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Spectral asymptotics with a remainder estimate of the Neumann Laplacian on horns: the case of the rapidly growing counting function

Published online by Cambridge University Press:  14 November 2011

S. I. Boyarchenko  and
S. Z. Levendorskiĭ
S. I. Boyarchenko
Affiliation:
Don State Technical University, Gagarin sq. 1, 344010, Rostov-on-Don, Russia
S. Z. Levendorskiĭ
Affiliation:
Rostov-on-Don Institute of National Economy, B. Sadovaya, 69, 344007, Rostov-on-Don, Russia
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Extract

We study the Neumann Laplacian in unbounded regions of the form Ω = {(t, x) | t >O,f(t)−1x ∊ Ω′}, where Ω′ ⊂ ℝn−1 is a bounded open set with the Lipschitz boundary and f decays in such a way that the spectrum of is discrete but the counting function N(λ, ) of the spectrum grows faster than a power of λ, a typical example being f(t) = exp (– t In … In t), for t ≧ t0. We compute the principal term of the asymptotics of N(λ, ), with a remainder estimate.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 128 , Issue 1 , 1998 , pp. 11 - 22
Copyright
Copyright © Royal Society of Edinburgh 1998

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References

1Berg, M. van der. On the spectrum of the Dirichlet Laplacian for horn-shaped regions in Rn with infinite volume. J. Fund. Anal. 58 (1984), 150–6.CrossRefGoogle Scholar
2Berger, G., Asymptotische Eigenverteilung des Laplace-Operators in bestimmten unbeschränkten Gebieten mit Neumannschen Randbedingungen und Restgliedabschatzungen. Z Anal. Anwendungen 4(1985), 85–96.CrossRefGoogle Scholar
3Davies, E. B.. Trace properties of the Dirichlet Laplacian. Math. Z. 188 (1985), 245–51.CrossRefGoogle Scholar
4Davies, E. B. and Simon, B.. Spectral properties of the Neumann Laplacian of horns. Geom. Fund. Anal. 2(1985), 105–17.CrossRefGoogle Scholar
5Evans, W. D. and Harris, D. J.. Sobolev embeddings for generalised ridge domains. Proc. London Math. Soc. (3) 54 (1987), 141–75.CrossRefGoogle Scholar
6Fleckinger, J.. Asymptotic distribution of eigenvalues of elliptic operators on unbounded domains. Lecture Notes in Math. 846 (1981), 119–28.CrossRefGoogle Scholar
7Hörmander, L.. The Weyl calculus of pseudodifferential operators. Comm. Pure Appl. Math. 32 (1979), 359–443.CrossRefGoogle Scholar
8Hempel, R., Seco, L. A. and Simon, B.. The essential spectrum of Neumann Laplacian on some bounded singular domains. J. Fund. Anal. 102 (1991), 448–83.CrossRefGoogle Scholar
9Jaksić, V.. On the spectrum of Neumann Laplacian of long range horns: A note on the Davies–Simon theorem. Proc. Amer. Math. Soc. 199 (1993), 663–9.Google Scholar
10Jaksić, V., Molchanov, S. and Simon, B.. Eigenvalue asymptotics of the Neumann Laplacian of regions and manifolds with cusps. J. Fund. Anal. 106 (1992), 59–79.CrossRefGoogle Scholar
11Levendorskiĭ, S. Z.. The method of approximate spectral projection. Izv. AN SSSR Ser. Mat. 49 (1985), 1177–229 (Russian transl. in Math. USSR Isv. 27 (1986)).Google Scholar
12Levendorskiĭ, S. Z.. The approximate spectral projection method. Ada Math. 7 (1988), 137–97.Google Scholar
13Levendorskiĭ, S. Z.. Asymptotic Distribution of Eigenvalues of Differential Operators (Dordrecht: Kluwer, 1990).CrossRefGoogle Scholar
14Levendorskiĭ, S. Z.. Spectral asymptotics with a remainder estimate for Schrödinger operators with slowly growing potentials. Proc. Roy. Soc. Edinburgh Sect. A 126 (1996), 829–36.CrossRefGoogle Scholar
15Molchanov, A. M.. On conditions for discreteness of the spectrum of self-adjoint differential equations of the second order. Trudy Moskov. Mat. Obshch. 2 (1953), 169 [in Russian].Google Scholar
16Reed, M. and Simon, B.. Methods of Modern Mathematical Physics IV. Analysis of Operators (New York: Academic Press, 1978).Google Scholar
17Rellich, R.. Das Eigenvert problem von Δλ + λu = 0. In Studies and Essays 329–44 (New York: Interscience, 1948).Google Scholar
18Rozenblum, G. V.. The eigenvalues of the first boundary value problem in unbounded domains. Math. USSR-Sb. 18 (1972), 235–48.CrossRefGoogle Scholar
19Rozenblum, G. V., Solomyak, M. Z. and Shubin, M. A.. Spectral theory of differential operators. Contemporary problems of mathematics (Itogi Nauki i Tekhniki VINITI), v.64 (Moscow: VINITI, 1989).Google Scholar
20Simon, B.. Nonclassical eigenvalue asymptotics. J. Fund. Anal. 53 (1983), 84–98.CrossRefGoogle Scholar
21Simon, B.. The Neumann Laplacian of a jelly roll. Proc. Amer. Math. Soc. 114 (1992), 783–6.CrossRefGoogle Scholar
22Tamura, H.. The asymptotic distribution of eigenvalues of the Laplace operator in an unbounded domain. Nagoya Math. J. 60 (1976), 7–33.CrossRefGoogle Scholar