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 (6)

• Nodal domains statistics - a criterion for quantum chaos
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• Can one count the shape of a drum?
• Nodal Domain Statistics for Quantum Maps, Percolation and SLE
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Cited by in corpus (13)

• On the connection between the number of nodal domains on quantum graphs and the stability of graph partitions
• Nodal portraits of quantum billiards: Domains, lines, and statistics
• Critical partitions and nodal deficiency of billiard eigenfunctions
• Dynamics of nodal points and the nodal count on a family of quantum graphs
• The Nodal Count {0, 1, 2, 3,...} Implies The Graph is a Tree
• Stability of nodal structures in graph eigenfunctions and its relation to the nodal domain count
• Isospectral graphs with identical nodal counts
• Nodal Statistics On Quantum Graphs
• Nodal domains of a non-separable problem - the right angled isosceles triangle
• Nodal domains of the equilateral triangle billiard
• Inverse Nodal Problems
• Isospectral discrete and quantum graphs with the same flip counts and nodal counts
• On the Nodal Count Statistics for Separable Systems in any Dimension