2 research outputs found
Dynamics of nodal points and the nodal count on a family of quantum graphs
We investigate the properties of the zeros of the eigenfunctions on quantum
graphs (metric graphs with a Schr\"odinger-type differential operator). Using
tools such as scattering approach and eigenvalue interlacing inequalities we
derive several formulas relating the number of the zeros of the n-th
eigenfunction to the spectrum of the graph and of some of its subgraphs. In a
special case of the so-called dihedral graph we prove an explicit formula that
only uses the lengths of the edges, entirely bypassing the information about
the graph's eigenvalues. The results are explained from the point of view of
the dynamics of zeros of the solutions to the scattering problem.Comment: 34 pages, 12 figure
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