On the connection between the number of nodal domains on quantum graphs and the stability of graph partitions

Abstract

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 ${\nu }_n$ of the $n$-th eigenfunction satisfies $n\ge {\nu }_n$. Here, we provide a new interpretation for the Courant nodal deficiency $d_n=n-{\nu }_n$ 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 $n$-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.

22 pages, 6 figures

References in corpus (7)

• Nodal domains on isospectral quantum graphs: the resolution of isospectrality ?
• Critical partitions and nodal deficiency of billiard eigenfunctions
• Dynamics of nodal points and the nodal count on a family of quantum graphs
• Stability of nodal structures in graph eigenfunctions and its relation to the nodal domain count
• Counting nodal domains on surfaces of revolution
• Inverse Nodal Problems
• Dependence of the spectrum of a quantum graph on vertex conditions and edge lengths

Cited by in corpus (22)

• Nodal count of graph eigenfunctions via magnetic perturbation
• On the ground state of quantum graphs with attractive $\delta$-coupling
• Critical partitions and nodal deficiency of billiard eigenfunctions
• Extremality conditions and regularity of solutions to optimal partition problems involving Laplacian eigenvalues
• The Nodal Count {0, 1, 2, 3,...} Implies The Graph is a Tree
• Stability of nodal structures in graph eigenfunctions and its relation to the nodal domain count
• Isospectral graphs with identical nodal counts
• Anomalous nodal count and singularities in the dispersion relation of honeycomb graphs
• Stability of eigenvalues of quantum graphs with respect to magnetic perturbation and the nodal count of the eigenfunctions
• On Pleijel's nodal domain theorem for quantum graphs
• Edge switching transformations of quantum graphs
• Courant-sharp eigenvalues of Neumann 2-rep-tiles
• Quantum graphs -- Generic eigenfunctions and their nodal count and Neumann count statistics
• Stability of spectral partitions and the Dirichlet-to-Neumann map
• Neumann Domains on Quantum Graphs
• Nodal domains, spectral minimal partitions, and their relation to Aharonov-Bohm operators
• Isospectral discrete and quantum graphs with the same flip counts and nodal counts
• Interlacing and Friedlander-type inequalities for spectral minimal partitions of metric graphs
• Concrete method for recovering the Euler characteristic of quantum graphs
• On the nodal structure of nonlinear stationary waves on star graphs
• Topics in elliptic problems: from semilinear equations to shape optimization
• Morse inequalities for ordered eigenvalues of generic self-adjoint families