Refraction of high-frequency waves density by sharp interfaces and semiclassical measures at the boundary
type: Item
instance of: scholarly article
title: Refraction of high-frequency waves density by sharp interfaces and semiclassical measures at the boundary
zbMATH Open document ID: 0963.35022
DOI: 10.1016/S0021-7824(00)00158-6
author: Luc Miller
published in: Journal de Mathématiques Pures et Appliquées. Neuvième Série
publication date: 2000-06-22T00:00:00Z
review text: 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 ``wave…
reviewed by: Otto Liess
Mathematics Subject Classification ID: 35B40
Mathematics Subject Classification ID: 78A45
Mathematics Subject Classification ID: 81Q20
Mathematics Subject Classification ID: 35A27
zbMATH DE Number: 1463875
zbMATH Keywords: Cauchy problem for the semiclassical Schrödinger equation
zbMATH Keywords: Schrödinger equation
zbMATH Keywords: semiclassical wave equation
zbMATH Keywords: transmission problem across the flat interface of codimension one
zbMATH Keywords: microlocal versions of the Snell-Descartes law of refraction
MaRDI profile type: Publication
full work available at URL: s0021-7824(00)00158-6
OpenAlex ID: W2037868571
Wikidata QID: Q127012800
description: scientific article; zbMATH DE number 1463875