33 citations · 34 across the 10 of their papers we have counts for
10 papers
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 o…
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…
Cycle intersection form and oscillation of graph eigenfunctions
Gregory Berkolaiko, Jared C. Bronski, Mark Goresky
For a real symmetric matrix strictly supported on a finite simple graph, it is shown that the inertia of a weighted intersection form on the cycle space of the graph, with weig…
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…
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,…
Stability of spectral partitions with corners
Gregory Berkolaiko, Yaiza Canzani, Graham Cox +2
A spectral minimal partition of a manifold is a decomposition into disjoint open sets that minimizes a spectral energy functional. While it is known that bipartite minimal partitio…
Homology of spectral minimal partitions
Gregory Berkolaiko, Yaiza Canzani, Graham Cox +1
A spectral minimal partition of a manifold is its decomposition into disjoint open sets that minimizes a spectral energy functional. It is known that bipartite spectral minimal par…
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…
Morse inequalities for ordered eigenvalues of generic self-adjoint families
Gregory Berkolaiko, Igor Zelenko
In many applied problems one seeks to identify and count the critical points of a particular eigenvalue of a smooth parametric family of self-adjoint matrices, with the parameter s…
Intermediate wave-function statistics
G. Berkolaiko, J. P. Keating, B. Winn
We calculate statistical properties of the eigenfunctions of two quantum systems that exhibit intermediate spectral statistics: star graphs and Seba billiards. First, we show that…