14 results on '"Wunsch, Jared"'
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2. Mode solutions to the wave equation on a rotating cosmic string background
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Morgan, Katrina and Wunsch, Jared
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Mathematics - Analysis of PDEs ,FOS: Mathematics ,FOS: Physical sciences ,General Relativity and Quantum Cosmology (gr-qc) ,Mathematical Physics (math-ph) ,General Relativity and Quantum Cosmology ,Mathematical Physics ,Analysis of PDEs (math.AP) - Abstract
A static rotating cosmic string metric is singular along a timelike line and fails to be globally hyperbolic; these features make it difficult to solve the wave equation by conventional energy methods. Working on a single angular mode at a time, we use microlocal methods to construct forward parametrices for wave and Klein--Gordon equations on such backgrounds., 15 pages; improvements to exposition in response to referee comments
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- 2022
3. On non-diffractive cones
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Galkowski, Jeffrey and Wunsch, Jared
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Mathematics - Analysis of PDEs ,Algebra and Number Theory ,FOS: Mathematics ,FOS: Physical sciences ,Mathematical Physics (math-ph) ,Geometry and Topology ,Mathematical Physics ,Analysis ,Analysis of PDEs (math.AP) - Abstract
A subject of recent interest in inverse problems is whether a corner must diffract fixed frequency waves. We generalize this question somewhat and study cones $[0,\infty)\times Y$ which do not diffract high frequency waves. We prove that if $Y$ is analytic and does not diffract waves at high frequency then every geodesic on $Y$ is closed with period $2\pi$. Moreover, we show that if $\dim Y=2$, then $Y$ is isometric to either the sphere of radius 1 or its $\mathbb{Z}^2$ quotient, $\mathbb{R}\mathbb{P}^2$., Comment: Updated in response to referee comments. 13 pages
- Published
- 2022
4. The $hp$-FEM applied to the Helmholtz equation with PML truncation does not suffer from the pollution effect
- Author
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Galkowski, Jeffrey, Lafontaine, David, Spence, Euan A., Wunsch, Jared, University College of London [London] (UCL), Institut de Mathématiques de Toulouse UMR5219 (IMT), Université Toulouse Capitole (UT Capitole), Université de Toulouse (UT)-Université de Toulouse (UT)-Institut National des Sciences Appliquées - Toulouse (INSA Toulouse), Institut National des Sciences Appliquées (INSA)-Université de Toulouse (UT)-Institut National des Sciences Appliquées (INSA)-Université Toulouse - Jean Jaurès (UT2J), Université de Toulouse (UT)-Université Toulouse III - Paul Sabatier (UT3), Université de Toulouse (UT)-Centre National de la Recherche Scientifique (CNRS), University of Bath [Bath], and Northwestern University [Evanston]
- Subjects
perfectly-matched layer ,finite element method ,high frequency ,error estimate ,Numerical Analysis (math.NA) ,Mathematics::Numerical Analysis ,Mathematics - Analysis of PDEs ,FOS: Mathematics ,[MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP] ,Helmholtz equation ,Mathematics - Numerical Analysis ,pollution effect ,[MATH.MATH-NA]Mathematics [math]/Numerical Analysis [math.NA] ,semiclassical analysis ,Analysis of PDEs (math.AP) - Abstract
We consider approximation of the variable-coefficient Helmholtz equation in the exterior of a Dirichlet obstacle using perfectly-matched-layer (PML) truncation; it is well known that this approximation is exponentially accurate in the PML width and the scaling angle, and the approximation was recently proved to be exponentially accurate in the wavenumber $k$ in [Galkowski, Lafontaine, Spence, 2021]. We show that the $hp$-FEM applied to this problem does not suffer from the pollution effect, in that there exist $C_1,C_2>0$ such that if $hk/p\leq C_1$ and $p \geq C_2 \log k$ then the Galerkin solutions are quasioptimal (with constant independent of $k$), under the following two conditions (i) the solution operator of the original Helmholtz problem is polynomially bounded in $k$ (which occurs for "most" $k$ by [Lafontaine, Spence, Wunsch, 2021]), and (ii) either there is no obstacle and the coefficients are smooth or the obstacle is analytic and the coefficients are analytic in a neighbourhood of the obstacle and smooth elsewhere. This $hp$-FEM result is obtained via a decomposition of the PML solution into "high-" and "low-frequency" components, analogous to the decomposition for the original Helmholtz solution recently proved in [Galkowski, Lafontaine, Spence, Wunsch, 2022]. The decomposition is obtained using tools from semiclassical analysis (i.e., the PDE techniques specifically designed for studying Helmholtz problems with large $k$)., arXiv admin note: text overlap with arXiv:2102.13081
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- 2022
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5. Wave propagation on rotating cosmic string spacetimes
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Morgan, Katrina and Wunsch, Jared
- Subjects
Mathematics - Analysis of PDEs ,FOS: Mathematics ,FOS: Physical sciences ,35Q75, 35L05, 58J47 ,General Relativity and Quantum Cosmology (gr-qc) ,Mathematical Physics (math-ph) ,General Relativity and Quantum Cosmology ,Mathematical Physics ,Analysis of PDEs (math.AP) - Abstract
A rotating cosmic string spacetime has a singularity along a timelike curve corresponding to a one-dimensional source of angular momentum. Such spacetimes are not globally hyperbolic: they admit closed timelike curves near the string. This presents challenges to studying the existence of solutions to the wave equation via conventional energy methods. In this work, we show that semi-global forward solutions to the wave equation do nonetheless exist, but only in a microlocal sense. The main ingredient in this existence theorem is a propagation of singularities theorem that relates energy entering the string to energy leaving the string. The propagation theorem is localized in the fibers of a certain fibration of the blown-up string, but global in time, which means that energy entering the string at one time may emerge previously., Comment: 58 pgs; Lemma 9.3 has been updated
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- 2022
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6. Semiclassical diffraction by conormal potential singularities
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Gannot, Oran and Wunsch, Jared
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Mathematics - Analysis of PDEs ,Mathematics::Analysis of PDEs ,FOS: Mathematics ,Mathematics::Spectral Theory ,Analysis of PDEs (math.AP) - Abstract
We establish propagation of singularities for the semiclassical Schr\"odinger equation, where the potential is conormal to a hypersurface. We show that semiclassical wavefront set propagates along generalized broken bicharacteristics, hence reflection of singularities may occur along trajectories reaching the hypersurface transversely. The reflected wavefront set is weaker, however, by a power of $h$ that depends on the regularity of the potential. We also show that for sufficiently regular potentials, wavefront set may not stick to the hypersurface, but rather detaches from it at points of tangency to travel along ordinary bicharacteristics., Comment: 99 pages; exposition revised in response to referee comments
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- 2018
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7. Diffractive Propagation on Conic Manifolds
- Author
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Wunsch, Jared
- Subjects
Mathematics - Analysis of PDEs ,FOS: Mathematics ,Mathematics::Differential Geometry ,Analysis of PDEs (math.AP) - Abstract
In this survey, we review some applications and extensions of the author's results with Richard Melrose on propagation of singularities for solutions to the wave equation on manifolds with conical singularities. These results mainly concern: the local decay of energy on noncompact manifolds with diffractive trapped orbits (joint work with Dean Baskin); singularities of the wave trace created by diffractive closed geodesics (joint work with G. Austin Ford); and the distribution of scattering resonances associated to such closed geodesics (joint work with Luc Hillairet)., Comment: 15 pages; contribution to Seminaire Laurent Schwartz proceedings
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- 2016
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8. Asymptotics of radiation fields in asymptotically Minkowski space
- Author
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Baskin, Dean, Vasy, András, and Wunsch, Jared
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Mathematics - Differential Geometry ,Mathematics - Spectral Theory ,Mathematics - Analysis of PDEs ,Differential Geometry (math.DG) ,FOS: Mathematics ,Spectral Theory (math.SP) ,35L05 (Primary), 35P25, 58J45 (Secondary) ,Analysis of PDEs (math.AP) - Abstract
We consider a non-trapping $n$-dimensional Lorentzian manifold endowed with an end structure modeled on the radial compactification of Minkowski space. We find a full asymptotic expansion for tempered forward solutions of the wave equation in all asymptotic regimes. The rates of decay seen in the asymptotic expansion are related to the resonances of a natural asymptotically hyperbolic problem on the "northern cap" of the compactification. For small perturbations of Minkowski space that fit into our framework, we show a rate of decay that improves on the Klainerman--Sobolev estimates., Comment: 67 pages, 2 figures; version 2 was a substantial revision with the main result (Theorem 1.1) improved; version 3 fixes a minor error in the iterative argument establishing the asymptotic expansion; version 4 includes a few cosmetic changes and adds a few references
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- 2012
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9. Strichartz estimates on exterior polygonal domains
- Author
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Baskin, Dean, Marzuola, Jeremy L., and Wunsch, Jared
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Mathematics - Analysis of PDEs ,010102 general mathematics ,0103 physical sciences ,Mathematics::Analysis of PDEs ,FOS: Mathematics ,010307 mathematical physics ,0101 mathematics ,35Q41, 35P25 ,01 natural sciences ,Analysis of PDEs (math.AP) - Abstract
Using a new local smoothing estimate of the first and third authors, we prove local-in-time Strichartz and smoothing estimates without a loss exterior to a large class of polygonal obstacles with arbitrary boundary conditions and global-in-time Strichartz estimates without a loss exterior to a large class of polygonal obstacles with Dirichlet boundary conditions. In addition, we prove a global-in-time local smoothing estimate in exterior wedge domains with Dirichlet boundary conditions and discuss some nonlinear applications., Comment: 15 pages. This paper has been substantially revised; the results have been improved to include global-in-time estimates in the Dirichlet case. Further revised to include more related scattering references thanks to a careful reading by Fabrice Planchon
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- 2012
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10. Local smoothing for the Schr��dinger equation with a prescribed loss
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Christianson, Hans and Wunsch, Jared
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FOS: Mathematics ,Spectral Theory (math.SP) ,Analysis of PDEs (math.AP) ,35J10, 35B34 - Abstract
We consider a family of surfaces of revolution, each with a single periodic geodesic which is degenerately unstable. We prove a local smoothing estimate for solutions to the linear Schr��dinger equation with a loss that depends on the degeneracy, and we construct explicit examples to show our estimate is saturated on a weak semiclassical time scale. As a byproduct of our proof, we obtain a cutoff resolvent estimate with a sharp polynomial loss., 26 pages, incorporated referee's comments
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- 2011
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11. Diffraction of singularities for the wave equation on manifolds with corners
- Author
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Melrose, Richard, Vasy, Andras, and Wunsch, Jared
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Mathematics - Analysis of PDEs ,35L20, 58J47 ,FOS: Mathematics ,FOS: Physical sciences ,Mathematical Physics (math-ph) ,Mathematics::Symplectic Geometry ,Mathematical Physics ,Analysis of PDEs (math.AP) - Abstract
We consider the fundamental solution to the wave equation on a manifold with corners of arbitrary codimension. If the initial pole of the solution is appropriately situated, we show that the singularities which are diffracted by the corners (i.e., loosely speaking, are not propagated along limits of transversely reflected rays) are smoother than the main singularities of the solution. More generally, we show that subject to a hypothesis of nonfocusing, diffracted wavefronts of any solution to the wave equation are smoother than the incident singularities. These results extend our previous work on edge manifolds to a situation where the fibers of the boundary fibration, obtained here by blowup of the corner in question, are themselves manifolds with corners., Comment: Revision adding significant exposition, correcting many small errors; index of notation added
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- 2009
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12. Microlocal analysis and evolution equations
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Wunsch, Jared
- Subjects
Mathematics - Analysis of PDEs ,35-02, 35S05, 35S30 ,FOS: Mathematics ,Analysis of PDEs (math.AP) - Abstract
Lecture notes from 2008 CMI/ETH Summer School on Evolution Equations. These notes are an informal introduction to the applications of microlocal methods in the study of linear evolution equations and spectral theory. Calculi of pseudodifferential operators and Fourier integral operators are discussed and axiomatized, but not constructed: the focus is on how to apply these tools., Comment: Corrects an error in statement of Theorem 9.2 kindly pointed out by Amir Vig
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- 2008
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13. On the structure of the Schr��dinger propagator
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Hassell, Andrew and Wunsch, Jared
- Subjects
FOS: Mathematics ,FOS: Physical sciences ,35A17 ,58J47, 35A21 ,Mathematical Physics (math-ph) ,Analysis of PDEs (math.AP) - Abstract
We discuss the form of the propagator $U(t)$ for the time-dependent Schr��dinger equation on an asyptotically Euclidean, or, more generally, asymptotically conic, manifold with no trapped geodesics. In the asymptotically Euclidean case, if $��\in \mathcal{C}_0^\infty$, and with $\mathcal{F}$ denoting Fourier transform, $\mathcal{F}\circ e^{-ir^2/2t} U(t) ��$ 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 Schr��dinger 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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- 2003
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14. The Schroedinger propagator for scattering metrics
- Author
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Hassell, Andrew and Wunsch, Jared
- Subjects
Mathematics - Analysis of PDEs ,FOS: Mathematics ,35A17 ,58J47, 35A21 ,Mathematics::Spectral Theory ,Analysis of PDEs (math.AP) - Abstract
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 $H = \half \Delta + V$, where $\Delta$ 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., Comment: 29 pages, no figures
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- 2003
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