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 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
nνn for sequences of eigenfunctions with K≤n≤K+ΔK where K,ΔK∈N. We show that the distribution approaches
a limit as K,ΔK→∞ (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