Commutative Jacobi fields in Fock space
Let \(H\) be a real separable Hilbert space with \(H=H_0\). By a rigging of Hilbert spaces, we have \(\Phi=\text{pr-}\lim_{r\in T}H_r\) and \(\Phi'= \text{ind-}\lim_{r\in T} H_{-r}\). Actually, \(\Phi'\) is the dual of \(\Phi\) and \(\Phi\subset H\subset \Phi'\) is a Gel'fand triple. A usual procedure to take the \(n\)-th symmetric tensor power of the above spaces with complexification gives the rigging of \(n\)-particle Fock spaces: \({\mathcal F}_n(\Phi) \subset {\mathcal F}_n(H) \subset {\mathcal F}_n (\Phi')\). A generalization of the white noise analysis was proposed by \textit{Yu. M. Berezanskij}, \textit{V. O. Livinskij} and \textit{E. V. Litvinov} [Ukr. Math. J. 46, No. 3, 183-203 (1994); translation from Ukr. Mat. Zh. 46, No. 3, 177-197 (1994; Zbl 0843.47012) and Methods Funct. Anal. Topol. 1, No. 1, 28-55 (1995; Zbl 0870.60037)], where a theory of distributions on an infinite-dimensional space is constructed in the case of a non-Gaussian measure. The spectral measure of some commuting family \(A=(\widetilde A(\varphi)\); \(\varphi \in\Phi)\) of selfadjoint operators \(\widetilde A(\varphi)\) plays an important role in this generalization. In fact, a Jacobi field is a family \((\widetilde A(\varphi)\); \(\varphi\in \Phi)\) of commuting selfadjoint operators \(\widetilde A(\varphi)\) acting in a Fock space \({\mathcal F} (H)\), having a Jacobi structure, and depending linearly on the test function \(\varphi\). While, the Fourier transform \(I\) connected with the family \(A\) maps the Fock space onto the space \(L^2\) on the spectrum of this family with respect to the measure \(\rho\). Moreover, the rigging of the Fock space is transferred to some rigging of the space \(L^2\), and the latter rigging generates the corresponding theory of generalized functions on this spectrum [cf. \textit{E. W. Litvinov}, ibid. 1, No. 1, 61-85 (1995; Zbl 0871.60034)]. Indeed, we have \[ (L^2)= L^2(\Phi',d\rho) =\bigoplus^\infty_{n=0} (L^2_n), \quad (L^2_n)= I\bigl({\mathcal F}_n(H) \bigr), \] and the transform \(I\) essentially corresponds to the Wiener-Itô-Segal isomorphism in white noise analysis [cf. \textit{N. Obata}, ``White noise calculus and Fock space'' (1994; Zbl 0814.60058)]. The author proposes a somewhat different axiomatization of a similar Jacobi field which is much simpler and is closely connected with a Hilbert rigging. Moreover, he constructs its spectral representation and discusses its applications to the theory of distributions on an infinite-dimensional space. Lastly three examples of Jacobi fields are also given, such as 1) classical field with Gaussian measure, 2) perturbed classical field with Poisson measure, and 3) the field constructed by Jacobi matrices.
distribution on infinite-dimensional spaces
scientific article; zbMATH DE number 1129330
scientific article