TL;DR
This paper demonstrates that the sequence of nodal points uniquely identifies certain geometric shapes like rectangles, tori, and Klein bottles up to scaling, revealing deep connections between spectral data and geometry.
Contribution
It establishes that nodal sequences determine the geometry of specific manifolds within certain classes, advancing inverse spectral theory.
Findings
Nodal sequences determine rectangles with Dirichlet conditions up to scale.
Nodal sequences determine separable 2D tori and Klein bottles.
Nodal sequences determine flat tori in 2D and 3D.
Abstract
It is shown that nodal sequences determine the underlying manifold up to scaling within classes of rectangles with Dirichlet boundary conditions, separable two dimensional tori, two-dimensional flat Klein bottles and flat tori in two and three dimensions.
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