B. Pour and . On-utilise-la-formule, 8) et on remarque que t N 0 F j (t) = ?N t N 0 +j?N ?1 P(t ?N ) + jt

. Dont-on-déduit-facilement, + j ? N ? 1 ? 0 et du fait que N 0 ? 1 + j ? N, que leurs valeurs absolues sont majorées respectivement par (N + j) sup [0,1] |P(x)| et, ] |P(x)| ce qui termine la démonstration, p.122

A. B. Intégralesint´intégrales-doubles-d-'opérateursop´opérateurs-bibliographie, ]. K. Andersson, and R. B. , Melrose : The propagation of singularities along gliding rays, Invent. math, vol.41, issue.1, pp.197-232, 1977.

S. Agmon, Spectral properties of Schrödinger operators and scattering theory, Ann. Scuola. Pisa, vol.2, issue.4, pp.151-218, 1975.

S. Agmon, Some new results in spectral and scattering theory of differential operators on R n , Sém. Goulaouic-Schwartz, Exp. II, pp.1-11, 1978.

M. Sh, M. G. Birman, and . Krein, On the theory of wave operators and scattering operators, Dokl. Akad. Nauk SSSR, vol.144, issue.3, pp.475-478, 1962.

M. Sh and M. Z. Birman, Solomyak : Remarks on the spectral shift function, Zap. Nauchn. Sem. Leningrad Math. Inst. Steklov (LOMI), vol.27, pp.33-46, 1972.

M. Sh and M. Z. Birman, Solomyak : Spectral theory of Self-Adjoint operators in Hilbert Space, Mathematics and its Applications, 1987.

V. Bruneau, Propriétés asymptotiques du spectre continu d'opérateurs de Dirac, 1995.

V. Bruneau and V. , Semiclassical Resolvent Estimates for Trapping Perturbations, Communications in Mathematical Physics, vol.213, issue.2, pp.413-432, 2000.
DOI : 10.1007/s002200000246

URL : http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.27.4582

J. Chazarain, Formule de Poisson pour les variétés riemanniennes, Invent. math, pp.65-82, 1974.
DOI : 10.1007/bf01418788

Y. Colin-deverdì-ere, Une formule de traces pour l'opérateur de Schrödinger dans R 3 , Ann, pp.27-39, 1981.

P. Cotta-ramusino, W. Krüger, and R. Schrader, Quantum scattering by external metrics and Yang-Mills potentials, Ann. Ins. H. Poincaré, Physique théorique, vol.XXXI, issue.1, pp.43-71, 1979.

M. Dimassi and J. Sjöstrand, Spectral asymptotics in the semi-classical limit, Lecture Note Series, vol.268, 1999.
DOI : 10.1017/CBO9780511662195

J. J. Duistermaat and V. Guillemin, The spectrum of positive elliptic operators and periodic bicharacteristics, Inventiones Mathematicae, vol.197, issue.1, pp.39-79, 1975.
DOI : 10.1007/BF01405172

C. Gérard and A. Martinez, Principe d'absorption limite pour des opérateurs de Schrödinger Schrödingerà longue portée, C.R. acad. sci. Paris Ser. I, vol.306, pp.121-123, 1988.

C. Gérard and A. Martinez, Prolongement méromorphe de la matrice de scattering pour des probì emesàemesà deux corpsàcorpsà longue portée, pp.81-110, 1989.

L. Guillopé, Une formule de trace pour l'opérateur de Schrödinger dans R n, Thèse de doctorat, 1981.

B. Helffer and D. Robert, Calcul fonctionnel par la transformation de Mellin et op??rateurs admissibles, Journal of Functional Analysis, vol.53, issue.3, pp.246-268, 1983.
DOI : 10.1016/0022-1236(83)90034-4

URL : http://doi.org/10.1016/0022-1236(83)90034-4

L. Hörmander, The spectral function of an elliptic operator, Acta Math, vol.124, pp.173-218, 1968.

L. Hörmander, Analysis of partial differential operators I-IV, Grundlehren, pp.256-257, 1983.

H. Isozaki and H. Kitada, Scattering matrices for two-body Schrödinger operators, Sci. papers college Arts and Sci, pp.85-107, 1985.

V. Ivrii, On the second term of the spectral asymptotics for the Laplace-Beltrami operator in manifolds with boundary, Funk. Anal. i pril, pp.25-34, 1982.

V. Ivrii, Microlocal analysis and precise spectral asymptotics, 1998.
DOI : 10.1007/978-3-662-12496-3

A. Jensen, Spectral properties of Schrödinger operators and time decay of the wave functions , results in L 2 (R m ), m ? 5 Duke mathematical journal, pp.57-80, 1980.

A. Jensen and T. Kato, Spectral properties of Schrödinger operators and time decay of the wave funcions, Duke mathematical journal, pp.583-611, 1979.

A. Jensen and G. , Nenciu : A unified approach to resolvent expansions at thresholds, 2000.

A. Jensen, E. Mourre, and P. Perry, Multiple commutator estimates and resolvent smoothness in quantum scattering theory, Annales de l'I.H.P. Physique théorique, pp.41-207, 1984.

L. S. Koplienko, Trace formula for nontrace-class perturbations, Siberian Math, J, vol.25, pp.62-71, 1984.
DOI : 10.1007/bf00968686

L. S. Koplienko, Regularized spectral shift function for one-dimensional Schrödinger operator with slowly decreasing potential, Siberian Math, J. vol, vol.26, pp.365-369, 1985.

E. Mourre, Absence of singular spectrum for certain selfadjoint operators, Comm. in math. phys, pp.391-400, 1981.

A. Neidhardt, Spectral Shift Function and Hilbert-Schmidt Perturbation: Extensions of Some Work of L. S. Koplienko, Mathematische Nachrichten, vol.2, issue.1, pp.7-25, 1988.
DOI : 10.1002/mana.19881380102

V. Petkov and G. Popov, Asymptotic behavior of the scattering phase for non trapping obstacles, pp.111-149, 1982.

V. Petkov and D. Robert, Asymptotique semi-classique d'hamiltoniens quantiques et trajectoires classiques périodiques, Comm, pp.365-390, 1985.

V. Petkov and M. Zworski, Breit-Wigner Approximation and the Distribution??of Resonances, Communications in Mathematical Physics, vol.204, issue.2, pp.329-351, 1999.
DOI : 10.1007/s002200050648

M. Reed and B. , Simon : Methods of modern mathematical physics I-IV, 1979.

D. Robert, Autour de l'approximation semi-classique, Progress in mathematics 68, Birkhäuser, 1987.

D. Robert, Asymptotique de la phase de diffusionàdiffusionà hauté energie pour des perturbations du second ordre du Laplacien, Ann. scient, E.N.S, vol.25, pp.107-134, 1992.
DOI : 10.24033/asens.1645

D. Robert, Relative Time-Delay for Perturbations of Elliptic Operators and Semiclassical Asymptotics, Journal of Functional Analysis, vol.126, issue.1, pp.36-82, 1994.
DOI : 10.1006/jfan.1994.1141

D. Robert, On the Weyl formula for obstacles, Partial differential equations and mathematical physics, L.Hörmander and A, pp.264-285, 1996.

D. Robert, Semi-Classical Approximation in Quantum Mechanics. A survey of old and recent Mathematical results, Helv, pp.44-116, 1998.

D. Robert, Semiclassical asymptotics for the spectral shift function, Differential operators and spectral theory, M. Sh. Birman's 70th Anniversary, pp.187-203, 1999.

D. Robert and H. Tamura, Semiclassical asymptotics for local spectral densities and timedelay in scattering processes, Journal of functional analysis, vol.80, issue.1, 1988.

A. Rybkin, On a trace formula of the Buslaev???Faddeev type for a long-range potential, Journal of Mathematical Physics, vol.40, issue.3, pp.1334-1343, 1999.
DOI : 10.1063/1.532805

J. T. Schwartz, Non-linear functional analysis, 1969.

G. Vodev, Exponential bounds of the resolvent for a class of noncompactly supported perturbations of the Laplacian, Mathematical Research Letters, vol.7, issue.3, pp.287-298, 2000.
DOI : 10.4310/MRL.2000.v7.n3.a4

D. Yafaev, Mathematical scattering theory, General theory Amer. Math. Soc., RI, vol.158, 1992.
DOI : 10.1090/surv/158

URL : https://hal.archives-ouvertes.fr/hal-00707502