Spectral analysis and Feller property for quantum Ornstein-Uhlenbeck semigroups
The object of the paper is the investigation of the evolution equation \[ {d\over dt}P_tx = {\mathcal L}P_tx, \qquad P_0 = I, \] determined by the Lindblad-type operator \[ {\mathcal L}x = -{\mu^2\over 2}(A^\star Ax-2A^\star xA+xA^\star A) -{\lambda^2\over 2}(AA^\star x-2AxA^\star+xAA^\star). \] The equation is for evolution of bounded linear operators on the complex Hilbert space \textit{l}\(^2({\mathbb Z}_+)\); the operators \(A\) and \(A^\star\) are the annihilation and creation operators of the usual representation of the canonical commutation relations associated with the quantum harmonic oscillator and the constants \(\lambda\) and \(\mu\) satisfy \( \mu > \lambda > 0\), \(\lambda/\mu\in (0,1)\). The semigroups are constructed, by means of noncommutative Dirichlet forms, on the full algebra of bounded operators on a separable Hilbert space. The explicit action of their generators on a core in the domain is used to demonstrate the Feller property of the semigroups, with respect to the \(C^\star\)-subalgebra of compact operators. The Dirichlet forms are analyzed and the \(L^2\)-spectrum together with eigenspaces are found. When reduced to certain maximal Abelian subalgebras, the semigroups give rise to the Markov semigroups of classical Ornstein-Uhlenbeck process on the one hand, and of classical birth-and-death process on the other.
scientific article; zbMATH DE number 1445814