Details of Grant 

EPSRC Reference:	GR/M66196/01
Title:	THE BOUNDARY VALUE PROBLEMS IN FOCK HILBERT MODULE ASSOCIATED TO QUANTUM STOCHASTIC DIFFERENTIAL EQUATION
Principal Investigator:	Belavkin, Professor V
Other Investigators:
Researcher Co-Investigators:
Project Partners:
Department:	Sch of Mathematical Sciences
Organisation:	University of Nottingham
Scheme:	Standard Research (Pre-FEC)
Starts:	09 June 1999	Ends:	08 October 1999	Value (£):	6,500
EPSRC Research Topic Classifications:	Mathematical Analysis Mathematical Physics
EPSRC Industrial Sector Classifications:	No relevance to Underpinning Sectors
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Panel History:
Summary on Grant Application Form
Until the recent papers [19, 21] it was not known that a unitary co-cycle satisfying the Quantum Stochastic Differential Equation (QSDE) can be described not by a symmetric operator but rather by a symmetrical boundary value problem in Fock Hilbert module. The boundary condition describes operator-valued jumps of the solution in amplitude and in phase. The jumps are supported by the set of singularities of the formal generator, and the selfadjointness of the boundary value problem is necessary and sufficient for the existence of the unitary solution of the appropriate QSDE. The first aim of the project is the study of conditions necessary and sufficient for the selfadjointness. Preliminary analysis promises to obtain new results for non-adaptive QSDE with unbounded operator-valued coefficients.The development of the theory value problems for quantum dynamical semi-groups described by the master equations in von Neumann algebras, and by QSDE in Hilbert spaces, is closely connected to explosion problems for classical stochastic processes. A study of condition necessary and sufficient for the existence of non-trivial boundary is connected to the problems of unitary and isometrical extensions for symmetrical operators in Hilbert spaceThese results were obtained by the explicit resolvent analysis of the class of solvable problems for the Schrodinger equation in Fock Hilbert module [19, 21, 24]. It is proved that the QSDE arise naturally as the strong resolvent limits of the Schrodinger evolution in Fock Hilbert module and derive the coefficients of QSDE. The proof assumes some restrictive commutation relations [18] which can be skipped in the frame of boundary value problem method. Thus the second aim of the project is the relaxation of commutation assumptions sufficient to derive rigorously the strong resolvent convergence to singularly perturbed Hamiltonians.The QSDE theory and related theory of boundary value problems in Hilbert Fock module gives a number of nontrivial examples of one-to-one correspondence between singularly perturbed Hamiltonians and well-posed symmetrical boundary problems. This correspondence allow to apply algebraic methods developed in the frame of QSDE theory to the class of singularly perturbed problems and on the other hand, to define spectrum and index of QSDE.The third aim is an application of the developed techniques to derive and analyse in the frame of QSDE and quantum measurement theory the realistic model of quantum detector of gravitational waves like LIGO project [27]. This problem is quite interesting because of its interdisciplinary value. The mathematical analysis involves advanced numerical quantum Monte-Carlo algorithms [25] and quantum trajectories method [26]. On the other hand, this is the problem which presents a deep interplay between the quantum evolution and the classical noise. The application to this problem of the theory of quantum filtering recently developed in [28]-[28] by V.P. Belavkin is one of the main objectives of this project. The advantage of the proposed collaboration is that it assumes joint theoretical and numerical analysis of the problem.
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