Об Осцилляционности Спектра Краевой Задачи На Графе

We investigate the properties of the zeros of the eigenfunctions on quantum graphs (metric graphs with a Schrödinger-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.

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Dynamics of Nodal Points and the Nodal Count on a Family of Quantum Graphs

Band

Berkolaiko

Smilansky

2011

Ann. Henri Poincaré

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Neumann Domains on Quantum Graphs

Alon

Band

2021

Ann. Henri Poincaré

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The Number of Nodal Domains on Quantum Graphs as a Stability Index of Graph Partitions

Band

Berkolaiko

Raz

et al. 2011

Commun. Math. Phys.

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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 ν n of the n-th eigenfunction satisfies n ≥ ν n . Here, we provide a new interpretation for the Courant nodal deficiency d n = n − ν 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.

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Об Осцилляционности Спектра Краевой Задачи На Графе

Cited by 9 publications

References 1 publication

Dynamics of Nodal Points and the Nodal Count on a Family of Quantum Graphs

Dynamics of Nodal Points and the Nodal Count on a Family of Quantum Graphs

Neumann Domains on Quantum Graphs

The Number of Nodal Domains on Quantum Graphs as a Stability Index of Graph Partitions

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