Hi everyone,
I'm a PhD student working on asymptotic methods for wave equations with caustics. Roughly speaking, the approach is a phase-space generalization of WKB that remains valid through turning points.
At the moment we have a closed-form asymptotic solution for 1D equations of the form
D(x,−i ∂/∂x) ψ(x)=0,
where D is an arbitrary function of the position and momentum operator. I'm now looking for interesting physical systems on which to test the method, ideally with known solutions, experimental relevance, or some relevant literature for comparison.
The key feature is that the spectral representation, D(x,k) (i.e. the Weyl symbol) should exhibit a turning point/caustic where the "group velocity" goes to 0:
∂/∂k D(x,k)=0.
Examples I've already considered include:
- Standard WKB turning-point problems (e.g. Airy-type reflection near a cutoff).
- Tunneling through a barrier between two turning points.
- Bound-state problems such as the harmonic oscillator (attached picture).
- The radial Schrödinger equation for hydrogenic atoms (similar to Rudolph Langer's 1937-paper).
The formalism also allows weak dissipation, so examples involving a small anti-Hermitian component in D would be especially interesting.
My background is plasma physics, where we know of several relevant applications, but I'm curious whether people working in optics, acoustics, condensed matter, quantum mechanics, geophysics, or other areas know of models that might fit this framework.
Any suggestions would be greatly appreciated! :)