Publications (35)
Fredholm determinants, Evans functions and Maslov indices for partial differential equations
Graham Cox, Yuri Latushkin, Alim Sukhtayev
The Evans function is a well known tool for locating spectra of differential operators in one spatial dimension. In this paper we construct a multidimensional analogue as the modif…
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…
Nodal deficiency, spectral flow, and the Dirichlet-to-Neumann map
Gregory Berkolaiko, Graham Cox, Jeremy L. Marzuola
It was recently shown that the nodal deficiency of an eigenfunction is encoded in the spectrum of the Dirichlet-to-Neumann operators for the eigenfunction's positive and negative n…
Eigenvalues of the MOTS stability operator for slowly rotating Kerr black holes
Liam Bussey, Graham Cox, Hari Kunduri
We study the eigenvalues of the MOTS stability operator for the Kerr black hole with angular momentum per unit mass . We prove that each eigenvalue depends analytically…
(Non)uniqueness of critical points in variational data assimilation
Graham Cox
In this paper we apply the 4D-Var data assimilation scheme to the initialization problem for a family of quasilinear evolution equations. The resulting variational problem is non-c…
A dynamical approach to semilinear elliptic equations
Margaret Beck, Graham Cox, Christopher Jones +2
A characterization of a semilinear elliptic partial differential equation (PDE) on a bounded domain in is given in terms of an infinite-dimensional dynamical system.…
Blowup solutions of Jang's equation near a spacetime singularity
Amir Babak Aazami, Graham Cox
We study Jang's equation on a one-parameter family of asymptotically flat, spherically symmetric Cauchy hypersurfaces in the maximally extended Schwarzschild spacetime. The hypersu…
Isoperimetric relations between Dirichlet and Neumann eigenvalues
Graham Cox, Scott Scott MacLachlan, Luke Steeves
Inequalities between the Dirichlet and Neumann eigenvalues of the Laplacian have received much attention in the literature, but open problems abound. Here, we study the number of N…
Estimation of CO flux from targeted satellite observations: a Bayesian approach
Graham Cox
We consider the estimation of carbon dioxide flux at the ocean-atmosphere interface, given weighted averages of the mixing ratio in a vertical atmospheric column. In particular we…
Manifold decompositions and indices of Schrödinger operators
Graham Cox, Christoper K. R. T. Jones, Jeremy L. Marzuola
The Maslov index is used to compute the spectra of different boundary value problems for Schrödinger operators on compact manifolds. The main result is a spectral decomposition fo…
Exponential dichotomies for elliptic PDE on radial domains
Margaret Beck, Graham Cox, Christopher Jones +2
It was recently shown by the authors that a semilinear elliptic equation can be represented as an infinite-dimensional dynamical system in terms of boundary data on a shrinking one…
Symmetry and instability of marginally outer trapped surfaces
Ivan Booth, Graham Cox, Juan Margalef-Bentabol
We consider an initial data set having a continuous symmetry and a marginally outer trapped surface (MOTS) that is not preserved by this symmetry. We show that such a MOTS is unsta…
Gaussian curvature and gyroscopes
Graham Cox, Mark Levi
We relate Gaussian curvature to the gyroscopic force, thus giving a mechanical interpretation of the former and a geometrical interpretation of the latter. We do so by considering…
Defining the spectral position of a Neumann domain
Ram Band, Graham Cox, Sebastian Egger
A Laplacian eigenfunction on a two-dimensional Riemannian manifold provides a natural partition into Neumann domains (a.k.a. a Morse--Smale complex). This partition is generated by…
Conformal invariants from nodal sets II. Manifolds with boundary
Graham Cox, Dmitry Jakobson, Mikhail Karpukhin +1
In this paper, we study conformal invariants that arise from nodal sets and negative eigenvalues of conformally covariant operators on manifolds with boundary. We also consider app…
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,…
The essential spectrum of the 2D Euler operator
Graham Cox
Even in two dimensions, the spectrum of the linearized Euler operator is notoriously hard to compute. In this paper we give a new geometric calculation of the essential spectrum fo…
Hamiltonian spectral flows, the Maslov index, and the stability of standing waves in the nonlinear Schrödinger equation
Graham Cox, Mitchell Curran, Yuri Latushkin +1
We use the Maslov index to study the spectrum of a class of linear Hamiltonian differential operators. We provide a lower bound on the number of positive real eigenvalues, which in…
The Morse and Maslov indices for multidimensional Schrödinger operators with matrix-valued potentials
Graham Cox, Christopher K. R. T. Jones, Yuri Latushkin +1
We study the Schrödinger operator on a star-shaped domain in with Lipschitz boundary . The operator is equipped with quite general Dirich…
A symplectic perspective on constrained eigenvalue problems
Graham Cox, Jeremy L. Marzuola
The Maslov index is a powerful tool for computing spectra of selfadjoint, elliptic boundary value problems. This is done by counting intersections of a fixed Lagrangian subspace, w…
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…
Remarks on a scalar curvature rigidity theorem of Brendle and Marques
Graham Cox, Pengzi Miao, Luen-fai Tam
In this paper, we provide some remarks on the scalar curvature rigidity theorem of Brendle and Marques in \cite{BrendleMarques}. The main result is that Brendle and Marques' theore…
A local test for global extrema in the dispersion relation of a periodic graph
Gregory Berkolaiko, Yaiza Canzani, Graham Cox +1
We consider a family of periodic tight-binding models (combinatorial graphs) that have the minimal number of links between copies of the fundamental domain. For this family we esta…
Limiting Eigenfunctions of Sturm-Liouville operators Subject to a Spectral Flow
Thomas Beck, Isabel Bors, Grace Conte +2
We examine the spectrum of a family of Sturm--Liouville operators with regularly spaced delta function potentials parametrized by increasing strength. The limiting behavior of the…
A Morse index theorem for elliptic operators on bounded domains
Graham Cox, Christopher K. R. T. Jones, Jeremy L. Marzuola
Given a selfadjoint, elliptic operator , one would like to know how the spectrum changes as the spatial domain is deformed. For a family of domains $\{Î…
Stability of spectral partitions and the Dirichlet-to-Neumann map
Gregory Berkolaiko, Yaiza Canzani, Graham Cox +1
The oscillation of a Laplacian eigenfunction gives a great deal of information about the manifold on which it is defined. This oscillation can be encoded in the nodal deficiency, a…
Black hole evolutions: Lessons from bifurcation theory
Ivan Booth, Graham Cox, Chiamaka Mary Okpala
We explore the role of the stability operator in regulating the evolution of marginally outer trapped surfaces (MOTSs). In 2005, Andersson, Mars and Simon showed that if the stabil…
Unique continuation principles for finite-element discretizations of the Laplacian
Graham Cox, Scott MacLachlan, Luke Steeves
Unique continuation principles are fundamental properties of elliptic partial differential equations, giving conditions that guarantee that the solution to an elliptic equation mus…
Entropy and entanglement in a bipartite quasi-Hermitian system and its Hermitian counterparts
Abed Alsalam Abu Moise, Graham Cox, Marco Merkli
We consider a quantum oscillator coupled to a bath of other oscillators. The total system evolves with a quasi-Hermitian Hamiltonian. Associated to it is a family of Hermitian…
Computing nodal deficiency with a refined Dirichlet-to-Neumann map
Gregory Berkolaiko, Graham Cox, Bernard Helffer +1
Recent work of the authors and their collaborators has uncovered fundamental connections between the Dirichlet-to-Neumann map, the spectral flow of a certain family of self-adjoint…
Instability of pulses in gradient reaction-diffusion systems: A symplectic approach
Margaret Beck, Graham Cox, Christopher Jones +3
In a scalar reaction-diffusion equation, it is known that the stability of a steady state can be determined from the Maslov index, a topological invariant that counts the state's c…
Large-time uniqueness in a data assimilation problem for Burgers' equation
Graham Cox
There is currently a great deal of interest in the 4D-Var data assimilation scheme, in which one uses observational data to find the optimal initial condition for a differential eq…
Generalized Maslov indices for non-Hamiltonian systems
Thomas John Baird, Paul Cornwell, Graham Cox +2
We extend the definition of the Maslov index to a broad class of non-Hamiltonian dynamical systems. To do this, we introduce a family of topological spaces--which we call Maslov-Ar…
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…
The ponderomotive Lorentz force
Graham Cox, Mark Levi
This paper describes a curious phenomenon: a particle in a rapidly varying potential is subject to an effective magnetic-like force. This force is in addition to the well-known pon…