In this paper, we present an extension of the classical Quantum ergodicity Theorem, due to Shnirelman, to the case of Laplacians with discontinous metrics along interfaces. The “geodesic flow” is then no more a flow, but a Markov process due to the fact that rays can by reflected or refracted at the interfaces. We give also an example build by gluing together two flat Euclidean disks.
@article{TSG_2012-2014__31__71_0, author = {Colin de Verdi\`ere, Yves}, title = {The semi-classical ergodic {Theorem} for discontinuous metrics}, journal = {S\'eminaire de th\'eorie spectrale et g\'eom\'etrie}, pages = {71--89}, publisher = {Institut Fourier}, address = {Grenoble}, volume = {31}, year = {2012-2014}, doi = {10.5802/tsg.295}, language = {en}, url = {http://www.numdam.org/articles/10.5802/tsg.295/} }
TY - JOUR AU - Colin de Verdière, Yves TI - The semi-classical ergodic Theorem for discontinuous metrics JO - Séminaire de théorie spectrale et géométrie PY - 2012-2014 SP - 71 EP - 89 VL - 31 PB - Institut Fourier PP - Grenoble UR - http://www.numdam.org/articles/10.5802/tsg.295/ DO - 10.5802/tsg.295 LA - en ID - TSG_2012-2014__31__71_0 ER -
%0 Journal Article %A Colin de Verdière, Yves %T The semi-classical ergodic Theorem for discontinuous metrics %J Séminaire de théorie spectrale et géométrie %D 2012-2014 %P 71-89 %V 31 %I Institut Fourier %C Grenoble %U http://www.numdam.org/articles/10.5802/tsg.295/ %R 10.5802/tsg.295 %G en %F TSG_2012-2014__31__71_0
Colin de Verdière, Yves. The semi-classical ergodic Theorem for discontinuous metrics. Séminaire de théorie spectrale et géométrie, Tome 31 (2012-2014), pp. 71-89. doi : 10.5802/tsg.295. http://www.numdam.org/articles/10.5802/tsg.295/
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