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David Borthwick

Publications and source records attributed to David Borthwick.

At least 19 recordsLinked to original sources
arXiv2023

Gaps between consecutive eigenvalues for compact metric graphs

On a compact metric graph, we consider the spectrum of the Laplacian defined with a mix of standard and Dirichlet vertex conditions. A Cheeger-type lower bound on the gap $λ_2 - λ_1$ is established, with a constant that depends only on the total length of the graph and minimum edge length. We also prove some improvements of known upper bounds for eigenvalue gaps and ratios for metric trees and extensions to certain other types of graphs.

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arXiv2022

The heat kernel on the diagonal for a compact metric graph

We analyze the heat kernel associated to the Laplacian on a compact metric graph, with standard Kirchoff-Neumann vertex conditions. An explicit formula for the heat kernel as a sum over loops, developed by Roth and Kostrykin, Potthoff, and Schrader, allows for a straightforward analysis of small-time asymptotics. We show that the restriction of the heat kernel to the diagonal satisfies a modified version of the heat equation. This observation leads to an "edge" heat trace formula, expressing the a sum over eigenfunction amplitudes on a single edge as a sum over closed loops containing that edge. The proof of this formula relies on a modified heat equation satisfied by the diagonal restriction of the heat kernel. Further study of this equation leads to explicit formulas for completely symmetric graphs.

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arXiv2022

Existence of resonances for Schrodinger operators on hyperbolic space

We prove existence results and lower bounds for the resonances of Schrödinger operators associated to smooth, compactly support potentials on hyperbolic space. The results are derived from a combination of heat and wave trace expansions and asymptotics of the scattering phase.

math.SP↗
arXiv2019

Sharp diameter bound on the spectral gap for quantum graphs

We establish an upper bound on the spectral gap for compact quantum graphs which depends only on the diameter and total number of vertices. This bound is asymptotically sharp for pumpkin chains with number of edges tending to infinity.

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arXiv2018

Existence and stability of Schrödinger solitons on noncompact manifolds

We consider the focusing nonlinear Schrödinger equation on a large class of rotationally symmetric, noncompact manifolds. We prove the existence of a solitary wave by perturbing off the flat Euclidean case. Furthermore, we study the stability of the solitary wave under radial perturbations by analyzing spectral properties of the associated linearized operator. Finally, in the L2-critical case, by considering the Vakhitov-Kolokolov criterion (see also results of Grillakis-Shatah-Strauss), we provide numerical evidence showing that the introduction of a nontrivial geometry destabilizes the solitary wave in a wide variety of cases, regardless of the curvature of the manifold. In particular, the parameters of the metric corresponding to standard hyperbolic space will lead to instability consistent with the blow-up results of Banica-Duyckaerts (2015). We also provide numerical evidence for geometries under which it would be possible for the Vakhitov-Kolokolov condition to suggest stability, provided certain spectral properties hold in these spaces

math-ph↗
arXiv2017

Improved fractal Weyl bounds for hyperbolic manifolds

We give a new fractal Weyl upper bound for resonances of convex co-compact hyperbolic manifolds in terms of the dimension $n$ of the manifold and the dimension $δ$ of its limit set. More precisely, we show that as $R\to\infty$, the number of resonances in the box $[R,R+1]+i[-β,0]$ is $O(R^{m(β,δ)+})$, where the exponent $m(β,δ)=\min(2δ+2β+1-n,δ)$ changes its behavior at $β=(n-1-δ)/2$. In the case $δ<(n-1)/2$, we also give an improved resolvent upper bound in the standard resonance free strip $\{\mathrm{Im}\ λ > δ-(n-1)/2\}$. Both results use the fractal uncertainty principle point of view recently introduced in [arXiv:1504.06589]. The appendix presents numerical evidence for the Weyl upper bound.

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arXiv2016

Symmetry reduction of holomorphic iterated function schemes and factorization of Selberg zeta functions

Given a holomorphic iterated function scheme with a finite symmetry group $G$, we show that the associated dynamical zeta function factorizes into symmetry-reduced analytic zeta functions that are parametrized by the unitary irreducible representations of $G$. We show that this factorization implies a factorization of the Selberg zeta function on symmetric $n$-funneled surfaces and that the symmetry factorization simplifies the numerical calculations of the resonances by several orders of magnitude. As an application this allows us to provide a detailed study of the spectral gap and we observe for the first time the existence of a macroscopic spectral gap on Schottky surfaces.

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arXiv2013

Distribution of resonances for hyperbolic surfaces

We study the distribution of resonances for geometrically finite hyperbolic surfaces of infinite area by countting resonances numerically. The resonances are computed as zeros of the Selberg zeta function, using an algorithm for computation of the zeta function for Schottky groups. Our particular focus is on three aspects of the resonance distribution that have attracted attention recently: the fractal Weyl law, the spectral gap, and the concentration of decay rates.

math.SP↗
arXiv2013

Upper bounds for the number of resonances on geometrically finite hyperbolic manifolds

On geometrically finite hyperbolic manifolds $Γ\backslash H^{d}$, including those with non-maximal rank cusps, we give upper bounds on the number $N(R)$ of resonances of the Laplacian in disks of size $R$ as $R\to \infty$. In particular, if the parabolic subgroups of $Γ$ satisfy a certain Diophantine condition, the bound is $N(R)= O(R^d (\log R)^{d+1})$.

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arXiv2013

Resonance asymptotics for Schrodinger operators on hyperbolic space

We study the asymptotic distribution of resonances for scattering by compactly supported potentials in hyperbolic space. We first establish an upper bound for the resonance counting function that depends only on the dimension and the support of the potential. We then establish the sharpness of this estimate by proving the a Weyl law for the resonance counting function holds in the case of radial potentials vanishing to some finite order at the edge of the support. As an application of the existence of potentials that saturate the upper bound, we derive additional resonance asymptotics that hold in a suitable generic sense. These generic results include asymptotics for the resonance count in sectors.

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arXiv2011

Did a 1-dimensional magnet detect a 248-dimensional Lie algebra?

About a year ago, a team of physicists reported in Science that they had observed "evidence for E8 symmetry" in the laboratory. This expository article is aimed at mathematicians and explains the chain of reasoning connecting measurements on a quasi-1-dimensional magnet with a 248-dimensional Lie algebra.

math-ph↗
arXiv2010

Sharp geometric upper bounds on resonances for surfaces with hyperbolic ends

We establish a sharp geometric constant for the upper bound on the resonance counting function for surfaces with hyperbolic ends. An arbitrary metric is allowed within some compact core, and the ends may be of hyperbolic planar, funnel, or cusp type. The constant in the upper bound depends only on the volume of the core and the length parameters associated to the funnel or hyperbolic planar ends. Our estimate is sharp in that it reproduces the exact asymptotic constant in the case of finite-area surfaces with hyperbolic cusp ends, and also in the case of funnel ends with Dirichlet boundary condtiions.

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arXiv2009

Sharp upper bounds on resonances for perturbations of hyperbolic space

For certain compactly supported metric and/or potential perturbations of the Laplacian on $\mathbb{H}^{n+1}$, we establish an upper bound on the resonance counting function with an explicit constant that depends only on the dimension, the radius of the unperturbed region in $\mathbb{H}^{n+1}$, and the volume of the metric perturbation. This constant is shown to be sharp in the case of scattering by a spherical obstacle.

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arXiv2008

Upper and lower bounds on resonances for manifolds hyperbolic near infinity

For a conformally compact manifold that is hyperbolic near infinity and of dimension $n+1$, we complete the proof of the optimal $O(r^{n+1})$ upper bound on the resonance counting function, correcting a mistake in the existing literature. In the case of a compactly supported perturbation of a hyperbolic manifold, we establish a Poisson formula expressing the regularized wave trace as a sum over scattering resonances. This leads to an $r^{n+1}$ lower bound on the counting function for scattering poles.

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arXiv2004

The semiclassical structure of low-energy states in the presence of a magnetic field

We consider a compact Riemannian manifold with a Hermitian line bundle whose curvature is non-degenerate. The Laplacian acting on high tensor powers (the semiclassical regime) of the bundle exhibits a cluster of low-energy states. We demonstrate that the orthogonal projectors onto these states are the Fourier components of an operator with the structure of the Szegö projector, i.e. a Fourier integral operator of Hermite type. This result yields semiclassical asymptotics for the low-energy eigenstates.

math.SP↗
arXiv2003

On the pseudospectra of Berezin-Toeplitz operators

We estimate the norm of the resolvent of non-selfadjoint Berezin Toeplitz operators in the semi-classical limit, under various assumptions on the Poisson bracket of the real and imaginary parts of the symbol. In case this bracket is negative, we symbolically construct pseudomodes well-localized in phase space. We also show that the numerical range converges to the convex hull of the image of the symbol. Our techniques involve the theory of Hermite distributions and classical subelliptic results of H\"ormander.

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