A LOCAL ORTHOGONAL TRANSFORM FOR ACOUSTIC WAVEGUIDES WITH AN INTERNAL INTERFACE

A numerical method is developed for solving the two-dimensional Helmholtz equation in a two layer region bounded by a flat top, a flat bottom and a curved interface. A local orthogonal transform is used to flatten the curved interface of the waveguide. The one-way reformulation based on the Dirichlet-to-Neumann (DtN) map is then used to reduce the boundary value problem to an initial value problem. Numerical implementation of the resulting operator Riccati equation uses a large range step method for discretizing the range variable and a truncated local eigenfunction expansion for approximating the operators. This method is particularly useful for solving long range wave propagation problems in slowly varying waveguides.

Control of Quantum Langevin Equations

The problem of controlling quantum stochastic evolutions arises naturally in several different fields such as quantum chemistry, quantum information theory, quantum engineering, etc. In this paper, we apply the recently discovered closed form of the unitarity conditions for stochastic evolutions driven by the square of white noise [9] to solve this problem in the case of quadratic cost functionals (cf. (5.5) below). The optimal control is explicitly given in terms of the solution of an operator Riccati equation. Under general conditions on the system Hamiltonian part of the stochastic evolution and on the system observable to be controlled, this equation admits solutions with the required properties and they can be explicitly described.