题目：High frequency vibration modes in a chaotic box

报告人：Stéphane Nonnenmacher，巴黎萨克雷大学

时间：4月15日（星期三），14：00-15：00

地点：东区第五教学楼5107教室

摘要：

The vibration of the membrane of a drum, the propagation of acoustic or electromagnetic waves inside a box, can be modelized by the wave equation inside a bounded domain (in 2 or 3 dimensions), with appropriate boundary conditions. Any solution can be decomposed into a sequence of stationary vibration modes with higher and higher eigenfrequencies. Those modes are identified with eigenfunctions of the Laplace operator in the box. 

For simple box shapes (rectangle, circle), those eigenfunctions can be written explicitly in terms of special functions, but for a generic box there is no such expression at hand. I will explain how one can nevertheless obtain some (less precise) information on the eigenfunctions in the high-frequency regime, a regime where wave propagation can be relative with the ray (or billiard) dynamics in the ball. In particular, if the billiard dynamics happens to be chaotic (in particular, ergodic), then one can prove that the energy of most eigenmodes equidistributes across the box in the high-frequency limit, a phenomenon referred to as quantum ergodicity.