arXiv subjects

Andrew Hassell

Publications and source records attributed to Andrew Hassell.

At least 37 records · Page 2Linked to original sources

arXiv2016

Comparable upper and lower bounds for boundary values of Neumann eigenfunctions and tight inclusion of eigenvalues

For smooth bounded domains in $\mathbb{R}$, we prove upper and lower $L^2$ bounds on the boundary data of Neumann eigenfunctions, and prove quasi-orthogonality of this boundary data in a spectral window. The bounds are tight in the sense that both are independent of eigenvalue; this is achieved by working with an appropriate norm for boundary functions, which includes a `spectral weight', that is, a function of the boundary Laplacian. This spectral weight is chosen to cancel concentration at the boundary that can happen for `whispering gallery' type eigenfunctions. These bounds are closely related to wave equation estimates due to Tataru. Using this, we bound the distance from an arbitrary Helmholtz parameter $E>0$ to the nearest Neumann eigenvalue, in terms of boundary normal-derivative data of a trial function $u$ solving the Helmholtz equation $(Δ-E)u=0$. This `inclusion bound' improves over previously known bounds by a factor of $E^{5/6}$. It is analogous to a recently improved inclusion bound in the Dirichlet case, due to the first two authors. Finally, we apply our theory to present an improved numerical implementation of the method of particular solutions for computation of Neumann eigenpairs on smooth planar domains. We show that the new inclusion bound improves the relative accuracy in a computed Neumann eigenvalue (around the $42000$th) from 9 digits to 14 digits, with little extra effort.

math.AP↗

arXiv2015

Completeness of boundary traces of eigenfunctions

In this paper, we study the boundary traces of eigenfunctions on the boundary of a smooth and bounded domain. An identity derived by Bäcker, Fürstburger, Schubert, and Steiner, expressing (in some sense) the asymptotic completeness of the set of boundary traces in a frequency window of size O(1), is proved both for Dirichlet and Neumann boundary conditions. We then prove a semiclassical generalization of this identity.

math.AP↗

arXiv2015

The distribution of phase shifts for semiclassical potentials with polynomial decay

This is the third paper in a series analyzing the asymptotic distribution of the phase shifts in the semiclassical limit. We analyze the distribution of phase shifts, or equivalently, eigenvalues of the scattering matrix, $S_h(E)$, for semiclassical Schrödinger operators on $\mathbb{R}^d$ which are perturbations of the free Hamiltonian by a potential $V$ with polynomial decay. Our assumption is that $V(x) \sim |x|^{-α} v(\hat x)$ as $x \to \infty$, for some $α> d$, with corresponding derivative estimates. In the semiclassical limit $h \to 0$, we show that the atomic measure on the unit circle defined by these eigenvalues, after suitable scaling in $h$, tends to a measure $μ$ on $\mathbb{S}^1$. Moreover, $μ$ is the pushforward from $\mathbb{R}$ to $\mathbb{R} / 2 π\mathbb{Z} = \mathbb{S}^1$ of a homogeneous distribution $ν$ of order $β$ depending on the dimension $d$ and the rate of decay $α$ of the potential function. As a corollary we obtain an asymptotic formula for the accumulation of phase shifts in a sector of $\mathbb{S}^1$. The proof relies on an extension of results of the second author and Wunsch on the classical Hamiltonian dynamics and semiclassical Poisson operator to the class of potentials under consideration here.

math.AP↗

arXiv2015

Resolvent and spectral measure on non-trapping asymptotically hyperbolic manifolds II: spectral measure, restriction theorem, spectral multiplier

We consider the Laplacian $Δ$ on an asymptotically hyperbolic manifold $X$, as defined by Mazzeo and Melrose. We give pointwise bounds on the spectral measure for the operator $(Δ- n^2/4)_+^{1/2}$ on such manifolds, under the assumptions that $X$ is nontrapping and there is no resonance at the bottom of the spectrum. This uses the construction of the resolvent given by Mazzeo and Melrose (valid when the spectral parameter lies in a compact set), Melrose, Sá Barreto and Vasy (high energy estimates for a perturbation of the hyperbolic metric) and the present authors (see also the work by Yiran Wang) in the general high-energy case. Under the additional assumption that $X$ has negative curvature everywhere, we give two applications of the spectral measure estimates. The first, following work due to Guillarmou and Sikora with the second author in the asymptotically conic case, is a restriction theorem, that is, a $L^p(X) \to L^{p'}(X)$ operator norm bound on the spectral measure. The second is a spectral multiplier result, that is, a bound on functions $F((Δ- n^2/4)_+^{1/2})$ of the square root of the Laplacian, in terms of norms of the function $F$. Compared to the asymptotically conic case, our spectral multiplier result is weaker, but the restriction estimate is stronger. In both cases, the difference can be traced to the exponential volume growth at infinity for asymptotically hyperbolic manifolds, as opposed to polynomial growth in the asymptotically conic setting. The pointwise bounds on the spectral measure established here will also be applied to Strichartz estimates in another work by the first author.

math.AP↗

arXiv2015

Spectral asymptotics for the semiclassical Dirichlet to Neumann operator

Let $M$ be a compact Riemannian manifold with smooth boundary, and let $R(λ)$ be the Dirichlet-to-Neumann operator at frequency $λ$. We obtain a leading asymptotic for the spectral counting function for $λ^{-1}R(λ)$ in an interval $[a_1, a_2)$ as $λ\to \infty$, under the assumption that the measure of periodic billiards on $T^*M$ is zero. The asymptotic takes the form \begin{equation*} N(λ; a_1,a_2) = \bigl(κ(a_2)-κ(a_1)\bigr)\mathsf{vol}'(\partial M) λ^{d-1}+o(λ^{d-1}), \end{equation*} where $κ(a)$ is given explicitly by \begin{equation*} κ(a) = \frac{ω_{d-1}}{(2π)^{d-1}} \biggl( -\frac{1}{2π} \int_{-1}^1 (1 - η^2)^{(d-1)/2} \frac{a}{a^2 + η^2} \, dη- \frac{1}{4} + H(a) (1+a^2)^{(d-1)/2} \biggr) \end{equation*} with the Heavyside function $H(a)$.

math.SP↗

arXiv2015

Resolvent and spectral measure on non-trapping asymptotically hyperbolic manifolds I: Resolvent construction at high energy

This is the first in a series of papers in which we investigate the resolvent and spectral measure on non-trapping asymptotically hyperbolic manifolds with applications to the restriction theorem, spectral multiplier results and Strichartz estimates. In this first paper, we use semiclassical Lagrangian distributions and semiclassical intersecting Lagrangian distributions, along with Mazzeo-Melrose 0-calculus, to construct the high energy resolvent on general non- trapping asymptotically hyperbolic manifolds, generalizing the work due to Melrose, Sa Barreto and Vasy. We note that there is an independent work by Y. Wang which also constructs the high-energy resolvent.

math.AP↗

arXiv2015

Wave propagation on Euclidean surfaces with conical singularities. I: Geometric diffraction

We investigate the singularities of the trace of the half-wave group, $\mathrm{Tr} \, e^{-it\sqrtΔ}$, on Euclidean surfaces with conical singularities $(X,g)$. We compute the leading-order singularity associated to periodic orbits with successive degenerate diffractions. This result extends the previous work of the third author \cite{Hil} and the two-dimensional case of the work of the first author and Wunsch \cite{ForWun} as well as the seminal result of Duistermaat and Guillemin \cite{DuiGui} in the smooth setting. As an intermediate step, we identify the wave propagators on $X$ as singular Fourier integral operators associated to intersecting Lagrangian submanifolds, originally developed by Melrose and Uhlmann \cite{MelUhl}.

math.AP↗

arXiv2015

Global-in-time Strichartz estimates on non-trapping asymptotically conic manifolds

We prove global-in-time Strichartz estimates without loss of derivatives for the solution of the Schroedinger equation on a class of non-trapping asymptotically conic manifolds. We obtain estimates for the full set of admissible indices, including the endpoint, in both the homogeneous and inhomogeneous cases. This result improves on the results in earlier papers of Hassell-Tao-Wunsch and Mizutani which were local in time.

math.AP↗

arXiv2014

Approximation and equidistribution of phase shifts: spherical symmetry

Consider a semiclassical Hamiltonian \begin{equation*} H_{V, h} := h^{2} Δ+ V - E \end{equation*} where $h > 0$ is a semiclassical parameter, $Δ$ is the positive Laplacian on $\mathbb{R}^{d}$, $V$ is a smooth, compactly supported central potential function and $E > 0$ is an energy level. In this setting the scattering matrix $S_h(E)$ is a unitary operator on $L^2(\mathbb{S}^{d-1})$, hence with spectrum lying on the unit circle; moreover, the spectrum is discrete except at $1$. We show under certain additional assumptions on the potential that the eigenvalues of $S_h(E)$ can be divided into two classes: a finite number $\sim c_d (R\sqrt{E}/h)^{d-1} $, as $h \to 0$, where $B(0, R)$ is the convex hull of the support of the potential, that equidistribute around the unit circle, and the remainder that are all very close to $1$. Semiclassically, these are related to the rays that meet the support of, and hence are scattered by, the potential, and those that do not meet the support of the potential, respectively. A similar property is shown for the obstacle problem in the case that the obstacle is the ball of radius $R$.

math.AP↗

arXiv2013

Equidistribution of phase shifts in semiclassical potential scattering

Consider a semiclassical Hamiltonian $H := h^{2} Δ+ V - E$ where $Δ$ is the positive Laplacian on $\mathbb{R}^{d}$, $V \in C^{\infty}_{0}(\mathbb{R}^{d})$ and $E > 0$ is an energy level. We prove that under an appropriate dynamical hypothesis on the Hamilton flow corresponding to $H$, the eigenvalues of the scattering matrix $S_{h}(V)$ define a measure on $\mathbb{S}^{1}$ that converges to Lebesgue measure away from $1 \in \mathbb{S}^{1}$ as $h \to 0$.

math.SP↗

arXiv2013

Exterior mass estimates and $L^2$ restriction bounds for Neumann data along hypersurfaces

We study the problem of estimating the $L^2$ norm of Laplace eigenfunctions on a compact Riemannian manifold $M$ when restricted to a hypersurface $H$. We prove mass estimates for the restrictions of eigenfunctions $ϕ_h$, $(h^2 Δ- 1)ϕ_h = 0$, to $H$ in the region exterior to the coball bundle of $H$, on $h^δ$-scales ($0\leq δ< 2/3$). We use this estimate to obtain an $O(1)$ $L^2$-restriction bound for the Neumann data along $H.$ The estimate also applies to eigenfunctions of semiclassical Schrödinger operators.

math.AP↗

arXiv2013

Improvement of eigenfunction estimates on manifolds of nonpositive curvature

Let $(M,g)$ be a compact, boundaryless manifold of dimension $n$ with the property that either (i) $n=2$ and $(M,g)$ has no conjugate points, or (ii) the sectional curvatures of $(M,g)$ are nonpositive. Let $Δ$ be the positive Laplacian on $M$ determined by $g$. We study the $L^{2}\to{}L^{p}$ mapping properties of a spectral cluster of $\sqrtΔ$ of width $1/\logλ$. Under the geometric assumptions above, \cite{berard77} Bérard obtained a logarithmic improvement for the remainder term of the eigenvalue counting function which directly leads to a $(\logλ)^{1/2}$ improvement for Hörmander's estimate on the $L^{\infty}$ norms of eigenfunctions. In this paper we extend this improvement to the $L^p$ estimates for all $p>\frac{2(n+1)}{n-1}$.

math.AP↗

arXiv2012

The Riesz transform for homogeneous Schrödinger operators on metric cones

We consider Schroedinger operators on metric cones whose cross section is a closed Riemannian manifold $(Y, h)$ of dimension $d-1 \geq 2$. Thus the metric on the cone $M = (0, \infty)_r \times Y$ is $dr^2 + r^2 h$. Let $Δ$ be the Friedrichs Laplacian on $M$ and $V_0$ be a smooth function on $Y$, such that $Δ_Y + V_0 + (d-2)^2/4$ is a strictly positive operator on $L^2(Y)$, with lowest eigenvalue $μ^2_0 $ and second lowest eigenvalue $μ^2_1$, with $μ_0, μ_1 > 0$. The operator we consider is $H = Δ+ V_0/r^2$, a Schrödinger operator with inverse square potential on $M$; notice that $H$ is homogeneous of degree -2. We study the Riesz transform $T = \nabla H^{-1/2}$ and determine the precise range of $p$ for which $T$ is bounded on $L^p(M)$. This is achieved by making a precise analysis of the operator $(H + 1)^{-1}$ and determining the complete asymptotics of its integral kernel. We prove that if $V$ is not identically zero, then the range of $p$ for $L^p$ boundedness is $$ d/ \Big(min(1+d/2+μ_0, d \Big) < p < d / \Big(max(d/2-μ_0, 0) \Big),$$ while if $V$ is identically zero, then the range is $$ 1 < p < d / \Big(max(d/2-μ_1, 0 \Big).$$ The result in the case $V$ identically zero was first obtained in a paper by H.-Q. Li.

math.AP↗

arXiv2012

Uniform Sobolev estimates for non-trapping metrics

We prove uniform Sobolev estimates $||u||_{L^{p'}} \leq C ||(Δ-α)u||_{L^{p}}$, where $p=2n/(n+2), p'=2n/(n-2)$, for the Laplacian $Δ$ on non-trapping asymptotically conic manifolds of dimension $n$. Here C is independent of $α$ which ranges over all complex numbers. This generalizes to non-constant coefficient Laplacians a result of Kenig-Ruiz-Sogge.

math.AP↗

arXiv2012

Resolvent at low energy III: the spectral measure

Let $M^\circ$ be a complete noncompact manifold and $g$ an asymptotically conic Riemaniann metric on $M^\circ$, in the sense that $M^\circ$ compactifies to a manifold with boundary $M$ in such a way that $g$ becomes a scattering metric on $M$. Let $Δ$ be the positive Laplacian associated to $g$, and $P = Δ+ V$, where $V$ is a potential function obeying certain conditions. We analyze the asymptotics of the spectral measure $dE(λ) = (λ/πi) \big(R(λ+i0) - R(λ- i0) \big)$ of $P_+^{1/2}$, where $R(λ) = (P - λ^2)^{-1}$, as $λ\to 0$, in a manner similar to that done previously by the second author and Vasy, and by the first two authors. The main result is that the spectral measure has a simple, `conormal-Legendrian' singularity structure on a space which is obtained from $M^2 \times [0, λ_0)$ by blowing up a certain number of boundary faces. We use this to deduce results about the asymptotics of the wave solution operators $\cos(t \sqrt{P_+})$ and $\sin(t \sqrt{P_+})/\sqrt{P_+}$, and the Schrödinger propagator $e^{itP}$, as $t \to \infty$. In particular, we prove the analogue of Price's law for odd-dimensional asymptotically conic manifolds. This result on the spectral measure has been used in a follow-up work by the authors (arXiv:1012.3780) to prove sharp restriction and spectral multiplier theorems on asymptotically conic manifolds.

math.AP↗

arXiv2012

Restriction and spectral multiplier theorems on asymptotically conic manifolds

The classical Stein-Tomas restriction theorem is equivalent to the statement that the spectral measure $dE(λ)$ of the square root of the Laplacian on $\RR^n$ is bounded from $L^p(\RR^n)$ to $L^{p'}(\RR^n)$ for $1 \leq p \leq 2(n+1)/(n+3)$, where $p'$ is the conjugate exponent to $p$, with operator norm scaling as $λ^{n(1/p - 1/p') - 1}$. We prove a geometric generalization in which the Laplacian on $\RR^n$ is replaced by the Laplacian, plus suitable potential, on a nontrapping asymptotically conic manifold, which is the first time such a result has been proven in the variable coefficient setting. It is closely related to, but stronger than, Sogge's discrete $L^2$ restriction theorem, which is an $O(λ^{n(1/p - 1/p') - 1})$ estimate on the $L^p \to L^{p'}$ operator norm of the spectral projection for a spectral window of fixed length. From this, we deduce spectral multiplier estimates for these operators, including Bochner-Riesz summability results, which are sharp for $p$ in the range above.

math.AP↗

arXiv2012

Potential scattering and the continuity of phase-shifts

Let $S(k)$ be the scattering matrix for a Schrödinger operator (Laplacian plus potential) on $\RR^n$ with compactly supported smooth potential. It is well known that $S(k)$ is unitary and that the spectrum of $S(k)$ accumulates on the unit circle only at 1; moreover, $S(k)$ depends analytically on $k$ and therefore its eigenvalues depend analytically on $k$ provided the values stay away from 1. We give examples of smooth, compactly supported potentials on $\RR^n$ for which (i) the scattering matrix $S(k)$ does not have 1 as an eigenvalue for any $k > 0$, and (ii) there exists $k_0 > 0$ such that there is an analytic eigenvalue branch $e^{2iδ(k)}$ of S(k)$ converging to 1 as $k \downarrow k_0$. This shows that the eigenvalues of the scattering matrix, as a function of $k$, do not necessarily have continuous extensions to or across the value 1. In particular this shows that a `micro-Levinson theorem' for non-central potentials in $\RR^3$ claimed in a 1989 paper of R. Newton is incorrect.

math.SP↗

arXiv2011

Fast computation of high frequency Dirichlet eigenmodes via the spectral flow of the interior Neumann-to-Dirichlet map

We present a new algorithm for numerical computation of large eigenvalues and associated eigenfunctions of the Dirichlet Laplacian in a smooth, star-shaped domain in $\mathbb{R}^d$, $d\ge 2$. Conventional boundary-based methods require a root-search in eigenfrequency $k$, hence take $O(N^3)$ effort per eigenpair found, using dense linear algebra, where $N=O(k^{d-1})$ is the number of unknowns required to discretize the boundary. Our method is O(N) faster, achieved by linearizing with respect to $k$ the spectrum of a weighted interior Neumann-to-Dirichlet (NtD) operator for the Helmholtz equation. Approximations $\hat{k}_j$ to the square-roots $k_j$ of all O(N) eigenvalues lying in $[k - ε, k]$, where $ε=O(1)$, are found with $O(N^3)$ effort. We prove an error estimate $$ |\hat k_j - k_j| \leq C \Big(\frac{ε^2}{k} + ε^3 \Big), $$ with $C$ independent of $k$. We present a higher-order variant with eigenvalue error scaling empirically as $O(ε^5)$ and eigenfunction error as $O(ε^3)$, the former improving upon the 'scaling method' of Vergini--Saraceno. For planar domains ($d=2$), with an assumption of absence of spectral concentration, we also prove rigorous error bounds that are close to those numerically observed. For $d=2$ we compute robustly the spectrum of the NtD operator via potential theory, Nyström discretization, and the Cayley transform. At high frequencies (400 wavelengths across), with eigenfrequency relative error $10^{-10}$, we show that the method is $10^3$ times faster than standard ones based upon a root-search.

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