Conservation equations for the semiclassical Schrödinger equation near caustics

Abstract

We derive an infinite hierarchy of conservation equations for the semiclassical Schrödinger equation, by employing the moments of the solutions of the Wigner (quantum Liouville) equation. The Wigner equation retains terms of any order with respect to the small parameter, and contrary to the moment equations derived form the classical Liouville equation, it provides a hierarchy of conservation equations which is uniformly valid when the wave field develops caustics. We confirm the validity of the equations with analytical examples for simple caustics, namely the focal points of a harmonic oscillator and the fold caustic generated from the initial data in the case of the free Schrödinger equation. We also make some new observations concerning the relation of the derived conservation equations with those coming from the application of Noether's variational theorem. We get some evidence that for general potentials, this standard variational approach provides the equations needed to handle only caustic-free wavefields, while the phase-space Wigner equation is able to provide as many equations as are necessary to handle wavefields with caustics.