The Damped Oscillator: A Locally Convex Formulation

Abstract

We formulate the quantum system of an oscillator driven by a quantum Wiener process, in the locally convex setting based on the rigged triple $\mathcal{S}(R)\subset L^2(R)\subset {\mathcal{S}}^{\prime }(R)$. The generalized observables are taken to be the elements of $\mathcal{L}(\mathcal{S}(R),{\mathcal{S}}^{\prime }(R))$. Pulling the dynamics back to phase space by means of the inverse of Weyl quantization, we prove that the time translations semigroup is equicontinuous of class $C_0$. Moreover, it is differentiable, and its generator is an extension to $\mathcal{L}(\mathcal{S}(R),{\mathcal{S}}^{\prime }(R))$ of the known result for bounded operators.