Vol. 51, No. 2, 1974

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ISSN: 0030-8730
On the eigenvalues of a second order elliptic operator in an unbounded domain

Denton Elwood Hewgill

Vol. 51 (1974), No. 2, 467–476
Abstract

Let E be an open set in Rn which satisfies the “narrowness at infinity” condition:

meas(E ∩ {x ∈ Rn : a ≦ |x| < a+ 1}) ≦ const(a +1)−β,

for all a > 0 and some β > 0. It is known that a uniformly strongly elliptic self-adjoint partial differential operator, on such a set E, has a discrete spectrum of eigenvalues {λj}. This paper is concerned with the growth rate of the function

        ∑
N (λ) =    1.
λn≦λ

The main result of the paper is to give an upper bound for N(λ). This upper bound will be a function of the β from the “narrowness” condition.

Mathematical Subject Classification 2000
Primary: 35P20
Milestones
Received: 14 February 1973
Revised: 25 June 1973
Published: 1 April 1974
Authors
Denton Elwood Hewgill