arXiv subjects

Christopher K. R. T. Jones

Publications and source records attributed to Christopher K. R. T. Jones.

23 records · Page 2Linked to original sources

arXiv2015

The Morse and Maslov indices for multidimensional Schrödinger operators with matrix-valued potentials

We study the Schrödinger operator $L=-Δ+V$ on a star-shaped domain $Ω$ in $\mathbb{R}^d$ with Lipschitz boundary $\partialΩ$. The operator is equipped with quite general Dirichlet- or Robin-type boundary conditions induced by operators between $H^{1/2}(\partialΩ)$ and $H^{-1/2}(\partialΩ)$, and the potential takes values in the set of symmetric $N\times N$ matrices. By shrinking the domain and rescaling the operator we obtain a path in the Fredholm-Lagrangian-Grassmannian of the subspace of $H^{1/2}(\partialΩ)\times H^{-1/2}(\partialΩ)$ corresponding to the given boundary condition. The path is formed by computing the Dirichlet and Neumann traces of weak solutions to the rescaled eigenvalue equation. We prove a formula relating the number of negative eigenvalues of $L$ (the Morse index), the signed crossings of the path (the Maslov index), the number of negative eigenvalues of the potential matrix evaluated at the center of the domain, and the number of negative squares of a bilinear form related to the boundary operator.

math.SP↗

arXiv2015

A Morse index theorem for elliptic operators on bounded domains

Given a selfadjoint, elliptic operator $L$, one would like to know how the spectrum changes as the spatial domain $Ω\subset \mathbb{R}^d$ is deformed. For a family of domains $\{Ω_t\}_{t\in[a,b]}$ we prove that the Morse index of $L$ on $Ω_a$ differs from the Morse index of $L$ on $Ω_b$ by the Maslov index of a path of Lagrangian subspaces on the boundary of $Ω$. This is particularly useful when $Ω_a$ is a domain for which the Morse index is known, e.g. a region with very small volume. Then the Maslov index computes the difference of Morse indices for the "original" problem (on $Ω_b$) and the "simplified" problem (on $Ω_a$). This generalizes previous multi-dimensional Morse index theorems that were only available on star-shaped domains or for Dirichlet boundary conditions. We also discuss how one can compute the Maslov index using crossing forms, and present some applications to the spectral theory of Dirichlet and Neumann boundary value problems.

math.AP↗

arXiv2013

Approximation of Random Slow Manifolds and Settling of Inertial Particles under Uncertainty

A method is provided for approximating random slow manifolds of a class of slow-fast stochastic dynamical systems. Thus approximate, low dimensional, reduced slow systems are obtained analytically in the case of sufficiently large time scale separation. To illustrate this dimension reduction procedure, the impact of random environmental fluctuations on the settling motion of inertial particles in a cellular flow field is examined. It is found that noise delays settling for some particles but enhances settling for others. A deterministic stable manifold is an agent to facilitate this phenomenon. Overall, noise appears to delay the settling in an averaged sense.

math.DS↗

arXiv2013

On the stability analysis of periodic sine-Gordon traveling waves

We study the spectral stability properties of periodic traveling waves in the sine-Gordon equation, including waves of both subluminal and superluminal propagation velocities as well as waves of both librational and rotational types. We prove that only subluminal rotational waves are spectrally stable and establish exponential instability in the other three cases. Our proof corrects a frequently cited one given by Scott.

math.AP↗

arXiv2009

Existence of multi-pulses of the regularized short-pulse and Ostrovsky equations

The existence of multi-pulse solutions near orbit-flip bifurcations of a primary single-humped pulse is shown in reversible, conservative, singularly perturbed vector fields. Similar to the non-singular case, the sign of a geometric condition that involves the first integral decides whether multi-pulses exist or not. The proof utilizes a combination of geometric singular perturbation theory and Lyapunov--Schmidt reduction through Lin's method. The motivation for considering orbit flips in singularly perturbed systems comes from the regularized short-pulse equation and the Ostrovsky equation, which both fit into this framework and are shown here to support multi-pulses.

nlin.PS↗