no. 1
Spectral asymptotics for manifolds with cylindrical ends
Annales de l'Institut Fourier, Tome 45 (1995) no. 1, pp. 251-263.

Le spectre du laplacien sur les variétés à bouts cyclindriques est composé d’un spectre continu à multiplicité localement finie et de valeurs propres plongées. Nous démontrons une formule asymptotique du type Weyl pour la somme du nombre de valeurs propres plongées et de la phase de diffusion. En particulier, nous obtenons la limite supérieure optimale du nombre de valeurs propres plongées inférieures ou égales à r 2 ,𝒪(r n ), où r est la dimension de la variété.

The spectrum of the Laplacian on manifolds with cylindrical ends consists of continuous spectrum of locally finite multiplicity and embedded eigenvalues. We prove a Weyl-type asymptotic formula for the sum of the number of embedded eigenvalues and the scattering phase. In particular, we obtain the optimal upper bound on the number of embedded eigenvalues less than or equal to r 2 ,𝒪(r n ), where n is the dimension of the manifold.

@article{AIF_1995__45_1_251_0,
     author = {Christiansen, Tanya and Zworski, Maciej},
     title = {Spectral asymptotics for manifolds with cylindrical ends},
     journal = {Annales de l'Institut Fourier},
     pages = {251--263},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {45},
     number = {1},
     year = {1995},
     doi = {10.5802/aif.1455},
     mrnumber = {96d:35100},
     zbl = {0818.58046},
     language = {en},
     url = {http://archive.numdam.org/articles/10.5802/aif.1455/}
}
TY  - JOUR
AU  - Christiansen, Tanya
AU  - Zworski, Maciej
TI  - Spectral asymptotics for manifolds with cylindrical ends
JO  - Annales de l'Institut Fourier
PY  - 1995
SP  - 251
EP  - 263
VL  - 45
IS  - 1
PB  - Association des Annales de l’institut Fourier
UR  - http://archive.numdam.org/articles/10.5802/aif.1455/
DO  - 10.5802/aif.1455
LA  - en
ID  - AIF_1995__45_1_251_0
ER  - 
%0 Journal Article
%A Christiansen, Tanya
%A Zworski, Maciej
%T Spectral asymptotics for manifolds with cylindrical ends
%J Annales de l'Institut Fourier
%D 1995
%P 251-263
%V 45
%N 1
%I Association des Annales de l’institut Fourier
%U http://archive.numdam.org/articles/10.5802/aif.1455/
%R 10.5802/aif.1455
%G en
%F AIF_1995__45_1_251_0
Christiansen, Tanya; Zworski, Maciej. Spectral asymptotics for manifolds with cylindrical ends. Annales de l'Institut Fourier, Tome 45 (1995) no. 1, pp. 251-263. doi : 10.5802/aif.1455. http://archive.numdam.org/articles/10.5802/aif.1455/

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

[2] T. Christiansen, Scattering theory for manifolds with asymptotically cylindrical ends, J. Funct. Anal. (to appear). | Zbl

[3] Y. Colin De Verdière, Pseudo Laplaciens, II, Ann. Inst. Fourier, 33-2 (1983), 89-113. | Numdam | MR | Zbl

[4] H. Donnelly, Eigenvalue estimates for certain noncompact manifolds, Michigan Math. J., 31 (1984), 349-357. | MR | Zbl

[5] R. Froese and M. Zworski, Finite volume surfaces with resonances far from the unitarity axis, Int. Math. Research Notices, 10 (1993), 275-277. | MR | Zbl

[6] B. Helffer, Semi-classical analysis for the Schrödinger operator and applications, Springer-Verlag, Berlin, 1980.

[7] L. Hörmander, The spectral function of an elliptic operator, Acta Math., 121 (1968), 193-218. | MR | Zbl

[8] P. Lax and R. Phillips, Scattering theory for automorpic functions, Ann. of Math. Studies, 87, Princeton University Press, 1976. | MR | Zbl

[9] R.B. Melrose, Weyl asymptotics for the phase in obstacle scattering, Comm. P.D.E., 13 (1988), 1421-1439. | MR | Zbl

[10] R.B. Melrose, The Atiyah-Patodi-Singer index theorem, A.K. Peters, Wellesley, 1993. | MR | Zbl

[11] W. Müller, Spectral geometry and scattering theory for certain complete surfaces of finite volume, Inv. Math., 109 (1992), 265-305. | MR | Zbl

[12] S. Patterson, The Selberg zeta function of a Kleinian group, in "Number theory, trace formulas and discrete group", p. 409-441, Academic Press, Boston, 1989. | MR | Zbl

[13] P. Perry, The Selberg zeta function and a local trace formula for Kleinian groups, J. Reine Angew. Math., 410 (1990), 116-152. | MR | Zbl

[14] D. Robert, A trace formula for obstacle problems and applications, to appear in "Mathematical results in quantum mechanics", Blossin Conference Proc., Berlin, 1993. | Zbl

Cité par Sources :