Refraction of high-frequency waves density by sharp interfaces and semiclassical measures at the boundary
The author considers the Cauchy problem for the semiclassical Schrödinger equation \(i h \partial_{t} \psi^{ h }=-( h^{2}/2) \Delta^{ h } + V(x) \psi^{h }\) in \({\mathbb R}_{t}\times {\mathbb R}^{d}_{x}\), \(\psi^{h }(t=0)= \psi^{h }_{0}\) in \({\mathbb R}^{d }_{x} ={\mathbb R }^{d -1}_{x'} \times {\mathbb R }_{x^{d } }\), respectively for the semiclassical wave equation \( \partial^{2}_{t} w^{h }- \nabla_{x}( c^{2}(x) \nabla_{x}) w^{h } =0\) in \({\mathbb R}_{t}\times {\mathbb R}^{d}_{x}\), \(( w^{h }, \partial_{t} w^{h }) (t=0)= ( w^{h }_{0}, \partial_{t} w^{h }_{0})\) in \({\mathbb R }^{d }_{x} ={\mathbb R }^{d -1}_{x'} \times {\mathbb R }_{x^{d } }\), with initial data \( \psi^{h }_{0} \), respectively \(( w^{h }_{0}, \partial_{t} w^{h }_{0})\), depending on a small parameter \(h\) and with a potential \(V\) and a celerity \(c\) which correspond to a transmission problem across the flat interface of codimension one \( \{ x^{d } =0\}\): \(V ( x)= V_{-}(x)\) for \( x^{d }<0\), \(V ( x)= V_{+}(x)\) for \( x^{d } \geq 0\), \(c(x)=c_{-}(x)\) for \( x^{d }<0\), \( c( x)= c_{+}(x)\) for \( x^{d } \geq 0\). The potential \(V\), respectively the celerity \(c\), are assumed to be smooth up to \( x^{d } =0\). In addition it is assumed that they have ``jumps'' of form \( V_{+}(x',0)>V_{-}(x',0)\), \(c_{+}(x',0)>c_{-}(x',0)\) on \( x^{d } =0\) and that \(V(x)>- \alpha |x |^{2}-\beta \), \( c(x) \geq \gamma >0\), for all \(x\). Also fix a sequence \( h_{j} \rightarrow 0+\). The problem is then to study the asymptotic behaviour when \( j \rightarrow \infty \) of the densities \(M^{h }_{t}=|\psi^{h } (\xi,t)|^{2} dx \), respectively \( E_{}^{h }= \{ |\partial_{t} w^{h }(\tau,x)|^{2}+|c(x) \nabla w^{h }(\tau,x)|^{2}\} dx\). This is described in terms of ``semiclassical'' measures, which measure ``asymptotic microlocal densities''. Microlocal versions of the Snell-Descartes law of refraction (which includes diffractive rays) are established and ``radiation phenomena'' for ``waves density propagation'' inside the interface along gliding rays are illustrated.
Cauchy problem for the semiclassical Schrödinger equation
transmission problem across the flat interface of codimension one
microlocal versions of the Snell-Descartes law of refraction
scientific article; zbMATH DE number 1463875