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Scattering and resolvent on geometrically finite hyperbolic manifolds with rational cusps

Colin Guillarmou (2005/2006)

Séminaire Équations aux dérivées partielles

These notes summarize the papers [8, 9] on the analysis of resolvent, Eisenstein series and scattering operator for geometrically finite hyperbolic quotients with rational non-maximal rank cusps. They complete somehow the talk given at the PDE seminar of Ecole Polytechnique in october 2005.

Semi-classical measures for generalized plane waves

Colin Guillarmou (2011/2012)

Séminaire Laurent Schwartz — EDP et applications

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

Semiclassical spectral estimates for Toeplitz operators

David Borthwick, Thierry Paul, Alejandro Uribe (1998)

Annales de l'institut Fourier

Let X be a compact Kähler manifold with integral Kähler class and L → X a holomorphic Hermitian line bundle whose curvature is the symplectic form of X . Let H ∈ C ∞ ( X , ℝ ) be a Hamiltonian, and let T k be the Toeplitz operator with multiplier H acting on the space ℋ k = H 0 ( X , L ⊗ k ) . We obtain estimates on the eigenvalues and eigensections of T k as k → ∞ , in terms of the classical Hamilton flow of H . We study in some detail the case when X is an integral coadjoint orbit of a Lie group.

Sharp bounds for the intersection of nodal lines with certain curves

Junehyuk Jung (2014)

Journal of the European Mathematical Society

Let Y be a hyperbolic surface and let φ be a Laplacian eigenfunction having eigenvalue - 1 / 4 - τ 2 with τ > 0 . Let N ( φ ) be the set of nodal lines of φ . For a fixed analytic curve γ of finite length, we study the number of intersections between N ( φ ) and γ in terms of τ . When Y is compact and γ a geodesic circle, or when Y has finite volume and γ is a closed horocycle, we prove that γ is “good” in the sense of [TZ]. As a result, we obtain that the number of intersections between N ( φ ) and γ is O ( τ ) . This bound is sharp.

Small eigenvalues of the Neumann realization of the semiclassical Witten Laplacian

D. Le Peutrec (2010)

Annales de la faculté des sciences de Toulouse Mathématiques

This article follows the previous works [HeKlNi, HeNi] by Helffer-Klein-Nier and Helffer-Nier about the metastability in reversible diffusion processes via a Witten complex approach. Again, exponentially small eigenvalues of some self-adjoint realization of Δ f , h ( 0 ) = - h 2 Δ + ∇ f ( x ) 2 - h Δ f ( x ) are considered as the small parameter h > 0 tends to 0 . The function f is assumed to be a Morse function on some bounded domain Ω with boundary ∂ Ω . Neumann type boundary conditions are considered. With these boundary conditions, some possible simplifications...

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