Abstract
Given any finite set of trajectories of a Lipschitzian quantum stochastic differential inclusion (QSDI), there exists a continuous selection from the complex-valued multifunction associated with the solution set of the inclusion, interpolating the matrix elements of the given trajectories. Furthermore, the difference of any two of such solutions is bounded in the seminorm of the locally convex space of solutions.
Highlights
Given any finite set of trajectories of a Lipschitzian quantum stochastic differential inclusion (QSDI), there exists a continuous selection from the complex-valued multifunction associated with the solution set of the inclusion, interpolating the matrix elements of the given trajectories
Establishment of continuous selections from the solution set multifunctions of differential inclusions defined on finite dimensional Euclidean spaces and their applications have been considered by many authors
The present paper is, concerned with the establishment of the existence of continuous selections from the complex valued multifunctions associated with the solution sets of QSDI interpolating the matrix elements of a given finite set of trajectories that start from distinct points
Summary
Establishment of continuous selections from the solution set multifunctions of differential inclusions defined on finite dimensional Euclidean spaces and their applications have been considered by many authors (see, e.g., Aubin and Cellina [1], Repovs and Semenov [2], Smirnov [3] and the references they contain). Theoretical and numerical aspects of QSDI have not enjoyed significant development in comparison with the classical cases there are some recent results along these directions (see, e.g., [4,5,6,7,8,9,10]). This situation remains in spite of the numerous practical problems in quantum dynamical systems, quantum open systems, quantum measurement theory, quantum optics, and quantum stochastic control theory for which methods of quantum stochastic inclusions are applicable. It is well known that discontinuous quantum stochastic differential equations can be numerically and theoretically treated by reformulating them as regularized inclusions (see [8,9,10,11,12,13])
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