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 ${\nu }_n$ 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 $\frac{{\nu }_n}{n}$ for sequences of eigenfunctions with $K\le n\le K+\Delta K$ where $K,\Delta K\in \mathbb{N}$. We show that the distribution approaches a limit as $K,\Delta K\to \infty$ (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).

36 pages, 8 figures

References in corpus (2)

• Can one count the shape of a drum?
• Nodal Domain Statistics for Quantum Maps, Percolation and SLE

Cited by in corpus (1)

• Inverse Nodal Problems