501 research outputs found
Fredholm realizations of elliptic symbols on manifolds with boundary II: fibered boundary
We consider two calculi of pseudodifferential operators on manifolds with
fibered boundary: Mazzeo's edge calculus, which has as local model the
operators associated to products of closed manifolds with asymptotically
hyperbolic spaces, and the phi calculus of Mazzeo and the second author, which
is similarly modeled on products of closed manifolds with asymptotically
Euclidean spaces. We construct an adiabatic calculus of operators interpolating
between them, and use this to compute the `smooth' K-theory groups of the edge
calculus, determine the existence of Fredholm quantizations of elliptic
symbols, and establish a families index theorem in K-theory
Diffractive Propagation on Conic Manifolds
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 proceeding
Generalized backscattering and the Lax-Phillips transform
Using the free-space translation representation (modified Radon transform) of
Lax and Phillips in odd dimensions, it is shown that the generalized
backscattering transform (so outgoing angle in terms of the
incoming angle with orthogonal and \Id-S invertible) may be further
restricted to give an entire, globally Fredholm, operator on appropriate
Sobolev spaces of potentials with compact support. As a corollary we show that
the modified backscattering map is a local isomorphism near elements of a
generic set of potentials.Comment: Minor changes, typos corrected, references adde
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