Resolving isospectral "drums" by counting nodal domains
Gnutzmann, Sven ; Smilansky, Uzy ; Sondergaard, Niels
arXiv, 0504050 / Harvested from arXiv
Several types of systems were put forward during the past decades to show that there exist {\it isospectral} systems which are {\it metrically} different. One important class consists of Laplace Beltrami operators for pairs of flat tori in $\mathbb{R}^n$ with $n\geq 4$. We propose that the spectral ambiguity can be resolved by comparing the nodal sequences (the numbers of nodal domains of eigenfunctions, arranged by increasing eigenvalues). In the case of isospectral flat tori in four dimensions - where a 4-parameters family of isospectral pairs is known- we provide heuristic arguments supported by numerical simulations to support the conjecture that the isospectrality is resolved by the nodal count. Thus - one can {\it count} the shape of a drum (if it is designed as a flat torus in four dimensions...).
Publié le : 2005-04-24
Classification:  Nonlinear Sciences - Chaotic Dynamics,  Mathematical Physics
@article{0504050,
     author = {Gnutzmann, Sven and Smilansky, Uzy and Sondergaard, Niels},
     title = {Resolving isospectral "drums" by counting nodal domains},
     journal = {arXiv},
     volume = {2005},
     number = {0},
     year = {2005},
     language = {en},
     url = {http://dml.mathdoc.fr/item/0504050}
}
Gnutzmann, Sven; Smilansky, Uzy; Sondergaard, Niels. Resolving isospectral "drums" by counting nodal domains. arXiv, Tome 2005 (2005) no. 0, . http://gdmltest.u-ga.fr/item/0504050/