On a class of operators of the second quantization
Copyright policy )On a class of operators of the second quantization
Let \(H\) be a real separable Hilbert space and let \(\Phi'\supset H \supset \Phi\) be the rigging of \(H\) by a nuclear Frechet space \(\Phi\) and its dual \(\Phi'\). Operators of quantum stochastic calculus in the functional realization \(L_2(\Phi',\mu)\) of the Fock space \({\mathcal F}(H)\), especially the differential second quantization \(d\Gamma(A)\) of the non-selfadjoint operator \(A\) from the special class, are investigated. It is proved that the image under the Segal isomorphism \({\mathcal F}(H)\to L_2(\Phi',\mu)\) of \(d\Gamma(A)\) coincides with the second order differential operator \[ (L_A f)(x)=-\tfrac 12 \text{Tr}_H A f''(x)+(Af'(x),x)_H, \quad x\in \Phi' \] as well as in the case of the selfadjoint \(A\) [see \textit{Yu. M. Berezanskij} and \textit{Yu. G. Kondrat'ev}, ``Spectral methods in infinite dimensional analysis'', Kiev, Naukova Dumka (1988; Zbl 0707.47001].
- IE University Spain
- Taras Shevchenko National University of Kyiv Ukraine
nuclear Fréchet space, Spaces of differentiable or holomorphic functions on infinite-dimensional spaces, PDEs on infinite-dimensional (e.g., function) spaces (= PDEs in infinitely many variables), rigging, infinite-dimensional differential operators, Quantizations, deformations for selfadjoint operator algebras, second order differential operator, Fock space, Linear symmetric and selfadjoint operators (unbounded), Spaces determined by compactness or summability properties (nuclear spaces, Schwartz spaces, Montel spaces, etc.), Locally convex Fréchet spaces and (DF)-spaces, Quantum stochastic calculus, (Generalized) eigenfunction expansions of linear operators; rigged Hilbert spaces, Segal isomorphism, quantum stochastic integrals, differential second quantization
nuclear Fréchet space, Spaces of differentiable or holomorphic functions on infinite-dimensional spaces, PDEs on infinite-dimensional (e.g., function) spaces (= PDEs in infinitely many variables), rigging, infinite-dimensional differential operators, Quantizations, deformations for selfadjoint operator algebras, second order differential operator, Fock space, Linear symmetric and selfadjoint operators (unbounded), Spaces determined by compactness or summability properties (nuclear spaces, Schwartz spaces, Montel spaces, etc.), Locally convex Fréchet spaces and (DF)-spaces, Quantum stochastic calculus, (Generalized) eigenfunction expansions of linear operators; rigged Hilbert spaces, Segal isomorphism, quantum stochastic integrals, differential second quantization
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