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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...

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...

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