A time-independent approach for the study of spectral shift function

Mouez Dimassi; Maher Zerzeri

doi:10.1016/j.crma.2012.03.016

Partial Differential Equations

A time-independent approach for the study of spectral shift function

Presented by: Jean-Michel Bony

Mouez Dimassi ¹; Maher Zerzeri ²

¹ IMB (UMR CNRS 5251), université de Bordeaux 1, 33405 Talence, France
² LAGA (UMR CNRS 7539), institut Galilée, université Paris nord 13, 93430 Villetaneuse, France

Comptes Rendus. Mathématique, Volume 350 (2012) no. 7-8, pp. 375-378.

Abstract
Résumé

In this Note, we give a new proof of a pointwise asymptotic expansion in powers of h of the derivative of the spectral shift function corresponding to the pair $(- h^{2} Δ + V (x), - h^{2} Δ)$ , near a non-trapping energy. Here the potential V is smooth, real-valued and $O ({| x |}^{- δ})$ for some $δ > n$ , and $h > 0$ is a small parameter. This result is originally due to D. Robert and H. Tamura and their proof is based on the construction of a long-time parametrix for the time-dependent Schrödinger equation. Here we give a time-independent method.

Dans cette Note, on donne une nouvelle preuve pour lʼasymptotique forte en puissances de h de la dérivée de la fonction de décalage spectral associée au couple $(- h^{2} Δ + V (x), - h^{2} Δ)$ , près dʼune énergie non captive. Ici le potentiel V est lisse, à valeurs réelles et $O ({| x |}^{- δ})$ pour un certain $δ > n$ , et $h > 0$ est un petit paramètre. Ce résultat est due à D. Robert et H. Tamura et leur preuve est basée sur la construction de paramétix pour des temps grands pour lʼéquation de Schrödinger dépendant du temps. Ici on donne une méthode indépendante du temps.

Received: 2012-02-15
Accepted: 2012-03-22
Published online: 2012-04-01

DOI: 10.1016/j.crma.2012.03.016

Author's affiliations:

Mouez Dimassi ¹; Maher Zerzeri ²

¹ IMB (UMR CNRS 5251), université de Bordeaux 1, 33405 Talence, France
² LAGA (UMR CNRS 7539), institut Galilée, université Paris nord 13, 93430 Villetaneuse, France

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     title = {A time-independent approach for the study of spectral shift function},
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     year = {2012},
     doi = {10.1016/j.crma.2012.03.016},
     language = {en},
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Mouez Dimassi; Maher Zerzeri. A time-independent approach for the study of spectral shift function. Comptes Rendus. Mathématique, Volume 350 (2012) no. 7-8, pp. 375-378. doi : 10.1016/j.crma.2012.03.016. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2012.03.016/

References
Cited by

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[2] M. Dimassi; J. Sjöstrand Spectral Asymptotics in the Semi-Classical Limit, London Mathematical Society Lecture Note Series, vol. 268, Cambridge University Press, Cambridge, 1999

[3] M. Dimassi, M. Zerzeri, A time-independent approach for the study of spectral shift function and applications, in preparation.

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[6] A. Martinez Resonance free domains for non globally analytic potentials, Ann. Henri Poincaré, Volume 3 (2002) no. 4, pp. 739-756

[7] D. Robert Semiclassical asymptotics for the spectral shift function, Amer. Math. Soc. Transl., Volume 189 (1999) no. 2, pp. 187-203

[8] D. Robert; H. Tamura Semi-classical asymptotics for local spectral densities and time delay problems in scattering processes, J. Funct. Anal., Volume 80 (1988) no. 1, pp. 124-147

[9] J. Sjöstrand; M. Zworski Fractal upper bounds on the density of semiclassical resonances, Duke Math. J., Volume 137 (2007) no. 3, pp. 381-459

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