Highly Oscillatory Partial Differential Equations

摘要

Partial differential equations with highly oscillatory solutions occur in many areas of science like quantum mechanics and acoustics, with important spinoffs to semiconductors, nanotechnology and low-temperature physics. These equations pose a great challenge to mathematical and numerical analysis. Recently a new mathematical technique has been developed to treat highly oscillatory PDEs, namely Wigner transforms. They allow deep new insights into high-frequency asymptotics. The state-of-the-art is reported here, with emphasis on applications like quantum semiconductor devices and Bose-Einstein condensation.