arXiv subjects

Andrew Hassell

Publications and source records attributed to Andrew Hassell.

At least 55 records · Page 3Linked to original sources

arXiv2011

Estimates on Neumann eigenfunctions at the boundary, and the "Method of Particular Solutions" for computing them

We consider the "Method of particular solutions" for numerically computing eigenvalues and eigenfunctions of the Laplacian $Δ$ on a smooth, bounded domain Omega in RR^n with either Dirichlet or Neumann boundary conditions. This method constructs approximate eigenvalues E, and approximate eigenfunctions u that satisfy $Δu=Eu$ in Omega, but not the exact boundary condition. An inclusion bound is then an estimate on the distance of E from the actual spectrum of the Laplacian, in terms of (boundary data of) u. We prove operator norm estimates on certain operators on $L^2(\partial Ω)$ constructed from the boundary values of the true eigenfunctions, and show that these estimates lead to sharp inclusion bounds in the sense that their scaling with $E$ is optimal. This is advantageous for the accurate computation of large eigenvalues. The Dirichlet case can be treated using elementary arguments and has appeared in SIAM J. Num. Anal. 49 (2011), 1046-1063, while the Neumann case seems to require much more sophisticated technology. We include preliminary numerical examples for the Neumann case.

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arXiv2011

Strichartz estimates without loss on manifolds with hyperbolic trapped geodesics

Doi proved that the $L^2_t H^{1/2}_x$ local smoothing effect for Schrödinger equation on a Riemannian manifold does not hold if the geodesic flow has one trapped trajectory. We show in contrast that Strichartz estimates and $L^1\to L^\infty$ dispersive estimates still hold without loss for $e^{itΔ}$ in various situations where the trapped set is hyperbolic and of sufficiently small fractal dimension.

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arXiv2010

Boundary quasi-orthogonality and sharp inclusion bounds for large Dirichlet eigenvalues

We study eigenfunctions and eigenvalues of the Dirichlet Laplacian on a bounded domain $Ω\subset\RR^n$ with piecewise smooth boundary. We bound the distance between an arbitrary parameter $E > 0$ and the spectrum $\{E_j \}$ in terms of the boundary $L^2$-norm of a normalized trial solution $u$ of the Helmholtz equation $(Δ+ E)u = 0$. We also bound the $L^2$-norm of the error of this trial solution from an eigenfunction. Both of these results are sharp up to constants, hold for all $E$ greater than a small constant, and improve upon the best-known bounds of Moler--Payne by a factor of the wavenumber $\sqrt{E}$. One application is to the solution of eigenvalue problems at high frequency, via, for example, the method of particular solutions. In the case of planar, strictly star-shaped domains we give an inclusion bound where the constant is also sharp. We give explicit constants in the theorems, and show a numerical example where an eigenvalue around the 2500th is computed to 14 digits of relative accuracy. The proof makes use of a new quasi-orthogonality property of the boundary normal derivatives of the eigenmodes, of interest in its own right.

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arXiv2010

Semiclassical L^p Estimates of Quasimodes on Curved Hypersurfaces

Let M be a compact manifold of dimension n, P = P(h) a semiclassical pseudodifferential operator on M, and u = u(h) an L^2 normalised family of functions such that Pu is O(h) in L^2(M) as h goes to 0. Let H be a compact submanifold of M. In a previous article, the second-named author proved estimates on the L^p norms, p > 2, of u restricted to H, under the assumption that the u are semiclassically localised and under some natural structural assumptions about the principal symbol of P. These estimates are of the form Ch^d(n;k;p) where k=dimH (except for a logarithmic divergence in the case k = n-2; p = 2). When H is a hypersurface, i.e. k = n-1, we have d(n;n-1;2)=1/4, which is sharp when M is the round n-sphere and H is an equator. In this article, we assume that H is a hypersurface, and make the additional geometric assumption that H is curved with respect to the bicharacteristic flow of P. Under this assumption we improve the estimate from d=1/4 to 1/6, generalising work of Burq-Gerard-Tzvetkov and Hu for Laplace eigenfunctions. To do this we apply the Melrose-Taylor theorem, as adapted by Pan and Sogge, for Fourier integral operators with folding canonical relations.

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arXiv2008

Ergodic billiards that are not quantum unique ergodic

Partially rectangular domains are compact two-dimensional Riemannian manifolds $X$, either closed or with boundary, that contain a flat rectangle or cylinder. In this paper we are interested in partially rectangular domains with ergodic billiard flow; examples are the Bunimovich stadium, the Sinai billiard or Donnelly surfaces. We consider a one-parameter family $X_t$ of such domains parametrized by the aspect ratio $t$ of their rectangular part. There is convincing theoretical and numerical evidence that the Laplacian on $X_t$ with Dirichlet or Neumann boundary conditions is not quantum unique ergodic (QUE). We prove that this is true for all $t \in [1,2]$ excluding, possibly, a set of Lebesgue measure zero. This yields the first examples of ergodic billiard systems proven to be non-QUE.

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arXiv2008

Eigenfunction concentration for polygonal billiards

In this note, we extend the results on eigenfunction concentration in billiards as proved by the third author in \cite{M1}. There, the methods developed in Burq-Zworski \cite{BZ3} to study eigenfunctions for billiards which have rectangular components were applied. Here we take an arbitrary polygonal billiard $B$ and show that eigenfunction mass cannot concentrate away from the vertices; in other words, given any neighbourhood $U$ of the vertices, there is a lower bound $$ \int_U |u|^2 \geq c \int_B |u|^2 $$ for some $c = c(U) > 0$ and any eigenfunction $u$.

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arXiv2008

Microlocal propagation near radial points and scattering for symbolic potentials of order zero

In this paper, the scattering and spectral theory of $H=Δ_g+V$ is developed, where $Δ_g$ is the Laplacian with respect to a scattering metric $g$ on a compact manifold $X$ with boundary and $V\in C^\infty(X)$ is real; this extends our earlier results in the two-dimensional case. Included in this class of operators are perturbations of the Laplacian on Euclidean space by potentials homogeneous of degree zero near infinity. Much of the particular structure of geometric scattering theory can be traced to the occurrence of radial points for the underlying classical system; a general framework for microlocal analysis at these points forms the main part of the paper.

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arXiv2007

Riesz transforms in one dimension

We study the boundedness on $L^p$ of the Riesz transform $\nabla L^{-1/2}$, where $L$ is one of several operators defined on $\R$ or $\R_+$, endowed with the measure $r^{d-1} dr$, $d > 1$, where $dr$ is Lebesgue measure. For integer $d$, this mimics the measure on Euclidean $d$-dimensional space, and in this case our setup is equivalent to looking at the Laplacian acting on radial functions on Euclidean space or variations of Euclidean space such as the exterior of a sphere (with either Dirichlet or Neumann boundary conditions), or the connected sum of two copies of $\R^d$. In this way we illuminate some recent results on the Riesz transform on asymptotically Euclidean manifolds. We are however interested in all real values of $d > 1$, and another goal of our analysis is to study the range of boundedness as a function of $d$; it is particularly interesting to see the behaviour as $d$ crosses 2. For example, in one of our cases which models radial functions on the connected sum of two copies of $\R^d$, the upper threshold for $L^p$ boundedness is $p=d$ for $d \ge 2$ and $p=d/(d-1)$ for $d < 2$. Only in the case $d=2$ is the Riesz transform actually bounded on $L^p$ when $p$ is equal to the upper threshold. We also study the Riesz transform when we have an inverse square potential, or a delta function potential; these cases provide a simple model for recent results of the first author and Guillarmou. Finally we look at the Hodge projector in a slightly more general setup.

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arXiv2007

Resolvent at low energy and Riesz transform for Schroedinger operators on asymptotically conic manifolds. II

Let $(M^\circ, g)$ be an asymptotically conic manifold, in the sense that $M^\circ$ compactifies to a manifold with boundary $M$ in such a way that $g$ becomes a scattering metric on $M$. A special case of particular interest is that of asymptotically Euclidean manifolds, where $\partial M = S^{n-1}$ and the induced metric at infinity is equal to the standard metric. We study the resolvent kernel $(P + k^2)^{-1}$ and Riesz transform of the operator $P = Δ_g + V$, where $Δ_g$ is the positive Laplacian associated to $g$ and $V$ is a real potential function $V$ that is smooth on $M$ and vanishes to some finite order at the boundary. In the first paper in this series we made the assumption that $n \geq 3$ and that $P$ has neither zero modes nor a zero-resonance and showed (i) that the resolvent kernel is conormal to the lifted diagonal and polyhomogeneous at the boundary on a blown up version of $M^2 \times [0, k_0]$, and (ii) the Riesz transform of $P$ is bounded on $L^p(M^\circ)$ for $1 < p < n$, and that this range is optimal unless $V \equiv 0$ and $M^\circ$ has only one end. In the present paper, we perform a similar analysis assuming again $n \geq 3$ but allowing zero modes and zero-resonances. We find the precise range of $p$ for which the Riesz transform (suitably defined) of $P$ is bounded on $L^p(M)$ when zero modes (but not resonances, which make the Riesz transform undefined) are present. Generically the Riesz transform is bounded for $p$ precisely in the range $(n/(n-2), n/3)$, with a bigger range possible if the zero modes have extra decay at infinity.

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arXiv2007

Resolvent at low energy and Riesz transform for Schrodinger operators on asymptotically conic manifolds, I

We analyze the resolvent $R(k)=(P+k^2)^{-1}$ of Schrödinger operators $P=Δ+V$ with short range potential $V$ on asymptotically conic manifolds $(M,g)$ (this setting includes asymptotically Euclidean manifolds) near $k=0$. We make the assumption that the dimension is greater or equal to 3 and that $P$ has no $L^2$ null space and no resonance at 0. In particular, we show that the Schwartz kernel of $R(k)$ is a conormal polyhomogeneous distribution on a desingularized version of $M\times M\times [0,1]$. Using this, we show that the Riesz transform of $P$ is bounded on $L^p$ for $1<p<n$ and that this range is optimal if $V$ is not identically zero or if $M$ has more than one end. We also analyze the case V=0 with one end. In a follow-up paper, we shall deal with the same problem in the presence of zero modes and zero-resonances.

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arXiv2006

The semiclassical resolvent and the propagator for nontrapping scattering metrics

Consider a compact manifold with boundary $M$ with a scattering metric $g$ or, equivalently, an asymptotically conic manifold $(M^\circ, g)$. (Euclidean $\mathbb{R}^n$, with a compactly supported metric perturbation, is an example of such a space.) Let $Δ$ be the positive Laplacian on $(M,g)$, and $V$ a smooth potential on $M$ which decays to second order at infinity. In this paper we construct the kernel of the operator $(h^2 Δ+ V - (λ_0 \pm i0)^2)^{-1}$, at a nontrapping energy $λ_0 > 0$, uniformly for $h \in (0, h_0)$, $h_0 > 0$ small, within a class of Legendre distributions on manifolds with codimension three corners. Using this we construct the kernel of the propagator, $e^{-it(Δ/2 + V)}$, $t \in (0, t_0)$ as a quadratic Legendre distribution. We also determine the global semiclassical structure of the spectral projector, Poisson operator and scattering matrix.

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arXiv2005

Eigenvalues of Schroedinger operators with potential asymptotically homogeneous of degree -2

We strengthen and generalise a result of Kirsch and Simon on the behaviour of the function $N_L(E)$, the number of bound states of the operator $L = Δ+V$ in $\R^d$ below $-E$. Here $V$ is a bounded potential behaving asymptotically like $P(ω)r^{-2}$ where $P$ is a function on the sphere. It is well known that the eigenvalues of such an operator are all nonpositive, and accumulate only at 0. If the operator $Δ_{S^{d-1}}+P$ on the sphere has negative eigenvalues $-μ_1,...,-μ_n$ less than $-(d-2)^2/4$, we prove that $N_L(E)$ may be estimated as \[ N_L(E)) = \frac{\log(E^{-1})}{2π}\sum_{i=1}^n \sqrt{μ_i-(d-2)^2/4} +O(1); \] thus, in particular, if there are no such negative eigenvalues then $L$ has a finite discrete spectrum. Moreover, under some additional assumptions including that $d=3$ and that there is exactly one eigenvalue $-μ_1$ less than -1/4, with all others $> -1/4$, we show that the negative spectrum is asymptotic to a geometric progression with ratio $\exp(-2π/\sqrt{μ_1 - \qtr})$.

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arXiv2005

Spreading of quasimodes in the Bunimovich stadium

We consider Dirichlet eigenfunctions $u_λ$ of the Bunimovich stadium $S$, satisfying $(Δ- λ^2) u_λ= 0$. Write $S = R \cup W$ where $R$ is the central rectangle and $W$ denotes the ``wings,'' i.e. the two semicircular regions. It is a topic of current interest in quantum theory to know whether eigenfunctions can concentrate in $R$ as $λ\to \infty$. We obtain a lower bound $C λ^{-2}$ on the $L^2$ mass of $u_λ$ in $W$, assuming that $u_λ$ itself is $L^2$-normalized; in other words, the $L^2$ norm of $u_λ$ is controlled by $λ^2$ times the $L^2$ norm in $W$. Moreover, if $u_λ$ is a $o(λ^{-2})$ quasimode, the same result holds, while for a $o(1)$ quasimode we prove that $L^2$ norm of $u_λ$ is controlled by $λ^4$ times the $L^2$ norm in $W$. We also show that the $L^2$ norm of $u_λ$ may be controlled by the integral of $w \abs{\partial_N u}^2$ along $\partial S \cap W$, where $w$ is a smooth factor on $W$ vanishing at $R \cap W$. These results complement recent work of Burq-Zworski which shows that the $L^2$ norm of $u_λ$ is controlled by the $L^2$ norm in any pair of strips contained in $R$, but adjacent to $W$.

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arXiv2004

Riesz transform and $L^p$ cohomology for manifolds with Euclidean ends

Let $M$ be a smooth Riemannian manifold which is the union of a compact part and a finite number of Euclidean ends, $\RR^n \setminus B(0,R)$ for some $R > 0$, each of which carries the standard metric. Our main result is that the Riesz transform on $M$ is bounded from $L^p(M) \to L^p(M; T^*M)$ for $1 < p < n$ and unbounded for $p \geq n$ if there is more than one end. It follows from known results that in such a case the Riesz transform on $M$ is bounded for $1 < p \leq 2$ and unbounded for $p > n$; the result is new for $2 < p \leq n$. We also give some heat kernel estimates on such manifolds. We then consider the implications of boundedness of the Riesz transform in $L^p$ for some $p > 2$ for a more general class of manifolds. Assume that $M$ is a $n$-dimensional complete manifold satisfying the Nash inequality and with an $O(r^n)$ upper bound on the volume growth of geodesic balls. We show that boundedness of the Riesz transform on $L^p$ for some $p > 2$ implies a Hodge-de Rham interpretation of the $L^p$ cohomology in degree 1, and that the map from $L^2$ to $L^p$ cohomology in this degree is injective.

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arXiv2004

Sharp Strichartz estimates on non-trapping asymptotically conic manifolds

We obtain the Strichartz inequalities $$ \| u \|_{L^q_t L^r_x([0,1] \times M)} \leq C \| u(0) \|_{L^2(M)}$$ for any smooth $n$-dimensional Riemannian manifold $M$ which is asymptotically conic at infinity (with either short-range or long-range metric perturbation) and non-trapping, where $u$ is a solution to the Schrödinger equation $iu_t + {1/2} Δ_M u = 0$, and $2 < q, r \leq \infty$ are admissible Strichartz exponents ($\frac{2}{q} + \frac{n}{r} = \frac{n}{2}$). This corresponds with the estimates available for Euclidean space (except for the endpoint $(q,r) = (2, \frac{2n}{n-2})$ when $n > 2$). These estimates imply existence theorems for semi-linear Schrödinger equations on $M$, by adapting arguments from Cazenave and Weissler \cite{cwI} and Kato \cite{kato}. This result improves on our previous result in \cite{HTW}, which was an $L^4_{t,x}$ Strichartz estimate in three dimensions. It is closely related to the results of Staffilani-Tataru, Burq, Tataru, and Robbiano-Zuily, who consider the case of asymptotically flat manifolds.

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arXiv2004

A Strichartz inequality for the Schroedinger equation on non-trapping asymptotically conic manifolds

We obtain an $L^4$ space-time Strichartz inequality for any smooth three-dimensional Riemannian manifold $(M,g)$ which is asymptotically conic at infinity and non-trapping, where $u$ is a solution to the Schrödinger equation $iu_t + {1/2} Δ_M u = 0$. The exponent $H^{1/4}(M)$ is sharp, by scaling considerations. In particular our result covers asymptotically flat non-trapping manifolds. Our argument is based on the interaction Morawetz inequality introduced by Colliander et al., interpreted here as a positive commutator inequality for the tensor product $U(t,z',z'') := u(t,z') u(t,z'')$ of the solution with itself. We also use smoothing estimates for Schrödinger solutions including a new one proved here with weight $r^{-1}$ at infinity and with the gradient term involving only one angular derivative.

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arXiv2003

Quantum ergodicity of boundary values of eigenfunctions

Suppose that Omega is a bounded, piecewise smooth Euclidean domain. We prove that the boundary values (Cauchy data) of eigenfunctions of the Laplacian on Omega with various boundary conditions are quantum ergodic if the classical billiard map (on the ball bundle of the boundary of Omega) is ergodic. Our proof is based on the classical observation that the boundary value of an interior eigenfunction is an eigenfunction of an operator F_h on the boundary of Omega, where h^{-2} is equal to the eigenvalue. In the case of the Neumann boundary condition, F_h is the boundary integral operator induced by the double layer potential. We show that F_h is a semiclassical Fourier integral operator quantizing the billiard map plus a `small' remainder. We then use the quantum dyanmics defined by F_h at the boundary to prove the result, much as the quantum dynamics generated by the wave group were exploited in the work of Zelditch-Zworski (1996). Novelties include the facts that F_h is not unitary and (consequently) the boundary values are equidistributed by measures which are not invariant under the billiard map and which depend on the boundary conditions.

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arXiv2003

On the structure of the Schrödinger propagator

We discuss the form of the propagator $U(t)$ for the time-dependent Schrödinger equation on an asyptotically Euclidean, or, more generally, asymptotically conic, manifold with no trapped geodesics. In the asymptotically Euclidean case, if $χ\in \mathcal{C}_0^\infty$, and with $\mathcal{F}$ denoting Fourier transform, $\mathcal{F}\circ e^{-ir^2/2t} U(t) χ$ is a Fourier integral operator for $t\neq 0.$ The canonical relation of this operator is a ``sojourn relation'' associated to the long-time geodesic flow. This description of the propagator follows from its more precise characterization as a ``scattering fibered Legendrian,'' given by the authors in a previous paper and sketched here. A corollary is a propagation of singularities theorem that permits a complete description of the wavefront set of a solution to the Schrödinger equation, restricted to any fixed nonzero time, in terms of the oscillatory behavior of its initial data. We discuss two examples which illustrate some extremes of this propagation behavior.

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