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Scheid, Open quantum Systems. Int. J. Mod. Phys. E, 3(2) (1994), 635–719. O. Bratteli and D. W. Robinson, Operator Algebras and Quantum Statistical Mechanics, Vol I. Springer (1987). M. Reed and B. Simon, Methods of Modern Mathematical Physics II. Academic Press, 6th edition (1986). T. Kato, Perturbation Theory for Linear Operators, Springer (1995). F. Fagnola and R. Rebolledo, Algebraic Conditions for Convergence to a Steady State. Preprint (2008). September 23, 2008 11:45 WSPC - Proceedings Trim Size: 9in x 6in QuantumProbability 49 HILBERT SPACE OF ANALYTIC FUNCTIONS ASSOCIATED WITH A ROTATION INVARIANT MEASURE N.
The position and momentum operators are defined on Dom(N 1/2 ). Both q and p have, by Nelson’s analytic vector theorem, self-adjoint extensions that will be still denoted by q and p. First we shall compare the domains of N and X. 1. Let r, s > 0 with rs > 1 and define R := r1/2 p + r−1/2 q, S := s1/2 q + s−1/2 p . Then, for all u ∈ Cc∞ (R) the following identities hold u, X 2 u = s− 1 r 2 u, q 4 u + 2 s − 1 r u, qR2 qu + u, R4 u − 2(rs − 1) u u, X 2 u = r− 1 s 2 u, p4 u + 2 r − 1 s 2 , 2 . u, pS 2 pu + u, S 4 u − 2(rs − 1) u September 23, 2008 32 11:45 WSPC - Proceedings Trim Size: 9in x 6in QuantumProbability A.
6 using PDE–tools. The extension to harmonic potentials with a small, smooth perturbation was recently obtained in Ref. 15 using fixed point arguments and spectral theory. Here we consider large perturbations of the harmonic potential. To this end we shall work in the density matrix formalism, using tools from operator theory. In the density matrix formalism a quantum state is described by a density matrix σ ∈ T1+ (h), the cone of positive trace class operators on some Hilbert space h. The time evolution with the initial state ρ0 = σ is governed by the linear QFP equation or master equation dρt = L∗ (ρt ) , dt (3) September 23, 2008 11:45 WSPC - Proceedings Trim Size: 9in x 6in QuantumProbability Quantum Fokker-Planck Models: the Lindblad and Wigner Approaches 25 with the Lindbladian i 2 p + ω 2 q 2 + V (q), ρ − iγ [q, {p, ρ}] 2 − Dqq [p, [p, ρ]] − Dpp [q, [q, ρ]] + 2Dpq [q, [p, ρ]] , L∗ (ρ) = − (4) where V (q) is the perturbation of the harmonic potential.