I was curious about the following vague question. In pseudodifferential calculus and semi-classical analysis there are various theorems that relate geodesics of the underlying space to the behavior of solutions to linear PDEs. For example, we know that the wave-front set propagates along characteristics or that we can construct quasi-modes of the Schroedinger operator using closed characteristics. So is it possible to see the reflection law from geometric optics in the case of manifolds with boundary? Is there a well-established type of semi-classical analysis that would allow to see how to choose the right geodesic after hitting the boundary?
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$\begingroup$ In Melrose and Taylor, "Boundary Problems for Wave Equations With Grazing and Gliding Rays", for the linear wave equation on a domain with boundary, and for the case of transversal rays, they very quickly sketch the construction in section 1.3. I am guessing the material is probably either well-known to the experts or standard. $\endgroup$– Willie WongCommented Oct 21, 2020 at 18:27
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