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arXiv · 0801.4873

Counting nodal domains on surfaces of revolution

Panos D. Karageorge·Uzy Smilansky
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Abstract

We consider eigenfunctions of the Laplace-Beltrami operator on special surfaces of revolution. For this separable system, the nodal domains of the (real) eigenfunctions form a checker-board pattern, and their number $ν_n$ is proportional to the product of the angular and the "surface" quantum numbers. Arranging the wave functions by increasing values of the Laplace-Beltrami spectrum, we obtain the nodal sequence, whose statistical properties we study. In particular we investigate the distribution of the normalized counts $\frac{ν_n}{n}$ for sequences of eigenfunctions with $K \le n\le K + ΔK$ where $K,ΔK \in \mathbb{N}$. We show that the distribution approaches a limit as $K,ΔK\to\infty$ (the classical limit), and study the leading corrections in the semi-classical limit. With this information, we derive the central result of this work: the nodal sequence of a mirror-symmetric surface is sufficient to uniquely determine its shape (modulo scaling).

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Panos D. Karageorge, Uzy Smilansky. 2008-01-31. Counting nodal domains on surfaces of revolution. https://doi.org/10.1088/1751-8113/41/20/205102

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