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Singular Hamiltonian systems arise in modeling high frequency or quantum waves through interfaces or barriers, in which the potential functions become discontinuous and the underlying Hamiltonian systems have singular forces, preventing the initial value problems from being well-posed. We first introduce a notion of solutions, by building in wave transmission and reflection mechanism at the barriers, allowing one to go beyond the singular point and defines a well-posed initial value problem consistent with physical laws of refractions. This also naturally induces numerical methods that build in the interface condition into the numerical fluxes.

Such ideas have found applications from partial transmissions, reflections and even diffractions of high frequency waves, semi-classical computation of quantum tunneling through barriers, and surface hopping in which the particles can tunnel through different quantum states due to degeneracy in the eigen-structures of the potential energy surfaces. The methods allow particle simulations, with the computational cost of classical mechanics, to capture certain important quantum phenomena.