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Moreover, the limit of the rescaled evolution operator Ut satisfies the white noise equation atUt = -ihtUt, with the white noise Hamiltonian of the form ht = D’bt (9) + bfD, where bt, bi are Fermi Fock white noise creation and annihilation operators. This means that they are operator valued distributions, acting on the Fock space r = F ( L 2 ( R ) )and satisfying the following relations: { bf, b”’} = y-6+(t - t’) ; {bt, bt’} = 0 . (11) where d+(t) is the causal &function (see Ref. 1) and ‘An operator is called even, if it commutes with the parity operator, and odd, if it anticommutes.

L. , New York . 3. L. Accardi and V. I. Bogachev: The Ornstein-Uhlenbeck process associated with the Lkvy Laplacian, and its Direchlet form, Probability and Math. 1 (1997), 95-114. 4. L. Accardi and 0. G. Smolyanov: On the Laplacians and traces, Rend. Sem. Mat. Bari, Ottobre 1993, Preprint- Volterra. 5. L. Accardi, P. G. Zamatki, 54 (1993), 144-148 6. L. Accardi and N. Obata: Derivation Property of the Ldvy-Laplacian, White noise analysis and quantum probability, (N. ) RIMS Kokyuroku 874, Publ.

2. 3 for the multiplicative renormalization of the pre-generating function cp(t,x) = ep(t)z. 4. Let p be a measure o n & and let e(x) = p ( { x } ) , x E & . A s s u m e that the multiplicative renormalization pntn is analytic with i s a generating function f o r p, Here p ( t ) = C,"==, # 0. Suppose &(x) is a probability m a s s function o n & f o r each n satisfying the conditions: p1 52 (8) 1 e(z)A:- [en(x)]E L~(P). (b) A:+ [&(x)] 5 0 at x = 0 , 00 for all 0 5 k < n. Then the orthogonal polynomial P,(x) associated with p is given b y with some constant k, if and only if with some constant do.