Asymptotic behavior of regularized scattering phases for long range perturbations
Journées équations aux dérivées partielles (2002), article no. 2, 12 p.

We define scattering phases for Schrödinger operators on ℝ d as limit of arguments of relative determinants. These phases can be defined for long range perturbations of the laplacian; therefore they can replace the spectral shift function (SSF) of Birman-Krein’s theory which can just be defined for some special short range perturbations (we shall recall this theory for non specialists). We prove the existence of asymptotic expansions for these phases, which generalize results on the SSF.

@article{JEDP_2002____A2_0,
     author = {Bouclet, Jean-Marc},
     title = {Asymptotic behavior of regularized scattering phases for long range perturbations},
     journal = {Journ\'ees \'equations aux d\'eriv\'ees partielles},
     eid = {2},
     pages = {1--12},
     publisher = {Universit\'e de Nantes},
     year = {2002},
     doi = {10.5802/jedp.600},
     mrnumber = {1968198},
     language = {en},
     url = {http://www.numdam.org/articles/10.5802/jedp.600/}
}
TY  - JOUR
AU  - Bouclet, Jean-Marc
TI  - Asymptotic behavior of regularized scattering phases for long range perturbations
JO  - Journées équations aux dérivées partielles
PY  - 2002
SP  - 1
EP  - 12
PB  - Université de Nantes
UR  - http://www.numdam.org/articles/10.5802/jedp.600/
DO  - 10.5802/jedp.600
LA  - en
ID  - JEDP_2002____A2_0
ER  - 
%0 Journal Article
%A Bouclet, Jean-Marc
%T Asymptotic behavior of regularized scattering phases for long range perturbations
%J Journées équations aux dérivées partielles
%D 2002
%P 1-12
%I Université de Nantes
%U http://www.numdam.org/articles/10.5802/jedp.600/
%R 10.5802/jedp.600
%G en
%F JEDP_2002____A2_0
Bouclet, Jean-Marc. Asymptotic behavior of regularized scattering phases for long range perturbations. Journées équations aux dérivées partielles (2002), article  no. 2, 12 p. doi : 10.5802/jedp.600. http://www.numdam.org/articles/10.5802/jedp.600/

[1] M. Sh. Birman, M. G. Krein, On the theory of wave operators and scattering operators, Dokl. Akad. Nauk SSSR, 144, 475-478 (1962). | MR | Zbl

[2] M. Sh. Birman, D. R. Yafaev, The spectral shift function. The work of M. G. Krein and its further development, St. Petersburg Math. J. 4, 833-870 (1993). | MR | Zbl

[3] D. Bollé, F. Gesztesy, H. Grosse, W. Schweiger, B. Simon, Witten index, axial anomaly and Krein's spectral function in supersymmetric quantum mechanics, J. Math. Phys. 28, 1512-1525 (1987). | MR | Zbl

[4] N. Borisov, W. Müller, R. Schrader, Relative index, axial anomaly and Krein's spectral shift function in supersymmetric scattering theory, Commun. Math. Phys. 114, 475-513 (1988). | MR | Zbl

[5] J.M. Bouclet, Distributions spectrales pour des opérateurs perturbés, PhD thesis, Nantes University (2000).

[6], Traces formulae for relatively Hilbert-Schmidt perturbations, Asymptotic Analysis, to appear. | MR | Zbl

[7], Spectral distributions for long range perturbations, preprint of Sussex University, (2002).

[8] V. Bruneau, Propriétés asymptotiques du spectre continu d'opérateurs de Dirac, PhD Thesis, University of Nantes, (1995).

[9] V. Bruneau, V. Petkov, Semiclassical estimates for trapping perturbations, Commun. Math. Phys. 213, No. 2, 413-432 (2000). | MR | Zbl

[10] N. Burq, Lower bounds for shape resonances widths of long range Schrödinger operators, American Journal of Math, to appear. | MR | Zbl

[11] V. S. Buslaev, Trace formulas and some asymptotic estimates of the resolvent kernel for the Schrödinger equation in three dimensional space, Problemy Mat. Fiz. 1, 82-101 (1966). | Zbl

[12] V. S. Buslaev, L. D. Faddeev, Formulas for traces for a singular SturmLiouville differential operator, Sov. Math. Dokl. 1, 451-454 (1960). II-11 | MR | Zbl

[13] Y. Colin-De-Verdière, Une formule de trace pour l'opérateur de Schrödinger dans R 3, Ann. Sci. E.N.S., IV sér. 14, 27-39 (1981). | Numdam | MR | Zbl

[14] C. Gérard, A. Martinez, Prolongement méromorphe de la matrice de scattering pour des problèmes à deux corps à longue portée, Ann. I.H.P., Phys. Théor. 51, No.1, 81-110, (1989). | Numdam | MR | Zbl

[15] L. Guillopé, Une formule de trace pour l'opérateur de Schrödinger dans R n, thèse de doctorat, Université de Grenoble (1981).

[16] M. Hitrik, I. Polterovich, Regularized traces and Taylor expansions for the heat semigroup, preprint, (2001). | MR

[17] H. Isozaki, H. Kitada, Modified wave operators with time independant modifiers, J. Fac. Sci., Univ. Tokyo, Sect. I A 32, 77-104 (1985). | MR | Zbl

[18] L.S. Koplienko, Trace formula for non trace-class perturbations, Sib. Math. J. 25, 735-743 (1984). | MR | Zbl

[19], Regularized spectral shift function for one dimensional Schrödinger operator with slowly decreasing potential, Sib. Math. J. 26, 365-369 (1985). | MR | Zbl

[20] M. G. Krein, Perturbation determinants and a formula for the trace of unitary and selfadjoint operators, Soviet Math. Dokl. 3, 707-710 (1962). | MR | Zbl

[21] A. Melin, Trace distributions associated to the Schrödinger operator, J. Anal. Math. 59, 133-160 (1992). | MR | Zbl

[22] W. Müller, Relative zeta functions, relative determinants and scattering theory, Commun. Math. Phys. 192 (2), 309-347 (1998). | MR | Zbl

[23] D. Robert, On long range scattering for perturbations of Laplace operators, J. Anal. Math. 59, 189-203 (1992). | MR | Zbl

[24], Asymptotique de la phase de diffusion à haute énergie pour des perturbations du second ordre du Laplacien, Ann. Sci. École Norm. Sup., 4ème série, No. 25, 107-134 (1992). | Numdam | MR | Zbl

[25], Relative time delay for perturbations of elliptic operators and semiclassical asymptotics, J. Funct. Anal. 126, No.1, 36-82 (1994). | MR | Zbl

[26] D. Yafaev, Mathematical scattering theory. General theory, vol. 105, A.M.S. Rhode Island (1992) | MR | Zbl

Cité par Sources :