Spectral asymptotics for manifolds with cylindrical ends
Annales de l'Institut Fourier, Tome 45 (1995) no. 1, p. 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},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {45},
     number = {1},
     year = {1995},
     pages = {251-263},
     doi = {10.5802/aif.1455},
     zbl = {0818.58046},
     mrnumber = {96d:35100},
     language = {en},
     url = {http://http://www.numdam.org/item/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://www.numdam.org/item/AIF_1995__45_1_251_0/

[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 25 #2447 | Zbl 0196.45004

[2] T. Christiansen, Scattering theory for manifolds with asymptotically cylindrical ends, J. Funct. Anal. (to appear). | Zbl 0837.58034

[3] Y. Colin De Verdière, Pseudo Laplaciens, II, Ann. Inst. Fourier, 33-2 (1983), 89-113. | Numdam | MR 84k:58222 | Zbl 0496.58016

[4] H. Donnelly, Eigenvalue estimates for certain noncompact manifolds, Michigan Math. J., 31 (1984), 349-357. | MR 86d:58120 | Zbl 0591.58033

[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 94i:58200 | Zbl 0797.58016

[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 58 #29418 | Zbl 0164.13201

[8] P. Lax and R. Phillips, Scattering theory for automorpic functions, Ann. of Math. Studies, 87, Princeton University Press, 1976. | MR 58 #27768 | Zbl 0362.10022

[9] R.B. Melrose, Weyl asymptotics for the phase in obstacle scattering, Comm. P.D.E., 13 (1988), 1421-1439. | MR 90a:35183 | Zbl 0686.35089

[10] R.B. Melrose, The Atiyah-Patodi-Singer index theorem, A.K. Peters, Wellesley, 1993. | MR 96g:58180 | Zbl 0796.58050

[11] W. Müller, Spectral geometry and scattering theory for certain complete surfaces of finite volume, Inv. Math., 109 (1992), 265-305. | MR 93g:58151 | Zbl 0772.58063

[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 91c:11029 | Zbl 0668.10036

[13] P. Perry, The Selberg zeta function and a local trace formula for Kleinian groups, J. Reine Angew. Math., 410 (1990), 116-152. | MR 92e:11057 | Zbl 0697.10027

[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 0818.35074