Remarks on positive semigroups

Let ℳ be a von Neumann algebra with a cyclic and separating vector Ω and let ω(·) denote the corresponding vector state, i.e., ω(A)=(Ω, AΩ) A ∈ ℳ. We have proved that a positive semigroup τ on ℳ can induce the dynamical semigroup in the GNS representation associated with ω if the state ω is a τ-invariant one. Some applications are given.     

Unbounded derivations and invariant states

Let be a von Neumann algebra with a cyclic and separating vector . Let =i[H, ·] be the spatial derivation implemented by a selfadjoint operatorH, such thatH=0. Let be the modular operator associated with the pair (, ). We prove the equivalence of the following three conditions:1)H is essential selfadjoint onD(), andH commutes strongly with .2) The restriction ofH toD() is essential selfadjoint onD(^1/2) equipped with the inner product(|)_#=(|)+(^1/2|^1/2), , D(^1/2).3) exp (itH) exp (–itH)= for anyt.We show by an example, that the first part of 1),H is essential selfadjoint onD(), does not imply 3). This disproves a conjecture due to Bratteli and Robinson [3].Part of this work was done while O.B. was a member of Zentrum für interdisziplinäre Forschung der Universität Bielefeld     

Unbounded derivations and invariant trace states

Let \(\mathfrak{M}\) be a von Neumann algebra with cyclic trace vector ?. Let δ(A)=i[H, A] be a spatial derivation of \(\mathfrak{M}\) implemented by an operatorH such thatH?=0 andH is essentially self-adjoint onD(δ)?. It follows that: $$e^{itH} \mathfrak{M}e^{ - itH} = \mathfrak{M},t \in \mathbb{R}.$$      

Completely positive dynamical semigroups and quantum resonance theory

Starting from a microscopic system–environment model, we construct a quantum dynamical semigroup for the reduced evolution of the open system. The difference between the true system dynamics and its approximation by the semigroup has the following two properties: It is (linearly) small in the system–environment coupling constant for all times, and it vanishes exponentially quickly in the large time limit. Our approach is based on the quantum dynamical resonance theory.     

Rotationally invariant bipartite states and bound entanglement

We consider rotationally invariant states in CN₁⊗CN₂${\mathbb{C}}^{N_1}\otimes {\mathbb{C}}^{N_2}$ Hilbert space with even N₁?4$N_1?4$ and arbitrary N₂?N₁$N_2?N_1$, and show that in such case there always exist states which are inseparable and remain positive after partial transposition, and thus the PPT criterion does not suffice to prove separability in such systems. We demonstrate it applying a map developed recently by Breuer [H.-P. Breuer, Phys. Rev. Lett. 97 (2006) 080501] to states that remain invariant after partial time reversal.     

Energy spectrum of extremal invariant states

For an extremal invariant state of a weakly asymptotically abelian dynamical system we prove that the corresponding energy spectrum is either one-sided or the whole reals, or a periodic subgroup. The latter case implies abelianness of the algebra in the representation generated by .     

Translationally invariant description of particle-hole states

A translationally invariant model is proposed which allows to calculate one-particle excitations and dipole transition probabilities of closed-shell nuclei with inclusion of the continuum. The connection of this model with the nuclear shell model is shown, and it is applied to the calculation of the photodisintegration cross section of⁴He.     

Generators of semigroups of completely positive maps

Any generator of a norm continuous semigroup of completely positive normal maps on a von Neumann algebraM can be decomposed into a sum of a completely positive map and a map of the formmx*m+mx.     

Non-translation invariant Gibbs states with coexisting phases

We investigate the spatially inhomogeneous states of two component,A - B, Widom-Rowlinson type lattice systems. When the fugacity of the two components are equal and large, these systems can exist in two different homogeneous (translation invariant) pure phases oneA-rich and oneB-rich. We consider now the system in a box with boundaries favoring the segregation of these two phases into top and bottom parts of the box. Utilizing methods due to Dobrushin we prove the existence, in three or more dimensions, of a sharp interface for the system which persists in the limit of the size of the box going to infinity. We also give some background on rigorous results for the interface problem in Ising spin systems.Supported in part by NSF Grant PHY 77-22302Supported by the Swiss National Foundation for Scientific Research     

Elliptic dark states: Explicit invariant form

An analysis is made of coherent population trapping as a result of resonant interaction of elliptically polarized light with atoms whose energy levels are degenerate with respect to the projection of the angular momentum and are coupled by a dipole transition. Explicit invariant expressions for dark states are obtained in tensor form for all transitions where population trapping occurs. A correspondence is established between the vector of the elliptic polarization and the pair of associated spinors. It is shown that all dark states can be constructed from these spinors by means of a multiple tensor product. For integer values of the angular momenta of the transitions these constructions reduce to spherical functions of a complex variable. As applications analytic expressions are obtained for the dark magneto-optic and geometric potentials, and the change in their profile with increasing angular momenta is analyzed.     

Resolvents,semigroups and Gibbs states for infinite coupling constant

The infinite coupling constant limit of the resolvent, the semigroup and the Gibbs state is obtained for a certain class of perturbations. As an example the infinite intrasite repulsion limit of the one-dimensional Hubbard model with nearest neighbour hopping terms is treated. This system exhibits a phase transition in the thermodynamic limit.     

Some remarks on dilating semigroups of completely positive mappings

We reconsider the problem of embedding a dissipative dynamical system in a conservative one; and we compare some of the partial solutions which have been recently proposed.     

Relative entropy of entanglement of rotationally invariant states

We calculate the relative entropy of entanglement for rotationally invariant states of spin- and arbitrary spin-j particles or of spin-1 particle and spin-j particle with integer j. A lower bound of relative entropy of entanglement and an upper bound of distillable entanglement are presented for rotationally invariant states of spin-1 particle and spin-j particle with half-integer j.     

Translation invariant Gibbs states in theq-state Potts model

We describe the set of all translation invariant Gibbs states in theq-state Potts model for the case ofq large enough and the other parameters to be arbitrary.     

Correction to: Completely positive dynamical semigroups and quantum resonance theory

Letters in Mathematical Physics - Correction. The bound (2.23) in Theorem 2.1.