2010/08/03 by Ram Band, Gregory Berkolaiko, Uzy Smilansky
Mathematics · Physics and Astronomy · #Quantum Mechanics and Non-Hermitian Physics #Quantum chaos and dynamical systems #Spectral Theory in Mathematical Physics #math-ph #math.MP #msc:34B45 #msc:35Q40
paper · pdf · doi:10.1007/s00023-011-0124-1
published as Ann. Henri Poincare, 13, 145-184 (2012) · 34 pages, 12 figures
arxiv created 2010/08/03 · openalex publication_date 2011/07/04 · arxiv updated 2013/03/06 · openalex created_date 2016/06/24 · openalex updated_date 2026/07/28
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.