Stability of eigenvalues of quantum graphs with respect to magnetic perturbation and the nodal count of the eigenfunctions

Abstract

We prove an analogue of the magnetic nodal theorem on quantum graphs: the number of zeros $\varphi$ of the $n$-th eigenfunction of the Schrödinger operator on a quantum graph is related to the stability of the $n$-th eigenvalue of the perturbation of the operator by magnetic potential. More precisely, we consider the $n$-th eigenvalue as a function of the magnetic perturbation and show that its Morse index at zero magnetic field is equal to $\varphi -(n-1)$.

19 pages, 3 figures

References in corpus (8)

• Quantum wires with magnetic fluxes
• On the connection between the number of nodal domains on quantum graphs and the stability of graph partitions
• Nodal count of graph eigenfunctions via magnetic perturbation
• Magnetic interpretation of the nodal defect on graphs
• Critical partitions and nodal deficiency of billiard eigenfunctions
• The Nodal Count {0, 1, 2, 3,...} Implies The Graph is a Tree
• 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

Cited by in corpus (8)

• Green's function approach for quantum graphs: an overview
• The Nodal Count {0, 1, 2, 3,...} Implies The Graph is a Tree
• Nodal Statistics On Quantum Graphs
• Anomalous nodal count and singularities in the dispersion relation of honeycomb graphs
• On Pleijel's nodal domain theorem for quantum graphs
• Lieb-Schultz-Mattis theorem in higher dimensions from approximate magnetic translation symmetry
• Neumann Domains on Quantum Graphs
• On the nodal structure of nonlinear stationary waves on star graphs