One can hear the corners of a drum

通过四处敲击聆听鼓的形状

Hearing the shape of a drum by knocking around

研究 Kac 问题变体，通过允许获取流形上每点局部 Weyl 计数函数来唯一恢复 Riemannian 度量，证明在 Laplace-Beltrami 谱简单时可确定鼓形状，并给出反例说明假设必要性。

How to hear the corners of a drum

We announce a new result which shows that under either Dirichlet, Neumann, or Robin boundary conditions, the corners in a planar domain are a spectral invariant of the Laplacian. For the case of polygonal domains, we show how a locality principle, in the spirit of Kac's "principle of not feeling the boundary" can be used together with calculations of explicit model heat kernels to prove the result. In the process, we prove this locality principle for all three boundary conditions. Albeit previously known for Dirichlet boundary conditions, this appears to be new for Robin and Neumann boundary c…

（未）听到某些三角形鼓被敲击的位置

(Not) hearing where certain triangular drums are struck

研究特殊平面三角形鼓被敲击的位置，通过证明得出能听到等边和等腰直角三角形被敲击位置，且30 - 60 - 90三角形有两个听觉不可区分点，为首例拓扑连通此类情况。

Can you hear the fractal dimension of a drum?

Electromagnetics and Acoustics on a bounded domain are governed by the Helmholtz's equation; when such a domain is a [pre-]fractal described by means of a `just-touching' Iterated Function System (IFS) spectral decomposition of the Helmholtz's operator is self-similar as well. Renormalization of the Green's function proves this feature and isolates a subclass of eigenmodes, called ``diaperiodic'', whose waveforms and eigenvalues can be recursively computed applying the IFS to the initiator's eigenspaces. The definition of ``spectral dimension'' is given and proven to depend on diaperiodic mode…

Analytical aspects of isospectral drums

We reexamine the proofs of isospectrality of the counterexample domains to Kac' question `Can one hear the shape of a drum?' from an analytical viewpoint. We reformulate isospectrality in a more abstract setting as the existence of a similarity transform intertwining two operators associated with elliptic forms, and give several equivalent characterizations of this property as intertwining the forms and form domains, the associated operators and operator domains, and the semigroups they generate. On a representative pair of counterexample domains, we use these criteria to show that the similar…

你能在流形上听出你的位置吗？

Can you hear your location on a manifold?

本文将“听鼓辨形”推广为“听声定位”，证明对一类通有度量，逐点谱计数函数可在对称等价意义下唯一确定流形上的点。