1,720,973 research outputs found
Generic quantum Markov semigroups: the Fock case
Full text linkWe introduce the class of generic quantum Markov semigroups. Within this class
we study the class corresponding to the Fock case which is further split into four sub-classes
each of which contains both bounded and unbounded generators, depending on some global
characteristics of the intensities of jumps. For the first two of these classes we find an explicit
solution which reduces the problem of finding the quantum semigroup to the calculation of two
classical semigroups, one of which is diagonal (in suitable basis) and the other one is triangular
(in the same basis). In the bounded case our formula gives the unique solution. In the unbounded
case it gives one solution, which we conjecture to be the minimal one
A quantum approach to Laplace operators
Full text linkIn this paper, a theory of stochastic processes generated by quantum extensions of Laplacians is developed. Representations of the associated heat semigroups are discussed by means of suitable time shifts. In particular the quantum Brownian motion associated to the Levy-Laplacian is obtained as the usual Volterra-Gross Laplacian using the Cesaro Hilbert space as initial space of our process as well as multiplicity space of the associated white noise
Cesaro Hilbert space and the Levy laplacian
Full text linkIn this paper we introduce a new scalar product on distribution spaces based on the Cesaro mean of a sequence. We then use this scalar product to construct a family of separable Hilbert spaces , called Cesaro Hilbert spaces and naturally associated to the Levy Laplacian. Finally we use the
essentially infinite dimensional character of the Levy Laplacian
to construct a class of solutions of the Levy heat equation which has no finite dimensional (or ``regular'' infinite dimensional) analogue
Module white noise calculus
Full text linkOur main result is an infinitesimal characterization of Hilbert module module flows, not necessarily of inner type, in terms of stochastic derivations from the initial algebra into the Itô algebra. We prove that any such derivation is the difference of a *-homomorphism and the trivial embedding
White noise Heisenberg evolution and Evans-Hudson flows
Full text linkWe study white noise Heisenberg equations giving rise to flows which are *-automorphisms of the observable algebra, but not necessarily inner automorphisms. We prove that the causally normally ordered form of these white noise Heisenberg equations are equivalent to Evans–Hudson flows. This gives in particular, the microscopic structure of the maps defining these flows, in terms of the original white noise derivations
White noise approach to stochastic integration
Full text linkIn order to prove the existence and the uniqueness of operator
solutions of some white noise stochastic differential equations,
we need some multidimensional non adapted estimates of white noise
stochastic integrals on a dense domains in bosonic Fock space.
These estimates are not sufficient to prove the fundamental
unitary conditions. For this we need to introduce the adapteness
and to restrict ourselves to 1-dimensional index set
On the Quadratic Heisenberg Group
No full text In this paper we introduce the quadratic Weyl operators canonically associated to the one mode renormalized square of white noise (RSWN) algebra as unitary operator acting on the one mode interacting Fock space {Γ, {ωn, n ∈ ℕ}, Φ} where {ωn, n ∈ ℕ} is the principal Jacobi sequence of the nonstandard (i.e. neither Gaussian nor Poisson) Meixner classes. We deduce the quadratic Weyl relations and construct the quadratic analogue of the Heisenberg Lie group with one degree of freedom. In particular, we determine the manifold structure of the group and introduce a local chart containing the identity on which the group law has a simple rational expression in the chart coordinates (see Theorem 6.3). </jats:p
Generic Fock Quantum Markov Semigroups with Instantaneous States
Full text linkWe construct a generic quantum Markov semigroup with instantaneous states exploiting the invariance of the diagonal algebra and the explicit form of the action of the pre-generator on off-diagonal matrix elements. Our semigroup acts on a unital C*-algebra and is strongly continuous on this algebra (Feller property). We discuss the generic hydrogenic atoms as an example
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