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We prove explicit asymptotics for the location of semiclassical scattering resonances in the setting of h-dependent delta-function potentials on \mathbb {R}. In the cases of two or three delta poles, we are able to show that resonances occur along specific lines of the form \operatorname {Im}z \sim -\gamma h \log (1/h). More generally, we use the method of Newton polygons to show that resonances near the real axis may only occur along a finite collection of such lines, and we bound the possible number of values of the parameter \gamma. We present numerical evidence of the existence of more and more possible values of \gamma for larger numbers of delta poles. [ABSTRACT FROM AUTHOR]

Maas (J. Fluid Mech., vol. 684, 2011, pp. 5–24) showed that, for an oscillating two-dimensional barotropic tide flowing over sub-critical topography of compact support, some topographic forms existed that produced non-radiating baroclinic disturbances. The problem is related to 'stealth' and 'cloaking' problems. Here Maas's result is derived using a simpler approach, not involving complicated mappings, but formally restricted to perturbation topography. Wider results come from the discussion of nearly compact support topographic disturbances provided by Schwartz functions with weak high-wavenumber radiation and by exploiting both a known functional equation formulation and Fourier methods. The problem is extended to disturbances on uniform slopes. A variety of non-radiating topographies can be found, although they are mathematically delicate and unlikely to be found in nature. Topography with weak radiation at high wavenumber is a much wider class of structures. Application of these solutions would lie with the ability to estimate dissipation over and near the topography from motions observed at a distance. [ABSTRACT FROM AUTHOR]

We consider three problems for the Helmholtz equation in interior and exterior domains in Rd (d = 2, 3): the exterior Dirichlet-to-Neumann and Neumann-to-Dirichlet problems for outgoing solutions, and the interior impedance problem. We derive sharp estimates for solutions to these problems that, in combination, give bounds on the inverses of the combined-field boundary integral operators for exterior Helmholtz problems. [ABSTRACT FROM AUTHOR]

Author(s): Jared Wunsch [1] by Paul C.Pasles Princeton University Press: 2007. 266 pp. $26.95, £15.95 Benjamin Franklin was a brilliant amateur scientist in an era when amateur science mattered. His [...]

We consider the possible concentration in phase space of a sequence of eigenfunctions (or, more generally, a quasimode) of an operator whose principal symbol has completely integrable Hamilton flow. The semiclassical wavefront set WF h of such a sequence is invariant under the Hamilton flow. In principle this may allow concentration of WF h along a single closed orbit if all frequencies of the flow are rationally related. We show that, subject to non-degeneracy hypotheses, this concentration may not in fact occur. Indeed, in the two-dimensional case, we show that WF h must fill out an entire Lagrangian torus. The main tools are the spreading of Lagrangian regularity previously shown by the author, and an analysis of higher order transport equations satisfied by the principal symbol of a Lagrangian quasimode. These yield a unique continuation theorem for the principal symbol of Lagrangian quasimode, which is the principal new result of the paper. [ABSTRACT FROM PUBLISHER]

Abstract: Bony and Häfner have recently obtained positive commutator estimates on the Laplacian in the low-energy limit on asymptotically Euclidean spaces; these estimates can be used to prove local energy decay estimates if the metric is non-trapping. We simplify the proof of the estimates of Bony–Häfner and generalize them to the setting of scattering manifolds (i.e. manifolds with large conic ends), by applying a sharp Poincaré inequality. Our main result is the positive commutator estimate where is a large parameter, I is a compact interval in , and its indicator function, and where A is a differential operator supported outside a compact set and equal to near infinity. The Laplacian can also be modified by the addition of a positive potential of sufficiently rapid decay—the same estimate then holds for the resulting Schrödinger operator. [Copyright &y& Elsevier]

Abstract: Consider a compact manifold with boundary M with a scattering metric g or, equivalently, an asymptotically conic manifold . (Euclidean , with a compactly supported metric perturbation, is an example of such a space.) Let Δ be the positive Laplacian on , and V a smooth potential on M which decays to second order at infinity. In this paper we construct the kernel of the operator , at a non-trapping energy , uniformly for , small, within a class of Legendre distributions on manifolds with codimension three corners. Using this we construct the kernel of the propagator, , as a quadratic Legendre distribution. We also determine the global semiclassical structure of the spectral projector, Poisson operator and scattering matrix. [Copyright &y& Elsevier]

We obtain the Strichartz inequality for any smooth three-dimensional Riemannian manifold ( M , g ) which is asymptotically conic at infinity and nontrapping, where u is a solution to the Schrödinger equation iu t + (1/2)? M u = 0. The exponent H 1/4 ( M ) is sharp, by scaling considerations. In particular our result covers asymptotically flat nontrapping 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 one (proved here) with weight r -1 at infinity and with the gradient term involving only one angular derivative. [ABSTRACT FROM AUTHOR]

It is well‐known that when the geometry and/or coefficients allow stable trapped rays, the outgoing solution operator of the Helmholtz equation grows exponentially through a sequence of real frequencies tending to infinity. In this paper we show that, even in the presence of the strongest possible trapping, if a set of frequencies of arbitrarily small measure is excluded, the Helmholtz solution operator grows at most polynomially as the frequency tends to infinity. One significant application of this result is in the convergence analysis of several numerical methods for solving the Helmholtz equation at high frequency that are based on a polynomial‐growth assumption on the solution operator (e.g. hp‐finite elements, hp‐boundary elements, and certain multiscale methods). The result of this paper shows that this assumption holds, even in the presence of the strongest possible trapping, for most frequencies. © 2021 The Authors. Communications on Pure and Applied Mathematics published by Wiley Periodicals LLC. [ABSTRACT FROM AUTHOR]