arXiv subjects

Blake Keeler

Publications and source records attributed to Blake Keeler.

6 recordsLinked to original sources

arXiv2024

Pointwise Weyl Laws for Quantum Completely Integrable Systems

The study of the asymptotics of the spectral function for self-adjoint, elliptic differential, or more generally pseudodifferential, operators on a compact manifold has a long history. The seminal 1968 paper of H\"ormander, following important prior contributions by G\"arding, Levitan, Avakumovi\'c, and Agmon-Kannai (to name only some), obtained pointwise asymptotics (or a "pointwise Weyl law") for a single elliptic, self-adjoint operator. Here, we establish a microlocalized pointwise Weyl law for the joint spectral functions of quantum completely integrable (QCI) systems, $\overline{P}=(P_1,P_2,\dots, P_n)$, where $P_i$ are first-order, classical, self-adjoint, pseudodifferential operators on a compact manifold $M^n$, with $\sum P_i^2$ elliptic and $[P_i,P_j]=0$ for $1\leq i,j\leq n$. A particularly important case is when $(M,g)$ is Riemannian and $P_1=(-\Delta)^\frac12$. We illustrate our result with several examples, including surfaces of revolution.

math.AP↗

arXiv2022

Asymptotics for the spectral function on Zoll manifolds

Let $(M,g)$ be a Zoll manifold, i.e., a smooth, compact, Riemannian manifold without boundary all of whose geodesics are closed with a minimal common period $T$. The positive definite Laplace-Beltrami operator has eigenvalues $\{\lambda_j^2\}_j$ which cluster around $\nu^2_\ell$ for some sequence $\nu_\ell\to \infty$. This article is concerned with the number of $\lambda_j$ in a window of fixed size $\mathrm{w}$ around $\nu_\ell$, denoted by $\mathbf{N}(\nu_\ell,\mathrm{w}):=\#\{j\,:\, \lambda_j\in[\nu_\ell-\mathrm{w},\nu_\ell+\mathrm{w}]\}.$ When the set of trajectories with period smaller than $T$ has zero measure, there is $c_{n}>0$, depending only on $n=\operatorname{dim} M$, such that $$ \mathbf{N}(\nu_\ell,\mathrm{w}) =c_n\operatorname{vol}_g(M)\nu_{\ell}^{n-1}+o(\nu_{\ell}^{n-1}), $$ as $\ell \to \infty$. However, for a general Zoll manifold this may not be the case. We show that, nevertheless, there is $N>0$, independent of $\ell$, such that $$ \sum_{j=0}^{N-1}\mathbf{N}(\nu_{\ell+j},\mathrm{w})= c_nN\operatorname{vol}_g(M)\nu_{\ell}^{n-1}+o(\nu_{\ell}^{n-1}), $$ as $\ell \to \infty$. In addition to asymptotics for the counting function, we study the kernel of the spectral projector for the Laplacian, $\Pi_{\ell,\mathrm{w}}(x,y)$ onto the spectrum in ${\bigcup_{j=0}^{N-1}[\nu_{\ell+j}-\mathrm{w},\nu_{\ell+j}+\mathrm{w}]}$. We show that for $x$ and $y$ in a shrinking neighborhood of a point with few loops of length smaller than $T$, $\Pi_{\ell,\mathrm{w}}(x,y)$ and its derivatives have the same asymptotics as those on the round sphere and flat torus.

math.AP↗

arXiv2022

Sharp exponential decay rates for anisotropically damped waves

In this article, we study energy decay of the damped wave equation on compact Riemannian manifolds where the damping coefficient is anisotropic and modeled by a pseudodifferential operator of order zero. We prove that the energy of solutions decays at an exponential rate if and only if the damping coefficient satisfies an anisotropic analogue of the classical geometric control condition, along with a unique continuation hypothesis. Furthermore, we compute an explicit formula for the optimal decay rate in terms of the spectral abscissa and the long-time averages of the principal symbol of the damping over geodesics, in analogy to the work of Lebeau for the isotropic case. We also construct genuinely anisotropic dampings which satisfy our hypotheses on the flat torus.

math.AP↗

arXiv2022

A logarithmic improvement in the two-point Weyl law for manifolds without conjugate points

In this paper, we study the two-point Weyl Law for the Laplace-Beltrami operator on a smooth, compact Riemannian manifold $M$ with no conjugate points. That is, we find the asymptotic behavior of the Schwartz kernel, $E_λ(x,y)$, of the projection operator from $L^2(M)$ onto the direct sum of eigenspaces with eigenvalue smaller than $λ^2$ as $λ\to\infty$. In the regime where $x,y$ are restricted to a compact neighborhood of the diagonal in $M\times M$, we obtain a uniform logarithmic improvement in the remainder of the asymptotic expansion for $E_λ$ and its derivatives of all orders, which generalizes a result of Bérard, who treated the on-diagonal case $E_λ(x,x)$. When $x,y$ avoid a compact neighborhood of the diagonal, we obtain this same improvement in an upper bound for $E_λ$. Our results imply that the rescaled covariance kernel of a monochromatic random wave locally converges in the $C^\infty$ topology to a universal scaling limit at an inverse logarithmic rate.

math.AP↗

arXiv2022

Pointwise dispersive estimates for Schrödinger operators on product cones

We investigate the dispersive properties of solutions to the Schrödinger equation with a weakly decaying radial potential on cones. If the potential has sufficient polynomial decay at infinity, then we show that the Schrödinger flow on each eigenspace of the link manifold satisfies a weighted $L^1\to L^\infty$ dispersive estimate. In odd dimensions, the decay rate we compute is consistent with that of the Schrödinger equation in a Euclidean space of the same dimension, but the spatial weights reflect the more complicated regularity issues in frequency that we face in the form of the spectral measure. In even dimensions, we prove a similar estimate, but with a loss of $t^{1/2}$ compared to the sharp Euclidean estimate.

math.AP↗

arXiv2016

Pseudospectra of Matrix Pencils for Transient Analysis of Differential-Algebraic Equations

To understand the solution of a linear, time-invariant differential-algebraic equation, one must analyze a matrix pencil (A,E) with singular E. Even when this pencil is stable (all its finite eigenvalues fall in the left-half plane), the solution can exhibit transient growth before its inevitable decay. When the equation results from the linearization of a nonlinear system, this transient growth gives a mechanism that can promote nonlinear instability. One might hope to enrich the conventional large-scale eigenvalue calculation used for linear stability analysis to signal the potential for such transient growth. Toward this end, we introduce a new definition of the pseudospectrum of a matrix pencil, use it to bound transient growth, explain how to incorporate a physically-relevant norm, and derive approximate pseudospectra using the invariant subspace computed in conventional linear stability analysis. We apply these tools to several canonical test problems in fluid mechanics, an important source of differential-algebraic equations.

math.NA↗