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

Vadim Kaloshin 氏
(University of Maryland, USA)

2019年7月4日(木) 15時00分

総合研究8号館講義室1

M. Kac popularized the following question `Can you hear the shape of a drum?’. Mathematically, consider a bounded planar domain $\Omega \subset \mathbb R^2$ with a smooth boundary and the associated Dirichlet problem.$\Delta u+\lambda u=0, u|_{partial \Omega}$. The set of $\lambda$'s this equation has a solution, is called the Laplace spectrum of $\Omega$. Does the Laplace spectrum determine $\Omega$ up to isometry? In general, the answer is negative. Consider the billiard problem inside $\Omega$. Call the length spectrum the closure of the set of perimeters of all periodic orbits of the billiard inside $\Omega$. Due to deep properties of the wave trace function, generically, the Laplace spectrum determines the length spectrum. We show that a generic axis symmetric domain is dynamically spectrally rigid, i.e. can't be deformed without changing the length spectrum. This partially answers a question of P. Sarnak.

Last modified: Wed Jan 16 18:03:30 JST 2019