Spectral theta series of operators with periodic bicharacteristic flow

Jens Marklof

Displaying similar documents to “Spectral theta series of operators with periodic bicharacteristic flow”

Long time asymptotics of the Camassa–Holm equation on the half-line

Anne Boutet de Monvel, Dmitry Shepelsky (2009)

Annales de l’institut Fourier

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We study the long-time behavior of solutions of the initial-boundary value (IBV) problem for the Camassa–Holm (CH) equation $u_{t} - u_{t x x} + 2 u_{x} + 3 u u_{x} = 2 u_{x} u_{x x} + u u_{x x x}$ on the half-line $x \geq 0$ . The paper continues our study of IBV problems for the CH equation, the key tool of which is the formulation and analysis of associated Riemann–Hilbert factorization problems. We specify the regions in the quarter space-time plane $x > 0$ , $t > 0$ having qualitatively different asymptotic pictures, and give the main terms of the asymptotics in terms of...

The higher transvectants are redundant

Abdelmalek Abdesselam, Jaydeep Chipalkatti (2009)

Annales de l’institut Fourier

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Let $A, B$ denote generic binary forms, and let $𝔲_{r} = {(A, B)}_{r}$ denote their $r$ -th transvectant in the sense of classical invariant theory. In this paper we classify all the quadratic syzygies between the ${𝔲_{r}}$ . As a consequence, we show that each of the higher transvectants ${𝔲_{r} : r \geq 2}$ is redundant in the sense that it can be completely recovered from $𝔲_{0}$ and $𝔲_{1}$ . This result can be geometrically interpreted in terms of the incomplete Segre imbedding. The calculations rely upon the Cauchy exact sequence of $S L_{2}$ -representations,...

Elementary linear algebra for advanced spectral problems

Johannes Sjöstrand, Maciej Zworski (2007)

Annales de l’institut Fourier

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We describe a simple linear algebra idea which has been used in different branches of mathematics such as bifurcation theory, partial differential equations and numerical analysis. Under the name of the Schur complement method it is one of the standard tools of applied linear algebra. In PDE and spectral analysis it is sometimes called the Grushin problem method, and here we concentrate on its uses in the study of infinite dimensional problems, coming from partial differential operators...

Bounded almost global solutions for non hamiltonian semi-linear Klein-Gordon equations with radial data on compact revolution hypersurfaces

Jean-Marc Delort, Jérémie Szeftel (2006)

Annales de l’institut Fourier

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This paper is devoted to the proof of almost global existence results for Klein-Gordon equations on compact revolution hypersurfaces with non-Hamiltonian nonlinearities, when the data are smooth, small and radial. The method combines normal forms with the fact that the eigenvalues associated to radial eigenfunctions of the Laplacian on such manifolds are simple and satisfy convenient asymptotic expansions.

On the distribution of the free path length of the linear flow in a honeycomb

Florin P. Boca, Radu N. Gologan (2009)

Annales de l’institut Fourier

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Consider the region obtained by removing from $ℝ^{2}$ the discs of radius $ε$ , centered at the points of integer coordinates $(a, b)$ with $b ≢ a (mod ℓ)$ . We are interested in the distribution of the free path length (exit time) $τ_{ℓ, ε} (ω)$ of a point particle, moving from $(0, 0)$ along a linear trajectory of direction $ω$ , as $ε \to 0^{+}$ . For every integer number $ℓ \geq 2$ , we prove the weak convergence of the probability measures associated with the random variables $ε τ_{ℓ, ε}$ , explicitly computing the limiting distribution. For $ℓ = 3$ , respectively $ℓ = 2$ , this...