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Generalized Backscattering and the Lax-Phillips Transform

Melrose, Richard;  Uhlmann, Gunther — 2008

Serdica Mathematical Journal

2000 Mathematics Subject Classification: 35P25, 35R30, 58J50. 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 w = Sq in terms of the incoming angle with S 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...

Star products and local line bundles

Richard Melrose — 2004

Annales de l’institut Fourier

The notion of a local line bundle on a manifold, classified by 2-cohomology with real coefficients, is introduced. The twisting of pseudodifferential operators by such a line bundle leads to an algebroid with elliptic elements with real-valued index, given by a twisted variant of the Atiyah-Singer index formula. Using ideas of Boutet de Monvel and Guillemin the corresponding twisted Toeplitz algebroid on any compact symplectic manifold is shown to yield the star products...

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