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Semi-classical measures for generalized plane waves

Colin Guillarmou

Séminaire Laurent Schwartz — EDP et applications

Following joint work with Dyatlov [], we describe the semi-classical measures associated with generalized plane waves for metric perturbation of d , under the condition that the geodesic flow has trapped set K of Liouville measure 0 .

Resolvent at low energy and Riesz transform for Schrödinger operators on asymptotically conic manifolds. II

Colin GuillarmouAndrew Hassell — 2009

Annales de l’institut Fourier

Let M be a complete noncompact manifold of dimension at least 3 and g an asymptotically conic metric on M , in the sense that M compactifies to a manifold with boundary M so that g becomes a scattering metric on M . We study the resolvent kernel ( P + k 2 ) - 1 and Riesz transform T of the operator P = Δ g + V , where Δ g is the positive Laplacian associated to g and V is a real potential function smooth on M and vanishing at the boundary. In our first paper we assumed that P has neither zero modes nor a zero-resonance...

Upper bounds for the number of resonances on geometrically finite hyperbolic manifolds

David BorthwickColin Guillarmou — 2016

Journal of the European Mathematical Society

On geometrically finite hyperbolic manifolds Γ d , including those with non-maximal rank cusps, we give upper bounds on the number N ( R ) of resonances of the Laplacian in disks of size R as R . In particular, if the parabolic subgroups of Γ satisfy a certain Diophantine condition, the bound is N ( R ) = 𝒪 ( R d ( log R ) d + 1 ) .

Conformal harmonic forms, Branson–Gover operators and Dirichlet problem at infinity

Erwann AubryColin Guillarmou — 2011

Journal of the European Mathematical Society

For odd-dimensional Poincaré–Einstein manifolds ( X n + 1 , g ) , we study the set of harmonic k -forms (for k < n / 2 ) which are C m (with m ) on the conformal compactification X ¯ of X . This set is infinite-dimensional for small m but it becomes finite-dimensional if m is large enough, and in one-to-one correspondence with the direct sum of the relative cohomology H k ( X ¯ , X ¯ ) and the kernel of the Branson–Gover [3] differential operators ( L k , G k ) on the conformal infinity ( X ¯ , [ h 0 ] ) . We also relate the set of C n - 2 k + 1 ( Λ k ( X ¯ ) ) forms in the kernel of d + δ g to the conformal...

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