(t) ∼ {(n + 2)(n + 1)[(etB1 )⊗nb ⊗(etB2 )⊗m
τ(t) τ1 (K1 ) 1 (K1 ) b b ⊗2,0 ϕn+2,m − ⊗2,0 ϕn+2,m ]}, 2t 2
τ(t) 2 (K2 ) b ⊗0,2 ϕn,m+2 − τ2 (K2 )b ⊗0,2 ϕn,m+2 ]}, t ∞ ⊗k b (t) ⊗l τ(t) (etB1 )⊗nb ⊗(etB2 )⊗m X (n + 2k)!(m + 2l)! 1 (K1 ) ⊗τ2 (K2 ) ∼{ × [ ]b ⊗2k,2l ϕn+2k,m+2l }, l+k n!m! t 2 k!l! k=2,l=0
CE
F2(t) ∼ {(m + 2)(m + 1)[(etB1 )⊗nb ⊗(etB2 )⊗m
∞ ⊗l ⊗k b (t) τ(t) (etB1 )⊗nb ⊗(etB2 )⊗m X (n + 2k)!(m + 2l)! 1 (K1 ) ⊗τ2 (K2 ) ∼{ × [ ]b ⊗2k,2l ϕn+2k,m+2l }. n!m! t 2l+k k!l! k=0,l=2
AC
F3(t)
F4(t)
From Lemma 5.1 we get
It remains to prove
tB tB
Γ(e 1 , e 2 )ϕ − ϕ
lim − N(B1 , B2 )ϕ
(θ t→0 t lim t→0
4 X
F (t)
i (θ ,θ i=1
1
= 0. 0 0 1 ,θ2 );(p1 ,p2 );(m1 ,m2 )
0 0 2 );(p1 ,p2 );(m1 ,m2 )
14
= 0.
ACCEPTED MANUSCRIPT
Step 1 : We shall prove
lim
F1(t)
(θ t→0
Observe that τ(t) (K ) (etB1 )⊗nb ⊗2,0 ϕn+2,m − ⊗(etB2 )⊗m 1 2t 1 b
τ1 (K1 ) b 2 ⊗2,0 ϕn+2,m
0 0 1 ,θ2 );(p1 ,p2 );(m1 ,m2 )
= 0.
p1 ,p2
p1 ,p2
CR IP T
τ(t) (K1 ) b ⊗2,0 ϕn+2,m − τ1 (K1 )b ⊗2,0 ϕn+2,m ]| p1 ,p2 |(etB1 )⊗nb ⊗(etB2 )⊗m [ 1 t h i + (etB1 )⊗nb ⊗((etB2 )⊗m − I ⊗m ) τ1 (K1 )b ⊗2,0 ϕn+2,m p1 ,p2 h i tB1 ⊗n ⊗n b ⊗m + ((e ) − I )⊗I τ1 (K1 )b ⊗2,0 ϕn+2,m .
≤
Then, by using the condition τ1 (K1 B1 ) = τ1 (αK1 ) which is equivalent to the condition τ1 (K1 ) ◦ (B1 ⊗ I) = ατ1 (K1 ), for any t , 0, we have
On the other hand, we have
− τ1 (K1 )b ⊗2,0 ϕn+2,m ]| p1 ,p2
≤ 2|t|em|t|kB2 k |τ1 (K1 )|−p1 kB1 ke(n+2)|t|kB1 k |ϕn+2,m | p1 ,p2 .
≤
m−1 X k=0
|[(etB1 )⊗nb ⊗((etB2 )⊗(m−1−k) ⊗ (etB2 − I) ⊗ I ⊗k )]ϕn,m | p1 ,p2 m−1 X
≤
|t|kB2 ken|t|kB1 k e|t|kB2 k
≤
|t|kB2 ken|t|kB1 k e|t|kB2 k m(e|t|kB2 k )m |ϕn,m | p1 ,p2
ED
|[(etB1 )⊗nb ⊗((etB2 )⊗m − I ⊗m )]ϕn,m | p1 ,p2
≤
Then, we deduce
AN US
τ(t) 1 (K1 ) b ⊗2,0 ϕn+2,m t
M
|(etB1 )⊗nb ⊗(etB2 )⊗m [
k=0
e|t|kB2 k(m−1−k) |ϕn,m | p1 ,p2
|t|kB2 ken|t|kB1 k e|t|kB2 k (2e|t|kB2 k )m |ϕn,m | p1 ,p2 .
PT
|[(etB1 )⊗nb ⊗((etB2 )⊗m − I ⊗m )]τ1 (K1 )b ⊗2,0 ϕn+2,m | p1 ,p2
≤ |t|kB2 k|τ1 (K1 )|−p1 en|t|kB1 k e|t|kB2 k (2e|t|kB2 k )m |ϕn+2,m | p1 ,p2 .
CE
Similarly, we can show that
|[((etB1 )⊗n − I ⊗n )b ⊗I ⊗m ]τ1 (K1 )b ⊗2,0 ϕn+2,m | p1 ,p2
≤ |t|kB1 k|τ1 (K1 )|−p1 e|t|kB1 k (2e|t|kB1 k )n |ϕn+2,m | p1 ,p2 .
AC
Hence, by using (12), (17), (16) and the obvious inequalities e|t|kBi k ≤ e2|t|kBi k ,
for q1 > p1 , q2 > p2 and m01 , m02 > 0, we get
(t)
F1 (θ ,θ 1
0 0 2 );(p1 ,p2 );(m1 ,m2 )
≤
en|t|kB1 k ≤ (2e|t|kB1 k )n ,
em|t|kB2 k ≤ (2e|t|kB2 k )m ,
|t| kϕk(θ1 ,θ2 );(q1 ,q2 );(m1 ,m2 ) [kB2 ke|t|kB2 k + 2kB1 ke2|t|kB1 k ]
|τ1 (K1 )|−p1 (8em1 kiq1 ,p1 kHS )2 (θ1 )2 ∞ X m1 m2 × { 0 16ee|t|kB1 k kiq1 ,p1 kHS }n { 0 2ee|t|kB2 k kiq2 ,p2 kHS }m . m m2 1 n,m=0
×
15
ACCEPTED MANUSCRIPT
The last summation converges with the assumption that m1 m2 |t|kB1 k kiq1 ,p1 kHS , 2 0 ee|t|kB2 k kiq2 ,p2 kHS } < 1. 0 16ee m1 m2
Therefore we deduce that
lim
F1(t)
(θ ,θ t→0
1
as desired. Step 2 : Similarly, using the same technic used in Step 1, we get
lim
F2(t)
(θ ,θ t→0
as desired. Step 3 : We shall prove
1
lim
F3(t)
(θ t→0
0 0 2 );(p1 ,p2 );(m1 ,m2 )
0 0 2 );(p1 ,p2 );(m1 ,m2 )
0 0 1 ,θ2 );(p1 ,p2 );(m1 ,m2 )
|
=0
= 0.
AN US
By using the inequality (n + 2k)! ≤ 2n+4k n!(k!)2 , we calculate
=0
CR IP T
max{
∞ (etB1 )⊗nb ⊗(etB2 )⊗m X (n + 2k)!(m + 2l)! (t) ⊗l b [τ1 (K1 )⊗kb ⊗τ(t) 2 (K2 ) ]⊗2k,2l ϕn+2k,m+2l | p1 ,p2 l+k k!l! n!m! 2 k=2,l=0
≤
∞ 1 |t|kB1 k n |t|kB2 k m X (t) k l (2e ) (2e ) k!l!(8|τ(t) 1 (K1 )|−p1 ) (8|τ2 (K2 )|−p2 ) |ϕn+2k,m+2l | p1 ,p2 . |t| k=2,l=0
ED
M
2|t|kBi k Then using the inequality (12) and |τ(t) |τi (Ki )|−pi , for any q1 > p1 , q2 > p2 , m1 , m2 > 0 and |t| < 1, i (Ki )|−pi ≤ |t|e we have ∞ (etB1 )⊗nb ⊗(etB2 )⊗m X (n + 2k)!(m + 2l)! (t) ⊗l b | [τ1 (K1 )⊗kb ⊗τ(t) 2 (K2 ) ]⊗2k,2l ϕn+2k,m+2l | p1 ,p2 l+k k!l! n!m! 2 k=2,l=0
≤
{4em2 kiq2 ,p2 kHS ekB2 k }m (θ2 )m ∞ √ X 2k { 32em1 ekB1 k |τ1 (K1 )|1/2 −p1 kiq1 ,p1 kHS } k!(θ1 )2k
PT
×
|t|kϕk(θ1 ,θ2 );(q1 ,q2 );(m1 ,m2 ) {4m1 ekiq1 ,p1 kHS ekB1 k }n (θ1 )n
CE
×
×
k=2
∞ √ X 2l { 32em2 ekB2 k |τ2 (K2 )|1/2 −p2 kiq2 ,p2 kHS } l!(θ2 )2l . l=0
Using (8) and the following assumptions
AC
√ max{ 32em1 ekB1 k |τ1 (K1 )|1/2 −p1 kiq1 ,p1 kHS , √ 32em2 ekB2 k |τ2 (K2 )|1/2 −p2 kiq2 ,p2 kHS } < 1.
the convergence of the product of the last two sums to a constant number c(F3 ) is assured. Therefore, using (5), for m01 , m02 > 0, we get
(t)
F3 (θ ,θ );(p ,p );(m0 ,m0 ) ≤ |t| kϕk(θ1 ,θ2 );(q1 ,q2 );(m1 ,m2 ) c(F3 ) 1
2
1
2
1
2
×
∞ X
n,m=0
{4
m1 m2 ekiq1 ,p1 kHS ekB1 k }n {4 0 ekiq2 ,p2 kHS ekB2 k }m . m01 m2 16
ACCEPTED MANUSCRIPT
Under the assumption max{4
m1 m2 kB1 k , 4 0 ekiq2 ,p2 kHS ekB2 k } < 1 0 ekiq1 ,p1 kHS e m1 m2
the last summation converges and we obtain
lim
F3(t)
(θ t→0
0 0 1 ,θ2 );(p1 ,p2 );(m1 ,m2 )
= 0.
lim
F4(t)
(θ t→0
0 0 1 ,θ2 );(p1 ,p2 );(m1 ,m2 )
To complete the proof, we use Theorem 4.3. References
AN US
References
= 0.
CR IP T
Step 4 : Using the same technic used in Step 3, we get
AC
CE
PT
ED
M
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