arXiv subjects

Andrew Hassell

Publications and source records attributed to Andrew Hassell.

61 records · Page 4Linked to original sources

arXiv2003

The Schroedinger propagator for scattering metrics

Let g be a scattering metric on a compact manifold X with boundary, i.e., a smooth metric giving the interior of X the structure of a complete Riemannian manifold with asymptotically conic ends. An example is any compactly supported perturbation of the standard metric on Euclidean space. Consider the operator $H = \half Δ+ V$, where $Δ$ is the positive Laplacian with respect to g and V is a smooth real-valued function on X vanishing to second order at the boundary. Assuming that g is non-trapping, we construct a global parametrix for the kernel of the Schroedinger propagator $U(t) = e^{-itH}$ and use this to show that the kernel of U(t) is, up to an explicit quadratic oscillatory factor, a class of `Legendre distributions' on $X \times X^{\circ} \times \halfline$ previously considered by Hassell-Vasy. When the metric is trapping, then the parametrix construction goes through microlocally in the non-trapping part of the phase space. We apply this result to obtain a microlocal characterization of the singularities of $U(t) f$, for any tempered distribution $f$ and any fixed $t \neq 0$, in terms of the oscillation of f near the boundary of X. If the metric is non-trapping, then we obtain a complete characterization; more generally we need to assume that f is microsupported in the nontrapping part of the phase space. This generalizes results of Craig-Kappeler-Strauss and Wunsch.

math.AP↗

arXiv2002

Upper and lower bounds for normal derivatives of Dirichlet eigenfunctions

Suppose that $M$ is a compact Riemannian manifold with boundary and $u$ is an $L^2$-normalized Dirichlet eigenfunction with eigenvalue $λ$. Let $ψ$ be its normal derivative at the boundary. Scaling considerations lead one to expect that the $L^2$ norm of $ψ$ will grow as $λ^{1/2}$ as $λ\to \infty$. We prove an upper bound of the form $\|ψ\|_2^2 \leq Cλ$ for any Riemannian manifold, and a lower bound $c λ\leq \|ψ\|_2^2$ provided that $M$ has no trapped geodesics (see the main Theorem for a precise statement). Here $c$ and $C$ are positive constants that depend on $M$, but not on $λ$. The proof of the upper bound is via a Rellich-type estimate and is rather simple, while the lower bound is proved via a positive commutator estimate.

math.AP↗

arXiv2001

Spectral and scattering theory for symbolic potentials of order zero

The spectral and scattering theory is investigated for a generalization, to scattering metrics on two-dimensional compact manifolds with boundary, of the class of smooth potentials on the Euclidean plane which are homogeneous of degree zero near infinity. The most complete results require the additional assumption that the restriction of the potential to the circle(s) at infinity be Morse. Generalized eigenfunctions associated to the essential spectrum at non-critical energies are shown to originate both at minima and maxima, although the latter are not germane to the $L^2$ spectral theory. Asymptotic completeness is shown, both in the traditional $L^2$ sense and in the sense of tempered distributions. This leads to a definition of the scattering matrix, the structure of which will be described in a future publication.

math.AP↗

arXiv2000

Intersecting Legendrians and blowups

The purpose of this note is to describe the relationship between two classes of Legendre distributions. These two classes are distributions associated to an intersecting pair of Legendre submanifolds, introduced by one of us by analogy with intersecting Lagrangian distributions of Melrose and Uhlmann, and Legendre distributions associated to a fibred scattering structure introduced by the authors. We prove a general result, and also give an example in two dimensions, which shows explicitly the relation between the two spaces in a simple setting.

math.AP↗

arXiv2000

The resolvent for Laplace-type operators on asymptotically conic spaces

Let X be a compact manifold with boundary, and g a scattering metric on X, which may be either of short range or `gravitational' long range type. Thus, g gives X the geometric structure of a complete manifold with an asymptotically conic end. Let H be an operator of the form $H = Δ+ P$, where $Δ$ is the Laplacian with respect to g and P is a self-adjoint first order scattering differential operator with coefficients vanishing at the boundary of X and satisfying a `gravitational' condition. We define a symbol calculus for Legendre distributions on manifolds with codimension two corners and use it to give a direct construction of the resolvent kernel of H, $R(σ+ i0)$, for $σ$ on the positive real axis. In this approach, we do not use the limiting absorption principle at any stage; instead we construct a parametrix which solves the resolvent equation up to a compact error term and then use Fredholm theory to remove the error term.

math.AP↗

arXiv2000

Determinants of Laplacians in Exterior Domains

We consider classes of simply connected planar domains which are isophasal, ie, have the same scattering phase $s(ł)$ for all $ł> 0$. This is a scattering-theoretic analogue of isospectral domains. Using the heat invariants and the determinant of the Laplacian, Osgood, Phillips and Sarnak showed that each isospectral class is sequentially compact in a natural $C$-infinity topology. In this paper, we show sequential compactness of each isophasal class of domains. To do this we define the determinant of the exterior Laplacian and use it together with the heat invariants (the heat invariants and the determinant being isophasal invariants). We show that the determinant of the interior and exterior Laplacians satisfy a Burghelea-Friedlander-Kappeler type surgery formula. This allows a reduction to a problem on bounded domains for which the methods of Osgood, Phillips and Sarnak can be adapted.

math.AP↗

arXiv1999

The spectral projections and the resolvent for scattering metrics

In this paper we consider certain asymptotically Euclidean spaces, namely compact manifolds with boundary X equipped with a scattering metric g, as defined by Melrose. We then consider Hamiltonians H which are `short-range' self-adjoint perturbations of the Laplacian of g. Melrose and Zworski have given a detailed description of the associated scattering matrix and Poisson operator as a Fourier integral operator and a (singular) Legendre distribution respectively. In this paper we describe the kernel of the spectral projections and the boundary value of the resolvent at the real axis. We define classes of Legendre distributions on certain types of manifolds with corners, and show that the kernels of the spectral projection and the resolvent are in these classes. We also discuss some applications of these results.

math.AP↗