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The Quantum Birkhoff Normal Form and Spectral Asymptotics

San Vũ Ngọc (2006)

Journées Équations aux dérivées partielles

In this talk we explain a simple treatment of the quantum Birkhoff normal form for semiclassical pseudo-differential operators with smooth coefficients. The normal form is applied to describe the discrete spectrum in a generalised non-degenerate potential well, yielding uniform estimates in the energy E . This permits a detailed study of the spectrum in various asymptotic regions of the parameters ( E , ℏ ) , and gives improvements and new proofs for many of the results in the field. In the completely resonant...

The Srní lectures on non-integrable geometries with torsion

Ilka Agricola (2006)

Archivum Mathematicum

This review article intends to introduce the reader to non-integrable geometric structures on Riemannian manifolds and invariant metric connections with torsion, and to discuss recent aspects of mathematical physics—in particular superstring theory—where these naturally appear. Connections with skew-symmetric torsion are exhibited as one of the main tools to understand non-integrable geometries. To this aim a a series of key examples is presented and successively dealt with using the notions of...

The symbol of a function of a pseudo-differential operator

Alfonso Gracia-saz (2005)

Annales de l'institut Fourier

We give an explicit formula for the symbol of a function of an operator. Given a pseudo-differential operator A ^ on L 2 ( ℝ N ) with symbol A ∈ 𝒞 ∞ ( T * ℝ N ) and a smooth function f , we obtain the symbol of f ( A ^ ) in terms of A . As an application, Bohr-Sommerfeld quantization rules are explicitly calculated at order 4 in ℏ .

The Weil algebra and the Van Est isomorphism

Camilo Arias Abad, Marius Crainic (2011)

Annales de l’institut Fourier

This paper belongs to a series of papers devoted to the study of the cohomology of classifying spaces. Generalizing the Weil algebra of a Lie algebra and Kalkman’s BRST model, here we introduce the Weil algebra W ( A ) associated to any Lie algebroid A . We then show that this Weil algebra is related to the Bott-Shulman complex (computing the cohomology of the classifying space) via a Van Est map and we prove a Van Est isomorphism theorem. As application, we generalize and find a simpler more conceptual...

Three results on the regularity of the curves that are invariant by an exact symplectic twist map

M.-C. Arnaud (2009)

Publications Mathématiques de l'IHÉS

A theorem due to G. D. Birkhoff states that every essential curve which is invariant under a symplectic twist map of the annulus is the graph of a Lipschitz map. We prove: if the graph of a Lipschitz map h:T→R is invariant under a symplectic twist map, then h is a little bit more regular than simply Lipschitz (Theorem 1); we deduce that there exists a Lipschitz map h:T→R whose graph is invariant under no symplectic twist map (Corollary 2).Assuming that the dynamic of a twist map restricted to a...

Through the analytic halo: Fission via irregular singularities

Philip Boalch (2009)

Annales de l’institut Fourier

This article is concerned with moduli spaces of connections on bundles on Riemann surfaces, where the structure group of the bundle may vary in different regions of the surface. Here we will describe such moduli spaces as complex symplectic manifolds, generalising the complex character varieties of Riemann surfaces.

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