34 research outputs found
On the connection between the number of nodal domains on quantum graphs and the stability of graph partitions
Courant theorem provides an upper bound for the number of nodal domains of
eigenfunctions of a wide class of Laplacian-type operators. In particular, it
holds for generic eigenfunctions of quantum graph. The theorem stipulates that,
after ordering the eigenvalues as a non decreasing sequence, the number of
nodal domains of the -th eigenfunction satisfies . Here,
we provide a new interpretation for the Courant nodal deficiency in the case of quantum graphs. It equals the Morse index --- at a
critical point --- of an energy functional on a suitably defined space of graph
partitions. Thus, the nodal deficiency assumes a previously unknown and
profound meaning --- it is the number of unstable directions in the vicinity of
the critical point corresponding to the -th eigenfunction. To demonstrate
this connection, the space of graph partitions and the energy functional are
defined and the corresponding critical partitions are studied in detail.Comment: 22 pages, 6 figure
Lieb-Robinson Bounds for Harmonic and Anharmonic Lattice Systems
We prove Lieb-Robinson bounds for the dynamics of systems with an infinite
dimensional Hilbert space and generated by unbounded Hamiltonians. In
particular, we consider quantum harmonic and certain anharmonic lattice
systems
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
Lieb-Robinson bounds for classical anharmonic lattice systems
We prove locality estimates, in the form of Lieb-Robinson bounds, for classical
oscillator systems defined on a lattice. Our results hold for the harmonic system and a
variety of anharmonic perturbations with long range interactions. The anharmonic estimates
are applicable to a special class of observables, the Weyl functions, and the bounds which
follow are not only independent of the volume but also the initial condition
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Lieb-Robinson bounds for classical anharmonic lattice systems
We prove locality estimates, in the form of Lieb-Robinson bounds, for classical
oscillator systems defined on a lattice. Our results hold for the harmonic system and a
variety of anharmonic perturbations with long range interactions. The anharmonic estimates
are applicable to a special class of observables, the Weyl functions, and the bounds which
follow are not only independent of the volume but also the initial condition