Can one count the shape of a drum?
arXiv:nlin/0606050 · doi:10.1103/PhysRevLett.97.090201
Abstract
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.
4 pages, 1 figure
References in corpus (3)
Cited by in corpus (18)
- Nodal portraits of quantum billiards: Domains, lines, and statistics
- Nodal count of graph eigenfunctions via magnetic perturbation
- Nodal domains on isospectral quantum graphs: the resolution of isospectrality ?
- The Nodal Count {0, 1, 2, 3,...} Implies The Graph is a Tree
- Dynamics of nodal points and the nodal count on a family of quantum graphs
- Isospectral graphs with identical nodal counts
- Nodal Statistics On Quantum Graphs
- Nodal domains of a non-separable problem - the right angled isosceles triangle
- Counting nodal domains on surfaces of revolution
- Nodal domains in open microwave systems
- Nodal domains on graphs - How to count them and why?
- Nodal domains of the equilateral triangle billiard
- Inverse Nodal Problems
- Quantum graphs -- Generic eigenfunctions and their nodal count and Neumann count statistics
- Isospectral discrete and quantum graphs with the same flip counts and nodal counts
- On the Nodal Count Statistics for Separable Systems in any Dimension
- WESD - Weighted Spectral Distance for Measuring Shape Dissimilarity
- Quantum Shape Effects