Principal Investigator: Professor V. P. Belavkin

In the frame of the project we studied:

1. The symmetrical boundary value problem in a Hilbert module corresponding to the single-jump Schroedinger stochastic equation in a Hilbert space.

2. The symmetrical boundary value problem in Fock Hilbert module corresponding to classical and quantum stochastic differential equations (SDE) in a Hilbert space.

3. The necessary and sufficient self-adjointness conditions for the above boundary value problems in the case of commuting and non-commuting coefficients.

4. The extension of the strong resolvent and inductive stochastic limit to the singularly perturbed Hamiltonians corresponding to the boundary conditions for the classical and quantum SDE.

5. The derivation and analysis of the dynamical boundary value models in this framework for quantum measurement and detection of gravitational waves.

Carrying out this program we achieved the following results:

1. It has been proven that the simplest discontinuous single-jump classical SDE satisfying the unitarity conditions in a Hilbert space is equivalent to a Dirac boundary value problem on the half-line in an extra-dimension. Thus the classical discontinuous stochastics generated by such jumps can be derived from a quantum continuous deterministic unitary evolution in Fock Hilbert modules by the second quantization of the Dirac boundary value problem.

2. A one-to-one correspondence has been found between the quantum SDE with bounded coefficients satisfying the Hudson-Parthasarathy unitarity conditions, and the self-adjoint boundary value problem in Fock module corresponding to the gradient and Laplacian types of quantum field free evolution Hamiltonians.

3. The stochastic limit is reviewed in an inductive convergence as the ultra relativistic approximation of positive free evolution Hamiltonians for the relativistic Schroedinger equations. It is proved that the rapidly oscillating solutions to the Schcroedinger input and output equations, which are connected by the boundary condition in the half-space, strongly converge to the corresponding solutions of the Dirac one-dimensional boundary value problem in the inductive Hardy class pre-Hilbert module.

4. The applications of Monte-Carlo method to the Belavkin-Schroedinger stochastic equation were studied and a training program for mathematics was developed for studying of the solutions to the corresponding boundary-value problem in the Hilbert module.

The results obtained and methods developed open a new interesting area belonging to both mathematical physics and stochastic analysis. In the nearest future one can expect a flow of further results on the equivalence of stochastic adapted and non-adapted cocycles generated by the classical or quantum Poisson and Wiener processes to the corresponding boundary value problems in Fock Hilbert modules. It is worth stressing as well that this area is has various applications: in classical and quantum probability, in theory of open systems, in quantum measurement and filtering theory, detection of gravitation waves and communication. The results of this programme need to be considered as a first step in the realisation of a long term project related to the these fields.

The results of the project are included into four new papers submitted for publication.

Acknowledgement: The Principle Investigator and the Visiting Fellow acknowledge the EPSRC funding of the four months research in the UK.