JournalsprimsVol. 41 , No. 1DOI 10.2977/prims/1145475403

The Damped Oscillator: A Locally Convex Formulation

  • Daniel A. Dubin

    Open University, Milton Keynes, UK
  • Mark A. Hennings

    Rugby School, UK
The Damped Oscillator: A Locally Convex Formulation cover

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 S(ℝ) ⊂ _L_2(ℝ) ⊂ S'(ℝ). The generalized observables are taken to be the elements of ℒ(S(ℝ), S'(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 ℒ(S(ℝ), S'(ℝ)) of the known result for bounded operators.