From the 1 of 6 linked papers with an AI index.
6 papers
Cycle intersection form and oscillation of graph eigenfunctions
Gregory Berkolaiko, Jared C. Bronski, Mark Goresky
The paper relates the inertia of a weighted cycle‑intersection form, built from a non‑zero eigenvector of a graph‑supported symmetric matrix, to oscillation properties such as sign…
The calculus of Duistermaat's triple index
Gregory Berkolaiko, Graham Cox, Yuri Latushkin +1
In this paper we develop a systematic calculus for the Duistermaat index, a symplectic invariant defined for triples of Lagrangian subspaces. Introduced nearly half a century ago,…
Quantum graph models of quantum chaos: an introduction and some recent applications
Gregory Berkolaiko, Sven Gnutzmann
Quantum graphs are a paradigmatic model for quantum chaos as well as for spectral theory. We give a concise didactical introduction to quantum graphs, or Schrödinger Hamiltonians…
The Duistermaat index and eigenvalue interlacing for self-adjoint extensions of a symmetric operator
Gregory Berkolaiko, Graham Cox, Yuri Latushkin +1
Eigenvalue interlacing is a useful tool in linear algebra and spectral analysis. In its simplest form, the interlacing inequality states that a rank-one positive perturbation shift…
Eigenvalues of the discrete p-Laplacian via graph surgery
Gregory Berkolaiko, Matthias Hofmann
We develop a Hellmann--Feynman type perturbation theory for the discrete signed -Laplacian and apply it to a parametrized perturbation by edge cuts. We show that the eigenvalues…
Smooth critical points of eigenvalues on the torus of magnetic perturbations of graphs
Lior Alon, Gregory Berkolaiko, Mark Goresky
Motivated by the nodal distribution universality conjecture for discrete operators on graphs and by the spectral analysis of their maximal abelian covers, we consider a family of H…