Can One Count the Shape of a Drum?

Sequences of nodal counts store information on the geometry (metric) of the domain where the wave equation is considered. To demonstrate this statement, we consider the eigenfunctions of the Laplace-Beltrami operator on surfaces of revolution. Arranging the wave functions by increasing values of the eigenvalues, and counting the number of their nodal domains, we obtain the nodal sequence whose properties we study. This sequence is expressed as a trace formula, which consists of a smooth (Weyl-like) part which depends on global geometrical parameters, and a fluctuating part, which involves the classical periodic orbits on the torus and their actions (lengths). The geometrical content of the nodal sequence is thus explicitly revealed.

Figure 1

Numerical checks of the fluctuating parts of trace formulae for the two ellipsoids [(a) and (b), see text] and the two tori [(c) and (d), see text]. Left: Double logarithmic plot of the integrated squared fluctuations (20) (arbitrary scale), the solid line has slope $7/2$. Right: Length spectra of the nodal counting functions (21). The solid line is obtained from the trace formulas (7, 19) and the points represent the numerical data.