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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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