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Tunnel effect for semiclassical random walk

Jean-François Bony, Frédéric Hérau, Laurent Michel (2014)

Journées Équations aux dérivées partielles

In this note we describe recent results on semiclassical random walk associated to a probability density which may also concentrate as the semiclassical parameter goes to zero. The main result gives a spectral asymptotics of the close to 1 eigenvalues. This problem was studied in [1] and relies on a general factorization result for pseudo-differential operators. In this note we just sketch the proof of this second theorem. At the end of the note, using the factorization, we give a new proof of the...

Two remarks about spectral asymptotics of pseudodifferential operators

Wojciech Czaja, Ziemowit Rzeszotnik (1999)

Colloquium Mathematicae

In this paper we show an asymptotic formula for the number of eigenvalues of a pseudodifferential operator. As a corollary we obtain a generalization of the result by Shubin and Tulovskiĭ about the Weyl asymptotic formula. We also consider a version of the Weyl formula for the quasi-classical asymptotics.

Une classe de symboles new-look

André Hirschowitz (1980)

Annales de l'institut Fourier

On construit par voie géométrique une classe de symboles classiques en dehors d’une sous-variété. La classe d’opérateurs pseudodifférentiels associée contient les paramétrix d’opérateurs tels que ∑ i = 1 n - 1 ∂ ∂ x i 4 + ∂ ∂ x n 3 ou ∂ ∂ x n 3 + ∑ i = 1 n - 1 ∂ ∂ x i 2 .

Une inégalité de Gårding à bord

Frédéric Hérau (2000)

Journées équations aux dérivées partielles

The aim of this work is to give a Gårding inequality for pseudodifferential operators acting on functions in L 2 ( ℝ n ) supported in a closed regular region F ⊂ ℝ n . A natural idea is to suppose that the symbol is non-negative in F × ℝ n . Assuming this, we show that this result is true for pseudo-differential operators of order one, when F is the half-space, and under a supplementary weak hypothesis of degeneracy of the symbol on the boundary.

Wave front set for positive operators and for positive elements in non-commutative convolution algebras

Joachim Toft (2007)

Studia Mathematica

Let WF⁎ be the wave front set with respect to C ∞ , quasi analyticity or analyticity, and let K be the kernel of a positive operator from C ₀ ∞ to ’. We prove that if ξ ≠ 0 and (x,x,ξ,-ξ) ∉ WF⁎(K), then (x,y,ξ,-η) ∉ WF⁎(K) and (y,x,η,-ξ) ∉ WF⁎(K) for any y,η. We apply this property to positive elements with respect to the weighted convolution u ∗ B φ ( x ) = ∫ u ( x - y ) φ ( y ) B ( x , y ) d y , where B ∈ C ∞ is appropriate, and prove that if ( u ∗ B φ , φ ) ≥ 0 for every φ ∈ C ₀ ∞ and (0,ξ) ∉ WF⁎(u), then (x,ξ) ∉ WF⁎(u) for any x.

Wave fronts of solutions of some classes of non-linear partial differential equations

P. Popivanov (1992)

Banach Center Publications

1. This paper is devoted to the study of wave fronts of solutions of first order symmetric systems of non-linear partial differential equations. A short communication was published in [4]. The microlocal point of view enables us to obtain more precise information concerning the smoothness of solutions of symmetric hyperbolic systems. Our main result is a generalization to the non-linear case of Theorem 1.1 of Ivriĭ [3]. The machinery of paradifferential operators introduced by Bony [1] together...

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