18 research outputs found
On the Nodal Count Statistics for Separable Systems in any Dimension
We consider the statistics of the number of nodal domains aka nodal counts
for eigenfunctions of separable wave equations in arbitrary dimension. We give
an explicit expression for the limiting distribution of normalised nodal counts
and analyse some of its universal properties. Our results are illustrated by
detailed discussion of simple examples and numerical nodal count distributions.Comment: 21 pages, 4 figure
Isospectral discrete and quantum graphs with the same flip counts and nodal counts
The existence of non-isomorphic graphs which share the same Laplace spectrum
(to be referred to as isospectral graphs) leads naturally to the following
question: What additional information is required in order to resolve
isospectral graphs? It was suggested by Band, Shapira and Smilansky that this
might be achieved by either counting the number of nodal domains or the number
of times the eigenfunctions change sign (the so-called flip count). Recently
examples of (discrete) isospectral graphs with the same flip count and nodal
count have been constructed by K. Ammann by utilising Godsil-McKay switching.
Here we provide a simple alternative mechanism that produces systematic
examples of both discrete and quantum isospectral graphs with the same flip and
nodal counts.Comment: 16 pages, 4 figure
Stability of nodal structures in graph eigenfunctions and its relation to the nodal domain count
The nodal domains of eigenvectors of the discrete Schrodinger operator on
simple, finite and connected graphs are considered. Courant's well known nodal
domain theorem applies in the present case, and sets an upper bound to the
number of nodal domains of eigenvectors: Arranging the spectrum as a non
decreasing sequence, and denoting by the number of nodal domains of the
'th eigenvector, Courant's theorem guarantees that the nodal deficiency
is non negative. (The above applies for generic eigenvectors. Special
care should be exercised for eigenvectors with vanishing components.) The main
result of the present work is that the nodal deficiency for generic
eigenvectors equals to a Morse index of an energy functional whose value at its
relevant critical points coincides with the eigenvalue. The association of the
nodal deficiency to the stability of an energy functional at its critical
points was recently discussed in the context of quantum graphs
[arXiv:1103.1423] and Dirichlet Laplacian in bounded domains in
[arXiv:1107.3489]. The present work adapts this result to the discrete case.
The definition of the energy functional in the discrete case requires a special
setting, substantially different from the one used in
[arXiv:1103.1423,arXiv:1107.3489] and it is presented here in detail.Comment: 15 pages, 1 figur