Dynamics of Open Quantum Systems II, Markovian Approximation

A finite-dimensional quantum system is coupled to a bath of oscillators in thermal equilibrium at temperature T>0. We show that for fixed, small values of the coupling constant λ, the true reduced dynamics of the system is approximated by the completely positive, trace preserving Markovian semigroup generated by the Davies-Lindblad generator. The difference between the true and the Markovian dynamics is O(|λ|1/4) for all times, meaning that the solution of the Gorini-Kossakowski-Sudarshan-Lindblad master equation is approximating the true dynamics to accuracy O(|λ|1/4) for all times. Our method is based on a recently obtained expansion of the full system-bath propagator. It applies to reservoirs with correlation functions decaying in time as 1/t4 or faster, which is a significant improvement relative to the previously required exponential decay.

Heat kernels of the discrete Laguerre operators

For the discrete Laguerre operators we compute explicitly the corresponding heat kernels by expressing them with the help of Jacobi polynomials. This enables us to show that the heat semigroup is ultracontractive and to compute the corresponding norms. On the one hand, this helps us to answer basic questions (recurrence, stochastic completeness) regarding the associated Markovian semigroup. On the other hand, we prove the analogs of the Cwiekel–Lieb–Rosenblum and the Bargmann estimates for perturbations of the Laguerre operators, as well as the optimal Hardy inequality.

Divisibility of Dynamical Maps with Time Independent Invariant State

A new condition for dynamical maps with time independent invariant state displaying Heisenberg divisibility is provided. This condition is trivially satisfied for a Markovian semigroup. However, beyond Markovian semigroups it provides a nontrivial characteristics of quantum Markovianity. Interestingly, when the invariant state is pure one recovers a condition based on Wigner–Yanase skew information.

STRONG FELLER CONTINUITY OF FELLER PROCESSES AND SEMIGROUPS

We study two equivalent characterizations of the strong Feller property for a Markov process and of the associated sub-Markovian semigroup. One is described in terms of locally uniform absolute continuity, whereas the other uses local Orlicz-ultracontractivity. These criteria generalize many existing results on strong Feller continuity and seem to be more natural for Feller processes. By establishing the estimates of the first exit time from balls, we also investigate the continuity of harmonic functions for Feller processes which enjoy the strong Feller property.

FLOWS AND INVARIANCE FOR DEGENERATE ELLIPTIC OPERATORS

LetSbe a sub-Markovian semigroup onL2(ℝd) generated by a self-adjoint, second-order, divergence-form, elliptic operatorHwithW1,∞(ℝd) coefficientsckl, and let Ω be an open subset of ℝd. We prove that ifeither C∞c(ℝd) is a core of the semigroup generator of the consistent semigroup onLp(ℝd) for somep∈[1,∞] or Ω has a locally Lipschitz boundary, thenSleavesL2(Ω) invariant if and only if it is invariant under the flows generated by the vector fields ∑dl=1ckl∂lfor allk. Further, for allp∈[1,2] we derive sufficient conditions on the coefficients for the core property to be satisfied. Then by combination of these results we obtain various examples of invariance in terms of boundary degeneracy both for Lipschitz domains and domains with fractal boundaries.