﻿ Can you hear the shape of a drum? and deformational spectral rigidity-东北师范大学数学与统计学院

Can you hear the shape of a drum? and deformational spectral rigidity

M. Kac popularized the following question "Can one hear the shape of a drum?" Mathematically, consider a bounded planar domain Ω ⊆ R2 with a smooth boundary and the associated Dirichlet problem

Δu + λu=0, u|∂Ω=0.

The set of λ's for which this equation has a solution is called the Laplace spectrum of Ω. Does the Laplace spectrum determine Ω up to isometry? In general, the answer is negative. Consider the billiard problem inside Ω. Call the length spectrum the closure of the set of perimeters of all periodic orbits of the billiard inside Ω. Due to deep properties of the wave trace function, generically, the Laplace spectrum determines the length spectrum. Jointly with J. De Simoi and Q. Wei, we show that an axially symmetric domain close to the circle is dynamically spectrally rigid, i.e. cannot be deformed without changing the length spectrum. This partially answers a question of P. Sarnak.

Vadim Kaloshin, 欧洲科学院院士，奥地利科学院教授, 曾获得美国科学院院士提名、西蒙斯数学奖等荣誉, 现担任Adv. Math., Ergodic Theory Dynam. Systems等杂志编委, 主要从事动力系统领域的研究, 在国际上最顶尖的四大综合性数学期刊Acta Math., Ann. of Math., J. Amer. Math. Soc., Invent. Math.上公开发表高质量学术论文8篇, 在Duke Math. J., Geom. Funct. Anal., J. Eur. Math. Soc. (JEMS), Comm. Pure Appl. Math., Arch. Ration. Mech. Anal.等国际权威期刊上公开发表高水平学术论文近70篇.