Properties of delta functions of a class of observables on white noise functionals

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1 J. Math. Anal. Appl. 39 7) Properties of delta functions of a class of observables on white noise functionals aishi Wang School of Mathematics and Information Science, Northwest Normal University, Lanzhou, Gansu 737, PR hina Received 3 January 6 Available online 8 August 6 Submitted by Steven G. Krantz Abstract Let δ a be the Dirac delta function at a R and E) L ) E) the canonical framework of white noise analysis over white noise space E,μ),whereE = S R).Forh H = L R) with h, denote by M h the operator of multiplication by W h =,h in L ). In this paper, we first show that M h is δ a - composable. Thus the delta function δ a M h ) makes sense as a generalized operator, i.e. a continuous linear operator from E) to E). We then establish a formula showing an intimate connection between δ a M h ) as a generalized operator and δ a W h ) as a generalized functional. We also obtain the representation of δ a M h ) as a series of integral kernel operators. Finally we prove that δ a M h ) depends continuously on a R. 6 Elsevier Inc. All rights reserved. Keywords: Delta function of observable; White noise analysis; Donsker s delta function. Introduction Let δ a be the Dirac delta function at a R and Q an observable, i.e. a self-adjoint operator in a Hilbert space. Then the delta function δ a Q) of Q is of physical significance cf. []), though it is highly singular from the mathematical point of view. In [5], we presented an idea to define the delta function of an observable within the context of white noise theory. More precisely, for a Schwartz generalized function ω S R), we introduced a notion of ω-composable observ- Supported by National Natural Science Foundation of hina 5765), Natural Science Foundation of Gansu Province, hina and NWNU-KJXG-. address: wangcs@nwnu.edu.cn. -47X/$ see front matter 6 Elsevier Inc. All rights reserved. doi:.6/j.jmaa.6.7.4

2 94. Wang / J. Math. Anal. Appl. 39 7) 93 9 ables on white noise functionals. And we defined the ω-function of such an observable as the generalized operator determined by the spectral density of the observable. Let E) L ) E) be the canonical framework of white noise analysis over E,μ), where E = S R) and μ the standard Gauss measure on E.Leth H = L R) with h. Then W h =,h is a Gauss random variable on E,μ). In particular, B t = W [,t] =, [,t], t, form a standard Brownian motion on E,μ). It is known that δ a W h ), of which Donsker s delta function δ a B t ) is a special case, may be realized as a generalized functional, i.e. an element of E) cf. [8]). Let M h be the operator of multiplication by W h in L ). Obviously, M h is an observable, which is closely intertwined with creation and annihilation operators in symmetric Fock space ƔH ) cf. [6]). In this paper, we apply the idea of [5] to observable M h. We first show that M h is δ a -composable for each h H = L R) with h and each a R. Thus the delta function δ a M h ) makes sense as a generalized operator, i.e. a continuous linear operator from E) to E). We then establish a formula showing an intimate connection between δ a M h ) as a generalized operator and δ a W h ) as a generalized functional. We also obtain the representation of δ a M h ) as a series of integral kernel operators. Finally we prove that δ a M h ) depends continuously on a R. We now fix some general notation. Throughout the paper, R and stand for the real line and complex plane, respectively. For a complex Hilbert space, we take its inner product to be linear in the first argument and conjugate linear in the second. For a real locally convex space V,we denote by V its complexification. If, is the canonical bilinear form on V V, then the canonical bilinear forms on V V and V n ) V n are still denoted by,. Similarly, if V is a real Hilbert space with norm, the norms of V and V n will be denoted by the same symbol.. Framework of white noise analysis In this section we briefly recall some notions, notation and facts in white noise analysis. For details see [3 5,8,]. Let H L R,dt; R) be the Hilbert space of real valued square integrable functions on R with norm and inner product,.leta = + t d /dt be the harmonic oscillator. Then A has a self-adjoint extension in H, which is still denoted by A. It is known that A has a bounded inverse operator A in H. And moreover, for α>, A α is a Hilbert Schmidt operator. For each integer p, lete p be the completion of Dom A p with respect to the Hilbertian norm p = A p. Then E p and E p can be regarded as each other s dual if we identify H with its dual. Let E = p E p be the projective limit of E p p and E the topological dual of E. Then E is a nuclear space and E = p E p is the inductive limit of E p p. Hence we have a real Gel fand triple E H E..) We denote by, the canonical bilinear form on E E, which is consistent with the inner product of H. It is known that E coincides with the Schwartz rapidly decreasing function space SR). Hence E coincides with the Schwartz generalized function space S R).

3 . Wang / J. Math. Anal. Appl. 39 7) Let μ be the standard Gaussian measure on E, i.e., its characteristic function is e i x,f μdx) = e f, f E..) E The measure space E,μ)is known as white noise space.letl ) L E,μ)be the Hilbert space of complex valued μ-square integrable functionals on E with the inner product, )) and norm. Then, by the well-known Wiener Itô Segal isomorphism theorem [3], for each ϕ L ) there exists a unique sequence f n ) n= with f n H n such that ϕ = n= I n f n ) in norm and ϕ = n! f n.3) n= where I n f n ) denotes the multiple Wiener integral of order n with kernel f n. Note that the harmonic oscillator A also has a self-adjoint extension in H, which is still denoted by A.LetƔA) be the second quantization operator of A defined by ƔA)ϕ = I n A n ) f n.4) n= where ϕ = n= I n f n ). Then ƔA) is a positive self-adjoint operator with Hilbert Schmidt inverse in L ). Similarly, for each integer p, lete p ) be the completion of Dom ƔA) p with respect to the Hilbertian norm p = ƔA) p. Then E p ) becomes a complex Hilbert space. In particular, E ) = L ).LetE) = p E p) be the projective limit of E p ) p and E) = p E p) the inductive limit of E p ) p. Then E) and E) can be regarded as each other s dual. Moreover, E) is a nuclear space and we come to a complex Gel fand triple E) L ) E).5) which is known as the canonical framework of white noise analysis over E,μ). Elements of E) respectively E) ) are called testing respectively generalized) functionals. In the following, we denote by, the canonical bilinear form on E) E). For ξ E, the exponential functional φ ξ associated with ξ is defined as φ ξ x) = e x,ξ ξ,ξ / = :x n :, n! ξ n, x E..6) n= It is known that the set φ ξ ξ E is total in the Hilbert space E p ) for each integer p. Hence Spanφ ξ ξ E is a dense subspace of E). ontinuous linear operators from E) to E) are called generalized operators in the framework E) L ) E), which are significant generalizations of bounded operators on Hilbert space L ). The space of all generalized operators is denoted by L[E), E) ]. For a generalized operator X L[E), E) ], its symbol X is defined as Xξ,η) =Xφ ξ,φ η, ξ,η E..7) The next lemma is known as the characterization theorem of generalized operators, which was originally proved in []. See also [,7,9,3] for similar discussions.

4 96. Wang / J. Math. Anal. Appl. 39 7) 93 9 Lemma.. Let G be a function on E E with values in. Then there exists a generalized operator X L[E), E) ] with Gξ, η) = Xξ,η) for all ξ,η E if and only if G satisfies the following two conditions: ) for any ξ,ξ,η,η E, the function s, t) Gξ + sξ,η+ tη ), s,t R, has an entire analytic extension to ; ) there exist constants, K, p such that Gξ, η) exp K ξ p p) + η, ξ,η E..8) In that case, the generalized operator X has the following norm estimate Xϕ q e K A q p) ϕ q, ϕ E), HS where q p and e K A q p) HS <. 3. Main results Let δ a be the Dirac delta function at a R, which is a Schwartz generalized function, i.e. δ a E.Leth H = L R) with h. Then W h =,h is a Gauss random variable on E,μ). In particular, B t = W [,t] =, [,t], t, form a standard Brownian motion on E,μ). It is known that δ a W h ), of which Donsker s delta function δ a B t ) is a special case, may be realized as a generalized functional, i.e. an element of E) see [8] for details). Let M h be the operator of multiplication by W h in L ). Obviously, M h is an observable. In this section, we apply the idea of [5] to observable M h. We first make some preparations. Let BR) be the Borel σ -field of the real line R and P[L )] the set of orthogonal projections in Hilbert space L ). The following two definitions were given originally in [5]. See also [4] for similar work. Definition 3.. An observable i.e. a self-adjoint operator) T in L ) is called an observable of Schwartz class if for each ξ,η E there exists a function ρ ξ,η E such that PS)φξ,φ η = ρ ξ,η t) dt, S BR), 3.) S where P : BR) P[L )] is the spectral measure of T cf. []). In that case, the function ρ ξ,η is called the spectral density of T corresponding to ξ and η. Definition 3.. Let ω E. An observable T in L ) is said to be ω-composable if it is an observable of Schwartz class and, moreover, there exists a generalized operator X L[E), E) ] such that Xξ,η) = ω,ρξ,η T, ξ,η E, 3.) where ρξ,η T is the spectral density of T. In that case, the generalized operator X, which is determined uniquely by 3.), is called the ω-function of the observable T and denoted by ωt),i.e. ωt) = X.

5 . Wang / J. Math. Anal. Appl. 39 7) By using Lemma., we can prove the next theorem which provides a criterion for checking whether or not an observable is ω-composable. Theorem 3.. Let ω E and T an observable of Schwartz class with spectral density ρ T ξ,η. Define Φ ω ξ, η) = ω,ρ T ξ,η for ξ,η E. Then T is ω-composable if and only if Φ ω satisfies the following two conditions: i) for any ξ,ξ,η,η E, the function s, t) Φ ω ξ + sξ,η+ tη ), s,t R, has an entire analytic extension to ; ii) there exist constants,k,p such that Φ ω ξ, η) exp K ξ p + η p), ξ,η E. 3.3) In that case, we have ωt )ξ, η) = ω,ρξ,η T, ξ,η E. We now investigate the δ a -composability of observable M h and properties of its delta function δ a M h ). Proposition 3.. For each h H with h, M h is an observable of Schwartz class with spectral density of the following form: ρξ,η Mh t) = exp ξ,η [ ] t h, ξ + η π h h, t R, ξ,η E. 3.4) Proof. Let BE ) be the Borel σ -field of E and Π : BE ) P[L )] the canonical spectral measure defined by ΠD)ψ = D ψ, ψ L ),D B E ), where D is the indicator of D. LetP h = Π Wh. Then it is easy to see that P h : BR) P[L )] is a spectral measure. Moreover, we can verify that M h = R tp hdt). In other words, P h : BR) P[L )] is just the spectral measure of M h. On the other hand, for ξ,η E and t R,wehave e its P h ds)φ ξ,φ η = e itw hω) Πdω)φ ξ,φ η R R = e itwhω) φ ξ ω)φ η ω)μdω) R = exp ξ,η +i h, ξ + η t h t, which implies that Ph S)φ ξ,φ η = Therefore the conclusion is true. S exp ξ,η [ ] t h, ξ + η π h h dt, S BR).

6 98. Wang / J. Math. Anal. Appl. 39 7) 93 9 Proposition 3.3. For each h H with h and each a R, M h is δ a -composable. Moreover, its delta function δ a M h ) satisfies the following condition: δ a M h ) ξ, η) = exp ξ,η [ ] a h, ξ + η π h h, ξ,η E. 3.5) Proof. Let Φ a ξ, η) = δ a,ρξ,η Mh =ρmh ξ,η a), ξ,η E. Then, by 3.4), we have Φ a ξ, η) = exp ξ,η [ ] a h, ξ + η π h h, ξ,η E. It is easy to see that the function s, t) Φ a ξ + sξ,η+ tη ), s,t R, has an entire analytic extension to for any ξ,ξ,η,η E. On the other hand, for any ξ,η E,wehave Φ a ξ, η) exp ξ,η + [ a + h, ξ + η ] π h π h exp h ξ + η ) + h a exp π h h + 5 ξ + η ). [ a + 4 h ξ + η )] Thus, by Theorem 3., the observable M h is δ a -composable and δ a M h ) ξ, η) = ρξ,η Mh a) = exp ξ,η [ ] a h, ξ + η π h h, ξ,η E. This proves the proposition. Example 3.. onsider δ a M t ), where M t = M [,t]. Since δ a B t ), where B t = W [,t], is known as Donsker s delta function cf. [3] or [8]), it is reasonable to call δ a M t ) quantum Donsker s delta function. Proposition 3.3 shows that quantum Donsker s delta function δ a M t ) makes sense as a generalized operator for t>. Theorem 3.4. Let h H with h and a R. Then the delta function δ a M h ) is positive in the sense that δa M h ) ϕ,ϕ, ϕ E). 3.6) In particular, quantum Donsker s delta function δ a M t ) is positive. Proof. It is known that δ a is a positive Schwartz generalized function. Hence the conclusion follows from Proposition 3.4 of [5]. The next theorem establishes a formula showing an intimate connection between δ a M h ) as a generalized operator and δ a W h ) as a generalized functional. Theorem 3.5. Let h H with h and a R. Then the delta function δ a M h ) admits the following property: δa M h ) ϕ,ψ = δ a W h ), ϕψ, ϕ,ψ E). 3.7)

7 . Wang / J. Math. Anal. Appl. 39 7) Proof. Since the pointwise multiplication ϕ, ψ) ϕψ is a continuous mapping from E) E) to E) and exponential functional set φ ξ ξ E is total in E), it suffices to verify that δa M h ) φ ξ,φ η = δa W h ), φ ξ φ η, ξ,η E. According to Theorem 7.3 of [8], as a generalized functional, δ a W h ) satisfies the condition below δa W h ), φ f = δ a, exp [ ] h, f π h h, f E. From this formula and in view of φ ξ φ η = e ξ,η φ ξ+η, we get δa W h ), φ ξ φ η = δa W h ), e ξ,η φ ξ+η = = This completes the proof. δ a, exp [ ] h, ξ + η e ξ,η π h h exp ξ,η [ ] a h, ξ + η π h h = δ a M h ) ξ, η) = δ a M h ) φ ξ,φ η. Remark 3.. For each generalized functional Φ E), it can be proved cf. [8,]) that there exists a generalized operator T Φ L[E), E) ], called the operator of multiplication by Φ, such that TΦ ϕ,ψ = Φ,ϕψ, ϕ,ψ E). Theorem 3.5 shows that the delta function δ a M h ) of observable M h coincides with the operator of multiplication by generalized functional δ a W h ). In the following, we denote by :u n : σ a Hermite polynomial with parameter σ defined by :u n : σ = σ ) n e u /σ dn du n e u /σ. It is well known [8] that exp ut σ t = n= :u n : σ t n. n! Let l,m and θ E l+m) ). Then there exists a generalized operator Ξ l,m θ) L[E), E) ] such that Ξ l,m θ)ξ, η) = θ,η l ξ m e ξ,η, ξ,η E. 3.8) Operator Ξ l,m θ) is called the integral kernel operator with kernel θ see [] for details). The next theorem gives the representation of δ a M h ) as a series of integral kernel operators.

8 9. Wang / J. Math. Anal. Appl. 39 7) 93 9 Theorem 3.6. Let h H with h and a R. Set Then θ n a, h) = δ a M h ) ψ = :a n : h π h n+ n! exp l,m= a h h n, n. 3.9) ) l + m Ξ l,m θl,m a, h) ) ψ, ψ E), 3.) m where the series on the right-hand side converges in E). Proof. According to Theorem 7.3 of [8], δ a W h ) admits the following Wiener Itô decomposition also called Wiener Itô expansion) δ a W h )x) = :x n :,θ n a, h). n= On the other hand, from Theorem 3.5 we see that δ a M h ) is actually the operator of multiplication by δ a W h ). Therefore, by Proposition of [], we get the desired representation of δ a M h ). Theorem 3.7. Let h H with h. Then the delta function δ a M h ) of observable M h depends continuously on a R in the sense that for each ϕ E) the mapping a δ a M h )ϕ is continuous from R to E). Proof. Let a R and a n R a sequence converging to a. Write T n = δ an M h ), n, for brevity. We need to prove that T n ϕ T ϕ for each ϕ E). It is easy to see that T n φ ξ,φ η T φ ξ,φ η for any ξ,η E. On the other hand, for any ξ,η E,wehave sup n T n ξ, η) = sup exp π h n a sup exp n n π h ξ,η h [ an h, ξ + η ] ξ + η ) h exp ξ + η ), where = exp π h h sup an. n Take q with 5e A q HS <, where A is the harmonic oscillator see Section ). Then, by Lemma., we get sup T n ϕ q n 5e A q p) ϕ q, ϕ E), 3.) HS

9 . Wang / J. Math. Anal. Appl. 39 7) which means that T n L[E q ), E q )] and, moreover, T n are uniformly bounded. Since exponential functional set φ ξ ξ E is total both in E q ) and in E q ), it follows that T n ϕ,ψ T ϕ,ψ, ϕ,ψ E). Hence T n ϕ T ϕ for each ϕ E). As a simple application of Theorem 3.7, we have the following interesting result. Proposition 3.8. Let h H with h. Then the generalized functional δ a W h ) depends continuously on a R, i.e. the mapping a δ a W h ) is continuous from R to E). In particular, Donsker s delta function δ a B t ) depends continuously on a R. Proof. It follows from 3.7) that δ a W h ) = δ a M h )φ. Thus, by Theorem 3.7, the mapping a δ a W h ) is continuous. Remark 3.. The continuity of the mapping a δ a B t ) was proved originally by Kuo [8] by using a different method. However, to the best of our knowledge, the continuity of the mapping a δ a W h ) for general h H has not been considered yet. References [] L. Accardi, Y.G. Lu, I.V. Volovich, Quantum Theory and Its Stochastic Limit, Springer-Verlag, Berlin,. [] D.M. hung, T.S. hung, U.. Ji, A characterization theorem for operators on white noise functionals, J. Math. Soc. Japan 5 999) [3] T. Hida, H.H. Kuo, J. Potthoff, L. Streit, White Noise An Infinite Dimensional alculus, Kluwer Academic, Dordrecht, 993. [4] H. Holden, B. Øksendal, J. Ubøe, T. Zhang, Stochastic Partial Differential Equations, Birkhäuser, Basel, 996. [5] Z.Y. Huang, J.A. Yan, Introduction to Infinite Dimensional alculus, Kluwer Academic, Dordrecht, 997. [6] R.L. Hudson, K.R. Parthasarathy, Quantum Itô s formula and stochastic evolutions, omm. Math. Phys ) [7] Yu.G. Kondratiev, L. Streit, Spaces of white noise distributions: onstructions, applications. I, Rep. Math. Phys ) [8] H.H. Kuo, White Noise Distribution Theory, R, Boca Raton, FL, 996. [9] S.L. Luo, Wick algebra of generalized operators involving quantum white noise, J. Operator Theory ) [] N. Obata, White Noise alculus and Fock Space, Springer-Verlag, Berlin, 994. [] N. Obata, An analytic characterization of symbols of operators on white noise functionals, J. Math. Soc. Japan ) [] K.R. Parthasarathy, An Introduction to Quantum Stochastic alculus, Birkhäuser, Basel, 99. [3] J. Potthoff, L. Streit, A characterization of Hida distributions, J. Funct. Anal. 99) 9. [4].S. Wang, Z.Y. Huang, X.J. Wang, δ-function of an operator: A white noise approach, Proc. Amer. Math. Soc. 33 5) [5].S. Wang, A new idea to define the δ-function of an observable in the context of white noise analysis, Infin. Dimens. Anal. Quantum Probab. Rel. Top. 8 5)