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Resolving isospectral 'drums' by counting nodal domains

Gnutzmann, S; Smilansky, U and Søndergaard, Niels LU (2005) In Journal of Physics A: Mathematical and General 38(41). p.8921-8933
Abstract
Several types of systems have been put forward during the past few decades to show that there exist isospectral systems which are metrically different. One important class consists of Laplace-Beltrami operators for pairs of flat tori in R-n with n >= 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 four-parameter 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 count the shape of a drum (if it is... (More)
Several types of systems have been put forward during the past few decades to show that there exist isospectral systems which are metrically different. One important class consists of Laplace-Beltrami operators for pairs of flat tori in R-n with n >= 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 four-parameter 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 count the shape of a drum (if it is designed as a flat torus in four dimensions). (Less)
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author
organization
publishing date
type
Contribution to journal
publication status
published
subject
in
Journal of Physics A: Mathematical and General
volume
38
issue
41
pages
8921 - 8933
publisher
IOP Publishing
external identifiers
  • wos:000233112500008
  • scopus:25844479881
ISSN
0305-4470
DOI
10.1088/0305-4470/38/41/006
language
English
LU publication?
yes
id
d6c7963c-7c9a-4d4f-9bbd-3bd313fb278d (old id 213834)
date added to LUP
2007-08-23 14:25:38
date last changed
2017-01-01 07:01:13
@article{d6c7963c-7c9a-4d4f-9bbd-3bd313fb278d,
  abstract     = {Several types of systems have been put forward during the past few decades to show that there exist isospectral systems which are metrically different. One important class consists of Laplace-Beltrami operators for pairs of flat tori in R-n with n >= 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 four-parameter 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 count the shape of a drum (if it is designed as a flat torus in four dimensions).},
  author       = {Gnutzmann, S and Smilansky, U and Søndergaard, Niels},
  issn         = {0305-4470},
  language     = {eng},
  number       = {41},
  pages        = {8921--8933},
  publisher    = {IOP Publishing},
  series       = {Journal of Physics A: Mathematical and General},
  title        = {Resolving isospectral 'drums' by counting nodal domains},
  url          = {http://dx.doi.org/10.1088/0305-4470/38/41/006},
  volume       = {38},
  year         = {2005},
}