Infinite Dimensional Analysis, Quantum Probability and Related Topics

There exists the well-known approximate expression describing the large time behavior of matrix elements of the evolution operator in quantum theory: < U(t)>≃ exp(at). This expression plays the crucial role in considerations of problems of quantum decoherence, radiation, decay, scattering theory, stochastic limit, derivation of master and kinetic equations etc. This expression was obtained in the Weisskopf–Wigner approximation and in the van Hove (stochastic) limit. We derive the exact general formula which includes the higher order corrections to the above approximate expression: <U(t)>=exp(At+B+C(t)). The constants A and B and the oscillating function C(t) are computed in perturbation theory. The method of perturbation of spectra and renormalized wave operators is used. The formula is valid for a general class of Hamiltonians used in statistical physics and quantum field theory.

We consider the quantum stochastic differential equation introduced by Hudson and Parthasarathy to describe the stochastic evolution of an open quantum system together with its environment. We study the (unbounded) Hamiltonian operator generating the unitary group connected, as shown by Frigerio and Maassen, to the solution of the equation. We find a densely defined restriction of the Hamiltonian operator; in some special cases we prove that this restriction is essentially self-adjoint and in one particular case we get the whole Hamiltonian with its full domain.

Quantum integral equation of Volterra type with generalized operator-valued kernel is introduced. Existence and uniqueness of solutions are established, explicit expression of the solution is given, the continuity, continuous dependence on free terms and other properties of the solution are proved.

In this paper we study the problem of dilating unital completely positive (CP) semigroups (quantum dynamical semigroups) to weak Markov flows and then to semigroups of endomorphisms (E₀-semigroups) using the language of Hilbert modules. This is a very effective, representation free approach to dilation. In this way we are able to identify the right algebra (maximal in some sense) for endomorphisms to act. We are led inevitably to the notion of tensor product systems of Hilbert modules and units for them, generalizing Arveson's notions for Hilbert spaces.

In the course of our investigations we are not only able to give new natural and transparent proofs of well-known facts for semigroups on , but also extend the results immediately to much more general setups. For instance, Arveson classifies E₀-semigroups on up to cocycle conjugacy by product systems of Hilbert spaces.⁵ We find that conservative CP-semigroups on arbitrary unital C*-algebras are classified up to cocycle conjugacy by product systems of Hilbert modules. Looking at other generalizations, it turns out that the role played by E₀-semigroups on in dilation theory for CP-semigroups on is now played by E₀-semigroups on , the full algebra of adjointable operators on a Hilbert module E. We have CP-semigroup versions of many results proved by Paschke²⁷ for CP maps.