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Counting nodal domains on surfaces of revolution

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 νn\nu_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 νnn\frac{\nu_n}{n} for sequences of eigenfunctions with KnK+ΔKK \le n\le K + \Delta K where K,ΔKNK,\Delta K \in \mathbb{N}. We show that the distribution approaches a limit as K,ΔKK,\Delta K\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).Comment: 36 pages, 8 figure

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    Last time updated on 01/04/2019